Unitary matrix integrals over symmetric polynomials play an important role in a wide variety of applications, including random matrix theory, gauge theory, number theory, and enumerative combinatorics. We derive novel results on such integrals and apply these and other identities to correlation functions of long-range random walks (LRRW) consisting of hard-core bosons. We generalize an identity due to Diaconis and Shahshahani which computes unitary matrix integrals over products of power sum polynomials. This allows us to derive two expressions for unitary matrix integrals over Schur polynomials, which can be directly applied to LRRW correlation functions. We then demonstrate a duality between distinct LRRW models, which we refer to as quasi-local particle-hole duality. We note a relation between the multiplication properties of power sum polynomials of degree $n$ and fermionic particles hopping by $n$ sites. This allows us to compute LRRW correlation functions in terms of auxiliary fermionic rather than hard-core bosonic systems. Inverting this reasoning leads to various results on long-range fermionic models as well. In principle, all results derived in this work can be implemented in experimental setups such as trapped ion systems, where LRRW models appear as an effective description. We further suggest specific correlation functions which may be applied to the benchmarking of such experimental setups.

§ INTRODUCTION

§.§ Background

This work presents a study of unitary matrix integrals over symmetric polynomials and their application to correlation function of long-range random walk (LRRW) models. We derive novel results on weighted unitary integrals over symmetric polynomials. These have a wide variety of mathematical and physical applications, including to Random Matrix Ensembles (RME's) and quantum chaos, gauge theories, number theory, and enumerative combinatorics. Connections between RME's and random walks have been explored extensively, see e.g. [1], [2] for early reviews on RME's and [3], [4] for more recent ones. Random walkers are called vicious when their paths are not allowed to intersect, i.e. when there exists the restriction that no two random walkers may occupy the same site. Vicious random walk models belong to the class of random-turns models when, at each time step, we move a single random walker. On the other hand, there are the lock-step models, where, at each time step, all random walkers are moved. For an early treatment, see [5], see also e.g. [6], [7],  [8], [9], [10], [11], [12]. This work will generally consider long-range versions of the random-turns model. One way in which a relation between RME's and vicious random walks can be seen to arise is from the fact that the joint eigenvalue distribution of unitary RME's is proportional to the squared Vandermonde determinant. This leads to a vanishing probability that any two eigenvalues coincide, which is known as level repulsion in random matrix theory. In physical terms, these non-intersecting random walkers can be interpreted as fermionic or hard-core bosonic particles, see e.g. the review [13]. This leads to an interpretation of the eigenvalue probability density of models related to classical Lie groups in terms of the ground state density of non-interacting and non-intersecting particles which are subject to confining potential and certain boundary conditions. This language naturally leads to a so-called $\tau$-functions of integrable hierarchies of differential equations with many deep results obtained over the years [14], [15], [16], [17], [18], [19]. Connection to integrability is widely used in the recent literature, see [20], [21] and also [22] for the case of unitary matrix models. Further, various correlation functions and important quantities in string theory are expressed in terms of these objects, see e.g. [23], [24] for recent developments. We also mention here applications in algebraic geometry and topology, see [4] for a review.

The connection between RME's and random walks appears in a different guise as well. Bogoliubov demonstrated [25] that the time-dependent correlation functions of the XX0-model are the generating functions of nearest neighbour vicious random walkers, i.e. those which can take only a unit size step to the left or right. Here, the XX0-model refers to the XX-model at zero magnetic field. These correlation functions can then be expressed as certain weighted unitary matrix integrals over Schur polynomials [25], [26]. In general, spin configurations which start with an infinite sequence of particles and end with an infinite sequence of holes correspond uniquely to certain Young diagrams. Time-dependent correlation functions of the XX0-model involving such configurations then take the form of unitary matrix integrals over a product of Schur polynomials associated to the corresponding Young diagrams. In the remainder of this work, we will refer to down spins as particles and up spins as holes. In the nearest neighbour case, the corresponding RME is given by the Gross-Witten-Wadia (GWW) model, where the inverse square of the coupling constant of the GWW model is proportional to the time parameter of the correlation function of the XX0-model. This relation between random walk correlation functions and matrix integrals over Schur polynomials was generalized to the case of LRRW models [27], where the particles can hop over greater distances. The particles in question behave as hard-core bosons rather than fermions, as the wave function does not acquire a minus sign when a particle hops over another one. In the long-range case, the weight function of the corresponding matrix model encodes the particular choice of LRRW model. In particular, the $n^\text{th}$ hopping parameter $a_n$, which allows particles to hop a distance of $n$ sites, equals the $n^\text{th}$ Fourier coefficient of the weight function. The hopping parameters are required to decay as $a_n \sim n^{-1-\epsilon} $ for $\epsilon >0$ as $n\to \infty$, corresponding to weight functions which satisfy the strong Szegő limit theorem.

The generalization to LRRW models allows for the application to long-range one-dimensional systems, such as those characterized by dipole-dipole interactions. For more recent examples of such systems, see e.g. [28]. Another physically motivated context where long-range systems are of interest is that of Anderson localization in long-range low-dimensional hopping models [29], [30]. These systems can be simulated experimentally with trapped ions, which have been demonstrated to exhibit great tunability in terms of the range of the the relevant hopping amplitude (see e.g. [31] for a review). Further applications include the dynamics of the Loschmidt echo in 1D systems, already considered in [25], see also [32], [33]. Dynamical phase transition in Loschmidt echo has been recently observed using the matrix models tools in [34]. The Loschmidt echo, a measure of chaoticity in quantum systems [35], can be regarded as a particular realization of the $\tau$-function mentioned above. The Loschmidt echo is also related to other important probes of many-body system dynamics, such as out-of-time-correlation functions. Importantly, the Loschmidt echo is an experimentally measurable quantity, see e.g. [36], [37], [38]. In the context of combinatorics, we mention also a relation to the plane partitions, see e.g. [33] for a recent account thereof.

In this paper, we consider physically relevant quantities, such as Loschmidt echo and various correlation functions in LRRW models, for which we employ identities on Young diagrams and the closely related theory of symmetric functions. We summarize the outline of the paper, including these results, in the remainder of the introduction.

§.§ Structure of the paper and main results

This work is structured as follows. We consider unitary matrix integrals over symmetric polynomials, which appear as correlation functions of long-range random walks of hard-core bosons. The basics of symmetric function theory, the relation between correlation functions of LRRW models and weighted unitary integrals over symmetric functions, and the evaluation of these objects are treated in appendices <ref>, <ref>, and <ref>, respectively. To be precise, we consider correlation functions between configurations which begin with an infinite string of particles in the form of hard-core bosons and end with an infinite string of holes, with a finite region of mixed particles and holes in between. These correlation functions are written as follows,

F_λ;μ(τ) = ∅σ_j_1^+ …σ_j_N^+ e^-τĤ σ_l_1^-  …σ_l_N^- ∅    ,    λ_r =j_r+r  ,   μ_s = l_s +s  ,

where $\ket{\emptyset}$ is a state with holes at all lattice sites, where we remind the reader that we refer to up spins as holes and down spins as particles. The hamiltonian $H$ is a long-range translationally-invariant hamiltonian whose hopping parameters $a_{m,n} = a_{m-n}$ decay faster than $\abs{a_k} \sim \abs{k}^{-1}$, see equation (<ref>). Defining the matrix model average $\avg{\dots}$ for some weight function $f$ as in (<ref>), it was found [25], [26], [27] that this can be written (in our notation) as

F_λ;μ(τ)/F_∅;∅(τ)  = s_λ(U) s_μ(U^-1) ,
where the dependence on the particular choice of LRRW model and the generalized time parameter $\tau$ are captured by the weight function $f$. Further, $s_\lambda$ and $ s_\mu$ are Schur polynomials, whose representations $\lambda$ and $\mu$ characterize the particle-hole configurations that occupy a finite interval in between the infinite strings of particles and holes.

Sections <ref> and <ref> present our results, starting in <ref> with the generalization of the result due to Diaconis and Shahshahani in [39]. Writing $p_\mu = p_{\mu_1}p_{\mu_2}\dots$, where $p_n$ is the $n^\text{th}$ power sum polynomial, our result in (<ref>) reads

p_ρ(U)p_μ(U^-1) =∑_n=0^ñ C_n  .

where $C_n$ and $\tilde{n}$ are the contribution arising from performing $n$ contractions in $\avg{p_\rho p_\mu}$ and the maximum number of such contractions, which are
defined in (<ref>) and (<ref>), respectively. Essentially, this result arises from the application of Wick's theorem and the fact that $\avg{p_n(U) p_k(U^{-1}) } - \avg{p_n(U)}   \avg{ p_k(U^{-1})} = n\delta_{n,k}$, which we derived a previous work [40]. In sections <ref> and <ref>, we derive expansions of general correlation functions of the form $\avg{s_\lambda (U) s_\nu(U^{-1})}$ in terms of Schur or power sum polynomials, respectively. Specifically, section <ref> establishes a relation between correlation functions $\avg{s_\lambda s_\nu}$ and $\avg{s_{\lambda\setminus \{\eta\} } },\avg{s_{\nu\setminus \{\eta\} } }^\ast$, where $\lambda\setminus \{\eta\}$ is a diagram obtained from $\lambda$ by the removal of border strips $\eta_j$ of sizes $\abs{\eta_j} = \alpha_j$. In section <ref>, we express $\avg{s_\lambda s_\nu}$ in terms of $\avg{p_\mu}$, where the expansion coefficients are determined by removing border strips from $\lambda$ and $\nu$ such that the resulting diagram is the same for both $\lambda$ and $\nu$. These results have a direct interpretation in terms of particle-hole configurations which is treated in section <ref>, for which reason we state these results with their particle-hole interpretations below.

We first consider the results derived in section <ref>, where we apply identities involving elementary and complete homogeneous symmetric polynomials to LRRW correlation functions. One result that arises in this way involves the well-known involution between elementary and homogeneous polynomials corresponding to the transposition of Young diagrams, which exchanges rows and columns. Although this result is mathematically trivial, its physical consequences are quite surprising and profound. Consider any two LRRW models, referred to as model A and B, whose hopping parameters are related by $a_k \to (-1)^{k+1} a_k$, and write their correlation functions as $F_{\lambda;\mu}^{(1)}(\tau)$ and $F_{\lambda;\mu}^{(2)}(\tau)$, respectively. Our result is given by equation (<ref>), which reads

F^(1)_λ;μ (τ)  = F^(2)_λ^t;μ^t (τ)   ,

where $^t$ refers to transposition of diagrams, which, combined with a parity transformation, implements a particle-hole transformation. Therefore, equation (<ref>) states that the correlation functions of models A and B are equal after performing local particle-hole and parity transformations, which only act non-trivially on the interval in between the infinite strings of particles and holes. For this reason, we refer to this result as Quasi-Local Particle-Hole duality (QLPH). In this way, a basic property of symmetric polynomials leads to a surprising result on LRRW models. Further such examples, involving e.g. the Pieri formula, are given in equations (<ref>), (<ref>), (<ref>), (<ref>), (<ref>).

Section <ref> treats the application of identities involving power sum polynomials and border strips, including results derived in section <ref>. We find that the multiplication of power sum polynomials of degree $n$ and the corresponding addition of border strips is related to fermionic particles hopping $n$ sites to the right, whereas removal of such a border strip corresponds to hopping $n$ sites to the left. This allows us to interpret various results in terms of an auxiliary fermionic (rather than hard-core bosonic) system. This reasoning can be applied to the computation of $\chi^\lambda_{~\alpha}$, which are the irreducible characters of the symmetric group, as we explain just above section <ref>. Further, combined with identities we found in [40], it leads immediately to results such as (<ref>) and (<ref>). In section <ref>, we use the relation between border strips and particle hopping to characterize the action of the hamiltonian in terms of Young diagrams.

In section <ref>, we use the aforementioned relation between power sums and fermions to derive two results on long-range fermionic models. First of all, from the fact that $\chi^{\lambda/\mu}_{~\alpha}$ does not depend on the order of the entries of $\alpha$ leads us to conclude the following. Consider a one-dimensional fermion configuration corresponding to some diagram $\mu$, and consider all ways of moving not necessarily distinct fermions to the right by $\alpha_1,\alpha_2,\dots$ sites, where $\alpha_j$ are unordered non-negative integers. We find that the order of the step sizes $\alpha_j$ by which we move fermions has no effect on the outcome of this process. That is, if we take a fermionic configuration and consecutively move fermions to the right by various step sizes, the outcome depends only on the distribution of the step sizes and not the order in which the steps are taken. Naturally, this statement also holds if we move fermions to the left instead of the right. The second result follows from the fact that $\chi^{\lambda/\rho}_{~\alpha}$ for $\alpha = (n^k)$  is cancellation-free [41], [42]. Here, $\rho$ is the so-called $n$-core of $\lambda$, which is the unique diagram that remains after removing the maximal number of size $n$ border strips from $\lambda$. In particular, start with a particle-hole configuration, which we call $\lambda$, and consider all ways to move (not necessarily distinct) particles by $n$ sites, where $n$ is any finite positive integer. By taking a particle and moving it by $n$ sites (from site $j$ to $j+n$), it may in the process `jump over' $\leq n-1$ other particles (occupying sites $j+1 $ to $j+n-1$). Our result then states that all ways to go from $\lambda$ to any other configuration $\nu$ involves jumping over either an even or an odd number of particles, depending only on the choice of $\lambda$ and $\nu$. When applied to a fermionic model where fermions can hop only by $\pm n$ sites for a single choice of $n$, this implies that there is no destructive interference between various ways to arrive at the same fermionic state, allowing for rapid spread through Hilbert space upon time evolution. The same reasoning can be applied to certain LRRW correlation functions, leading to (<ref>) and (<ref>).

In sections <ref> and <ref>, we apply our results from section <ref> to LRRW correlation functions. The expansion in <ref>, including its particle-hole interpretation in <ref>, given in (<ref>) and (<ref>), reads

\begin{align}
\avg{s_\lambda (U)  s_\nu (U^{-1}) }_c  %& \sum_{\alpha} \frac{1}{z_\alpha}  \prod_{j\geq 1} C(n,\alpha_j ;\lambda)C(n,\alpha_j;\nu) \notag \\
& =    \sum_{\alpha} \frac{1}{z_\alpha} \left( \sum_{\{\eta\}} (-1)^{\hg(T_{\eta})} s_{\lambda\setminus \{\eta\} }(y) \right)\times \left( { y \to x \atop \lambda \to \nu } \right) \\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! =   \sum_\alpha \frac{1}{z_\alpha} & \left(  (-1)^{P} s_{\lambda\setminus \{\eta\} }(y) \left\{ \begin{tabular}{lll}
Distinct ways to take particles in $\lambda$\\ and move them $\alpha_1, \alpha_2,\dots$ sites to the\\
left, thereby hopping over $P$ particles
\end{tabular}
\right\} \right)\times \left( { y \to x \atop \lambda \to \nu } \right) ~. \notag
\end{align}

The expansion in terms of power sum polynomials derived in section <ref>, including its particle-hole interpretation in <ref>, is given by equations (<ref>) and (<ref>) as

\begin{align}
\label{summpsp}
\avg{ s_\lambda (U) s_\nu(U^{-1})} = & \sum_{\omega, \gamma} \frac{p_\omega(y)p_\gamma(x)}{z_\omega z_\gamma} \sum_{\{\eta\} , \{ \xi\} } (-1)^{\hg(T_\eta) +\hg(T_\xi)} \delta_{\lambda\setminus \{\eta\}, \nu \setminus \{ \xi\}} \notag\\
= & \sum_{\omega, \gamma} \frac{p_\omega(y)p_\gamma(x)}{z_\omega z_\gamma} (-1)^{P}
\left\{ \begin{tabular}{llll}
Distinct ways to move particles in $\lambda$ and $\nu$\\ to the left by $\gamma_1, \gamma_2,\dots$  and $\omega_1, \omega_2,\dots$ sites,\\
respectively, so hopping over $P$ other particles\\
and ending up in the same configuration.
\end{tabular}
\right\}  ~.
\end{align}

The expansion in (<ref>) is particularly convenient in applications where one has access to the power sum polynomials, such as for LRRW models, where they are given by $p_k =  k \tau a_{\pm k}$, in terms of the hopping parameters $a_k$ and the generalized time parameter $\tau$. Then, in section <ref> we suggest applications of some of our results to experimental benchmarking. We finish with our conclusions, as well as possible generalizations and applications of our results, which include both theoretical and experimental suggestions.

§ UNITARY MATRIX INTEGRALS AND WICK'S THEOREM

We now proceed with the derivation of our results. In this section, we will derive expansions for unitary matrix integrals over symmetric polynomials. The reader may consult appendix <ref> for more information on these objects. Taking the limit $N\to \infty$ allows us to apply an expression we found in [40]. For $k,n \in \mathds{N}^+ $, and $f(z)=H(x;z)H(y;z^{-1})$ satisfying the strong Szegő limit theorem, we demonstrated that

U^n U^-k  = nδ_n,k + p_n(y) p_k (x)  .
For the CUE, $x_j =0 =y_j$ for all $j$, so that $p_n(x)=0$ for all $n$. Therefore, $\avg{\tra U^n \tra U^{-k} }_{\text{CUE}} = n\delta_{n,k}$ [43]. We briefly summarize the derivation of our more general result in (<ref>) as we will be using similar arguments throughout the remainder of this work. We repeat that, for $n \in \mathds{N}^+$,

\begin{equation}
\tra U^n =  p_n(e^{i\theta_j}) = \sum_{r = 0}^{n-1} (-1)^r s_{(n-r,1^r)}(U) ~.
\end{equation}

It is clear that this also holds when we replace $U$ with $U^{-1}$. Using (<ref>), we have

p_n(y)  ,     U^-n= 	p_n(x) .

U^n U^-k   =∑_ν∑_r = 0^n-1  (-1)^r s_(n-r,1^r)/ν (y)   ∑_s = 0^k-1  (-1)^s s_(k-s,1^s)/ν (x)   .
The first sum on the right hand side of runs over all representations $\nu$ satisfying $\nu \subseteq (n-r,1^r)$ as well as $\nu \subseteq (k-s,1^s)$. We list below the contributions arising for various choices of $\nu$. 	We will often omit the variables $x$ and $y$ henceforth.

* If $\nu$ is the empty partition $\nu = \emptyset$, $s_{\lambda/\nu} =s_\lambda$ i.e. the skew Schur polynomials reduces to the usual (non-skew) Schur polynomial. This contributes
∑_r = 0^n-1  (-1)^r s_(n-r,1^r)(y)
∑_s = 0^k-1  (-1)^s s_(k-s,1^s)(x)=  p_n(y) p_k(x)  .

* If $\nu = \lambda$, the skew Schur polynomial $s_{\lambda/\nu } = s_{\lambda/\lambda} = 1$. For $\lambda=(n-r,1^r)$ and $\mu=(k-s,1^s)$, one can only have $\lambda=\nu =\mu$ if $k=n$ and $r=s$. For $k=n$ and $\nu = (n-r,1^r)$ and we sum over $r$, we get a contribution of the following form
∑_r=0^n-1 (s_∅)^2 = n .
This is where the term $ n\delta_{n,k}$ in (<ref>) originates.

* Take $k \leq n$ without loss of generality. The only remaining choice for $\nu$ is $\nu \neq \emptyset$ and $\nu \neq (n-r,1^r)$. For $\lambda = (a,1^b)$ and $\nu = (c,1^d)$ such that $\nu \subseteq \lambda$, we have $\lambda /\nu = (a-c) \otimes (1^{b-d})$, e.g. for $\lambda = (4,1^2)$ and $\nu = (2,1)$, we have the following.

4,1,1   / 2,1    =     1   2

Fixing a single such $\nu=(a,1^b)$ and considering only the sum over $r$ in (<ref>), we have

∑_r = 0^n-1  (-1)^r s_(n-r,1^r)/(a,1^b) =
∑_r=b^n-a(-1)^r h_n-r-ae_r-b = 0  ,
where we applied (<ref>). That is, we get zero contribution for all $\nu \neq \emptyset, (n-r,1^r)$.

Combining the above arguments leads to equation (<ref>). Of course, the same reasoning can be applied to other expectation values involving $\tra U^n$, such as

U^-n s_λ(U)_c = ∑_ν≠∅  ∑_r=0^n-1 (-1)^r s_(n-r,1^r)/ν     s_λ/ν  .
As mentioned in appendix <ref>, the connected expectation value $\avg{\dots}_c$ in the above equation is given by $\avg{AB}_c = \avg{AB}-\avg{A}\avg{B}$ . The sum in (<ref>) is over all $\nu \neq \emptyset$ such that $\nu\subseteq \lambda, (n-r,1^r)$. Fixing any $\nu \subseteq (n-r,1^r)$ in (<ref>) with $\nu\neq (n-r,1^r)$ gives zero when summing over $r$ by application of (<ref>). Therefore, we only get a non-zero answer for terms for which $\nu = (n-1,1^r) \subseteq \lambda$. That is,

U^-n  s_λ(U)_c   =   ∑_r
(-1)^r s_λ/(n-r,1^r)    ,

where the sum is over $\min(0,  n-\lambda_1) \leq r \leq \min( n-1,\lambda_1^t+1)$. Using (<ref>), we can express this as

U^-n  s_λ(U)_c   = ∑_ν(-1)^ht (λ/ν)  s_ν ,
where the sum is over all $\nu$ such that $\lambda/\nu$ is a border strip $\eta$ of size $n$, which we also write as $ \nu = \lambda \setminus \eta$. More information on border strips can be found in appendix <ref>, see equation (<ref>) and especially (<ref>) and below.  As an example, take $\lambda =(6,4,3^2,2) $ and $n=4$. We show the diagrams $\nu$ appearing in (<ref>) below, where the cells that are removed are again given in black.

*(white)     *(white)     *(white)     *(black)      *(black)      *(black)
(white)      *(white)        *(white)      *(black)
(white)      *(white)      *(white)
(white)     *(white)     *(white)
(white)      *(white)

*(white)     *(white)     *(white)      *(white)     *(white)     *(white)
(white)      *(white)       *(black)      *(black)
(white)      *(white)        *(black)
(white)     *(white)       *(black)
(white)      *(white)

*(white)     *(white)     *(white)      *(white)     *(white)     *(white)
(white)      *(white)     *(white)      *(white)
(white)      *(white)        *(white)
(white)       *(black)      *(black)
(black)      *(black)

§.§ Applying Wick's theorem

We will now combine equation (<ref>) with Wick's theorem to compute more complicated objects. In particular, for terms of the form $\avg{ \tra U^{n_1} \tra U^{n_2} \dots \tra U^{-k_1} \tra U^{-k_2}\dots}$, we sum over all ways to contract between copies of $\tra U^{n_j}$ and $\tra U^{k_m}$. This contraction is done using the connected version of equation (<ref>), which equals the CUE result,
U^n U^-k_c = nδ_n,k .

We first consider how Wick's theorem arises from the properties of Young diagrams for some relatively simple cases. For example, we have

(U^2)^2 U^-1 _c = 2 U U^-2_c  U^-2=0 ,
where the factor two in the second expression arises from the fact that there are two ways to contract between $(\tra U^{2})^2$ and $ \tra U^{-1}$, which both contribute a term proportional to $\avg{\tra U \tra U^{-2}}_c=0$. We will now express the same equation using Young diagrams. Using (<ref>), the diagrams corresponding to $\tra U^2$ are as follows:

2   $\mathrm{-}$  1,1

We take the square of the above expression and apply equation (<ref>) (or, in this case, the Pieri formulas (<ref>) and (<ref>)) to find the diagrams contributing to $(\tra U^2)^2$. These are as follows.

4   $\mathrm{-}$  3,1 $+   $  $2$    2,2  $\mathrm{-}$  2,1,1  +  1,1,1,1

We will denote the above sum over diagrams as $ \sum_\lambda b_\lambda  s_\lambda$, i.e. $b_{(4)} = 1 ,~ b_{(3,1)}=-1 ,~b_{(2,2)}=2,~ b_{(2,1,1)}=-1 ,~ b_{(1^4)}=1 $. Applying (<ref>), we have

(U^2)^2 U^-1 _c =  ∑_λb_λs_λ/

That is, we take $  \sum_\lambda b_\lambda  s_\lambda$ and find the skew diagrams $\lambda/ {\small \Box}$, found by removing a single cell from $\lambda$ which has no cells to its right or below it. From these constraints, it follows that the resulting object after removing an cell is still a valid, non-skew diagram. This gives the following sum over diagrams, which evidently mutually cancel.

\begin{align}
\label{wickidxmp}
\begin{ytableau}
*(white) & *(white) & *(white) & *(black)
\end{ytableau}
\hspace{.05cm} ~  & - ~
\begin{ytableau}
*(white) & *(white) & *(white) \\
\end{ytableau}
\hspace{.05cm} ~  -  ~
\begin{ytableau}
*(white) & *(white) & *(black)  \\
\end{ytableau}
\hspace{.05cm} ~  + ~  2 ~~
\begin{ytableau}
*(white) & *(white) \\
*(white)   & *(black)
\end{ytableau}
\notag\\[15pt]
\hspace{.05cm}  & - ~
\begin{ytableau}
*(white) & *(white) \\
*(white) \\
\end{ytableau}
\hspace{.05cm} ~  - ~
\begin{ytableau}
*(white) & *(black) \\
*(white) \\
\end{ytableau}
\hspace{.05cm}  ~  + ~
\begin{ytableau}
*(white)  \\
*(white) \\
*(white) \\
\end{ytableau}
\hso  ~  = ~ 0
\end{align}

We thus explicitly confirm that Wick's theorem is satisfied for the case of equation (<ref>). Consider now $\avg{(\tra U^2)^2 \tra U^{-2}}_c $. Using (<ref>), this should give

(U^2)^2 U^-2_c=2U^2 U^-2_cU^2= 4 U^2
We will check this explicitly as well. Applying (<ref>) to $ \sum_\lambda b_\lambda  s_\lambda $ gives

(U^2)^2 U^-2_c=∑_λb_λ[  s_λ/ (2) -s_λ/(1^2) ] = ∑_λb_λ∑_ν(-1)^ht(λ/ν) s_ν ,

where the sum is over all $\nu$ such that $\lambda/\nu $ is a border strip of size 2, i.e. $\lambda/\nu = $ 2  or $\lambda/\nu = $ 1,1  . In terms of diagrams, this is given below. The first three diagrams corresponds to $\lambda/\nu = $ 2  whereas the latter three correspond to $\lambda/\nu = $ 1,1  , which appear with a minus sign due to the factor $(-1)^{\text{ht}(\lambda/\nu)}$.

*(white)     *(white)      *(black)      *(black)

*(white)      *(black)     *(black)

+      2

*(white)     *(white)
(black)       *(black)

-      2

*(white)     *(black)
(white)       *(black)

*(white)     *(white)

=    4

-   4

We see that $\avg{(\tra U^2)^2 \tra U^{-2}}_c=2\avg{\tra U^2 \tra U^{-2}}_c\avg{\tra U^2} = 4 $ 2  $- $ $4$  1,1  $ = 4 \avg{\tra U^2}$, thus confirming (<ref>). Lastly, we will briefly check

(U^2)^2 (U^-1)^2_c = 0 .
We have

1 $ \otimes$ 1  =
2 + 1,1

Note that  1,1   appears with a positive sign, instead of with a minus sign as it does for $\tra U^2$. Then,

(U^2)^2 (U^-1)^2_c=s_  0∑_λb_λs_λ/  + ∑_λb_λ[  s_λ/ (2) + s_λ/(1^2) ]  ,
where we applied the fact that $ \sum_\lambda b_\lambda   s_{\lambda / \Box} =0 $, see the diagrams in equation (<ref>). Note that (<ref>) gives an equation very similar to (<ref>), but now $ s_{\lambda/(1^2)}$ carries a positive sign. This gives the same six diagrams as in the top line of equation (<ref>), except that the last three diagrams are multiplied by $-1$, as can be seen below.

*(white)     *(white)      *(black)      *(black)

*(white)      *(black)     *(black)

$+   $  $2$

*(white)     *(white)
(black)       *(black)

$+$     $2$

*(white)     *(black)
(white)       *(black)

*(white)     *(white)

= 0

This confirms equation (<ref>). From these relatively simple examples, one can explicitly see how Wick's theorem arises from equation (<ref>) and the multiplication rules for Young diagrams given in appendix <ref>. We proceed to apply Wick's theorem to the computation of more general objects in the remainder of this section.

§.§ Generalization of an identity due to Diaconis and Shahshahani

We use Wick's theorem and equation (<ref>) to generalize an identity due to Diaconis and Shahshahani [39], see also [43]. Writing $p_\rho=p_{\rho_1}p_{\rho_2}\dots p_{\rho_{\ell(\rho)}} $  and $ m_j(\rho)=\text{Card} \{k:\rho_k=j\} $, as in equations (<ref>) and (<ref>), respectively, we wish to calculate $\avg{p_\rho (U) p_\mu(U^{-1})}$. This is written out as follows,

p_ρ(U) p_μ(U^-1) = (U^j_1)^m_j_1(ρ)  (U^j_2)^m_j_2(ρ)…(U^-k_1)^m_k_1(μ) (U^-k_2)^m_k_2(μ)…  .
We start with a simpler object that we can easily apply Wick's theorem to. We see that performing $n$ contractions on $\avg{(\tra U^j)^a (\tra U^{-j})^b} $ leads to the following expression

C_n,j a!b!j^n/(a-n)!(b-n)!n!p_j(x)^b-n p_j(y)^a-n
,   n≤Min(a,b)  .
Further, we have $C_{n,j}=0$ for $n \geq   \text{Min}(a,b) +1$. Equation (<ref>) arises as follows. There are $\frac{a!b!}{(a-n)!(b-n)!n!}$ ways to perform $n$ contractions between  $(\tra U^j)^a $ and $(\tra U^{-j})^b$, and the contracted terms give a contribution equal to $\left(\avg{\tra U^j \tra U^{-j}}_c\right)^n= j^n $. The $a+b-2n$ uncontracted traces contribute $\avg{\tra U^j}^{a-n}\avg{\tra U^{-j}}^{a-n} =p_j(x)^{b-n} p_j(y)^{a-n}$. We now consider all possible ways to perform $n$ contractions between $p_\rho(U)$ and $p_\mu(U^{-1})$. This leads to a sum over $\alpha= (\alpha_1,\alpha_2,\dots) $ which are partitions of $n$, which specify the contractions that are performed. In particular, $m_j(\alpha)$ gives the number of $\tra U^j$ and $\tra U^{-j}$ which are contracted. The contribution coming from $n$ contractions in $\avg{p_\rho(U)p_\mu(U)^{-1}}$ can then be written as

C_n =  ∑_α∏_j C_m_j(α),j ,

where the sum is over $\alpha$ that are partitions of $n$. We denote by $\tilde{n}$ is the maximal number of contractions one can perform, which is given by

ñ= Max(n) = ∑_j≥1 Min(m_j(ρ),m_j(μ)) .
Summing over all possible contractions and applying (<ref>) and (<ref>), we arrive at our result

p_ρ(U)p_μ(U^-1) =∑_n=0^ñ C_n  .

As mentioned before, this is a generalization of a result in [39], which considered the CUE, where $f=1$ so that $p_j(x) =0$ for all $j \neq 0$. Therefore, in the CUE case, one only gets a non-zero result when $ \rho=\mu$. In our notation, their result reads

p_ρ(U)p_μ(U^-1)_CUE = z_ρδ_ρ,μ  ,
where $\delta_{\rho ,\mu } =1 $ for $\rho=\mu$ and zero otherwise. Note that (<ref>) is only the last term in the full expansion in (<ref>), corresponding to $n=\tilde{n} = \sum_j m_j(\rho)$, so that all power sums in $p_\rho(U) $ and $p_\mu(U^{-1})$ are contracted.

§.§ Border strips and their application to unitary matrix integrals over Schur polynomials

We will derive two expressions for general $\avg{s_\lambda(U)s_\nu(U^{-1})}$ which rely on removing border strips from $\lambda $ and $\nu$. The first of these, equation (<ref>), relates $\avg{s_\lambda(U)s_\nu(U^{-1})}$ to sums over $\avg{s_\mu (U)}\avg{s_\rho (U^{- 1})} = s_\mu(y)s_\rho(x)$, where $\mu$ and $\rho$ are related to $\lambda$ and $\nu$ by the removal of border strips, respectively. The second expression, in equation (<ref>), provides an expansion of $\avg{s_\lambda(U)s_\nu(U^{-1})}$ in terms of the power sums $p_k(x)$ and $p_k(y)$. The latter expression appears to be particularly useful, as power sums are simpler objects than general Schur polynomials. In the context of LRRW models, $p_k(x)$ and $p_k(y)$ are given by $\pm \tau k a_{\pm k}$, where $a_{\pm k}$ are the hopping parameters in (<ref>). Equation (<ref>) therefore provides an expansion in $\tau$, where the expansion coefficients depend on $a_k$ which can be read off from the hamiltonian. We will treat the application of these formulas to LRRW models in sections <ref> and <ref>.

§.§.§ Expansion in Schur polynomials

From (<ref>), (<ref>), and (<ref>), we have

s_λp_n_c = ∑_η(-1)^(η) s_λ∖η = ∑_μ χ^λ_ μ/z_μ p_μ-(n) n m_n(μ) ,
where the sum is over all $\mu$ containing a row of size $n$ and $\mu-(n)$ is the remainder of $\mu$ after removing a row of size $n$. We remind the reader that $\lambda \setminus \eta$ is the diagram that results from $\lambda$ after removing border strip $\eta$ with (in this case) $\abs{\eta} = n$. We can also use the recursive definition of $\chi^\lambda_{~\mu}$ in equation (<ref>) to see that the second equality in (<ref>) should hold. From (<ref>), it is clear that

z_μ= z_μ- (n) n m_n(μ) .

Plugging this into the rightmost expression in (<ref>) and applying (<ref>) leads to

s_λp_n_c = ∑_μ∑_η(-1)^(η) χ^λ∖η_ μ-(n)/z_μ- (n) p_μ- (n) .

We can then apply equation (<ref>), which leads to the second equality in (<ref>). If we insert two identical power sums, we find the following

\begin{align}
\label{sppavg}
\avg{s_\lambda p_n^2}_c = & \sum_{ \eta ,\zeta}  (-1)^{\hg (\eta)+\hg (\zeta)} \avg{s_{(\lambda \setminus \eta ) \setminus  \zeta }} +  2\avg{p_n}\avg{s_\lambda p_n}_c   \notag \\
= & \sum_{\mu} \frac{\chi^\lambda_{~\mu}}{z_\mu } \avg{p_{\mu-(n^2)}} 2 n^2 m_n(\mu)(m_n(\mu)-1)+     2\avg{p_n}\avg{s_\lambda p_n}_c   ~,
\end{align}

where we consecutively remove border strips $\eta $ and $\zeta$ satisfying $\abs{\eta} = n = \abs{\zeta}$, resulting in the partition $(\lambda \setminus \eta ) \setminus  \zeta$. The sum on the right hand side runs over all $\mu$ containing (at least) two rows of length $n$, and $\mu-(n^2)$ is the remainder of $\mu$ after removing two such rows. The term $ 2\avg{p_n}\avg{s_\lambda p_n}_c $ arises from a single contraction between $p_n$ and $s_\lambda$. We can again apply equation (<ref>) to demonstrate the second equality in (<ref>), where we focus on the term arising from two contractions. Plugging

z_μ= z_μ- (n^2) n^2  m_n(μ) ( m_n(μ) -1 )  .

into the first term on the right of (<ref>) leads to

∑_μ∑_ζ,η (-1)^(ζ) + ( η)  χ^(λ∖η) ∖ζ_  μ- (n^2) / z_μ- (n^2)   p_μ- (n^2)  ,

which, by (<ref>), recovers the top line of (<ref>).

We then take $\avg{s_\lambda (p_n )^k} $ and perform all $k$ contractions. This gives an object we denote by $C(n,k;\lambda)$, which equals

C(n,k;λ) = ∑_μ χ^λ_ μ/z_μ p_μ-(n^k) k! n^k m_n(μ)!/(m_n(μ)-k)! = ∑_{η} (-1)^(T_η) s_λ∖{η}   .
The sum on the left is over all $\mu$ containing at least $k$ rows of length $n$. The sum on the right is over $k$ border strips $\{\eta\} = \{\eta_1 ,\dots \eta_k\}$ satisfying $\abs{\eta_j} = n$ for all $j$, where $\lambda\setminus \{\eta\} $ is the diagram obtained after removing all $\eta_j$ from $\lambda$, and $T_{\eta}$ is the BST consisting of the union of $\eta_1,\dots,\eta_k$. It follows that the term in (<ref>) gives zero if it is not possible to construct a subdiagram of $\lambda$ with $k$ border strips of size $n$, e.g. simply if $\abs{\lambda} < nk$. Note that $C(n,1;\lambda) = \avg{s_\lambda p_n}_c $ and $C(n,2;\lambda) = \avg{s_\lambda p_n^2 }_c - 2\avg{p_n}\avg{s_\lambda p_n}_c $.

We now consider

s_λ(U)s_ν(U^-1)_c = ∑_μ,ρ χ_ μ^λ/z_μχ_ ρ^ν/z_ρ p_μ(U) p_ρ(U^-1)  .

When we consider those $\mu$ and $\rho$ that contain a row of size $n$ and we contract a single copy of $p_n$ between $p_\mu(U)$ and $ p_\rho(U^{-1})$, we get an object we denote by $A (n,1; \lambda , \nu  )$, which is given by

A (n,1; λ, ν) = ∑_μ,ρ χ_ μ^λ/z_μχ_ ρ^ν/z_ρ n m_n(μ) m_n(ρ) p_μ-(n)(U)  p_ρ-(n)(U^-1)   = 1/n s_λp_n_c s_νp_n_c   .

Using (<ref>) gives
A (n,1;λ, ν) = 1/n  C(n,1;λ)C(n,1;ν)  = 1/n (  ∑_η(-1)^(η) s_λ∖η )  (  ∑_ζ(-1)^(ζ) s_ν∖ζ )    ,

where the sums are again over border strips satisfying  $\abs{\eta} = n = \abs{\zeta}$. Consider now $\mu$ and $\rho$ that contain at least two rows of length $n$ and contract two copies of $p_n$ between $p_\mu(U)$ and $ p_\rho(U^{-1})$,

\begin{align}
A (n,2 ; \lambda , \nu  ) = & \sum_{\mu,\rho} \frac{\chi_{~\mu}^\lambda}{z_\mu}\frac{\chi_{~\rho}^\nu}{z_\rho} \frac{n^2}{2} m_n(\mu)(m_n(\mu)-1) m_n(\rho)(m_n(\rho)-1) \avg{p_{\mu-(n)}(U)} \avg{ p_{\rho-(n)}(U^{-1}) } \notag \\
= &  \frac{1}{2n^2} C(n,2;\lambda)C(n,2;\nu)
\end{align}

More generally, performing $k$ contractions between $ (\tra U^n)^k$ and $(\tra U^{-n})^k$ results in
A (n,k ; λ, ν) = 1/k!n^kC(n,k;λ)C(n,k;ν) .
Consider a partition $\alpha$  and, as above, contract over $\alpha_1 $ copies of $p_1(U)$ and $p_1(U^{-1})$, $\alpha_2 $ copies of $p_2(U)$ and $p_2(U^{-1})$, etc. This gives the following.
A (n,α; λ, ν)= 1/z_α  ∏_j≥1 C(n,α_j ;λ)C(n,α_j;ν) .

We can apply the above expression and (<ref>), leading to

s_λs_ν_c =  ∑_α 1/z_α  ∏_j≥1 C(n,α_j ;λ)C(n,α_j;ν) ,

which can be written out as

s_λ(U)  s_ν(U^-1)_c =    ∑_α 1/z_α ( ∑_{η}  (-1)^(T_η) s_λ∖{η} (y) )×(  y →x λ→ν )  .

The second sum above is over all border strips $\eta_j$ satisfying $\abs{\eta_j} = \alpha_j$. The above equation can be interpreted as follows. For any $\alpha$, consider all ways to remove $\alpha_1$ border strips of unit size (single cells) from $\lambda $ and $\nu$, $\alpha_2$ border strips of size 2, $\alpha_3 $ of size 3, and so on. The resulting diagrams are written as $\lambda\setminus \{\eta\}$ and idem for $\lambda \to \nu$. Remember from equation (<ref>) and the comments below that the resulting diagrams do not depend on the order of the entries of $\alpha$, that is, on the order in which we remove border strips of various sizes. Indeed, the diagrams resulting from this procedure only depend on $\lambda$ and the set of cardinalities $m_j(\alpha)= \text{Card} \{k:\lambda_k=j\}$.

Equation (<ref>) expresses general expectation values $\avg{s_\lambda s_\nu}_c $ in terms of the non-skew Schur polynomials corresponding to $\lambda\setminus \{\eta \}$ and $\nu \setminus \{\eta \}$. On the other hand, equation (<ref>), which is our starting point, gives an expansion in terms of skew Schur polynomials. There are various applications where an expression in terms of non-skew Schur polynomials is desirable. In general, this may be due to the fact that there are many more skew diagrams than non-skew ones, so that an expansion in non-skew diagrams may reveal underlying structures that are otherwise difficult to discern. This is also the case for the LRRW correlation functions we will be considering in the next section, where equation (<ref>) will reveal relations between various correlation functions.

We consider now the special case where one can form BST's of shapes $\lambda$ and $\nu$ from $ \alpha_j$ border strips of size $j$, summed over $j$, such that we can fully contract between $p_\mu(U)$ and $p_\rho(U^{-1})$. That is, we consider the case where we can completely tile both $\lambda$ and $\nu$ with $\alpha_1$ single cells, $ \alpha_2 $ border strips of size 2 (dominoes), and so on, for the same choice of $\alpha$. This clearly requires $\abs{\lambda} = \abs{\nu} = \sum_{j\geq 1} j \alpha_j$, which is a necessary (but far from sufficient) condition for $\chi^\lambda_{~\alpha} , \chi^\nu_{~\alpha}  \neq 0$. Consider the CUE, where $\avg{s_{\lambda/\mu}(U^{\pm 1})}=0$ for any $\lambda/\mu \neq \emptyset$. Applying the Murnaghan-Nakayama formula given in equation (<ref>),

∑_α_1,…,α_k (-1)^(T_α)  = χ^λ_ α ,

we then find that

s_λs_ν_CUE = ∑_αz_α^-1 χ^λ_ α χ^ν_ α= δ_λ, ν .
This is just the orthonormality property of symmetric group characters, see e.g. [Proposition 7.17.6b, [44]]. Another way to arrive at the same expression is to directly plug equation (<ref>),
p_μp_ρ_CUE =z_μδ_μ,ρ ,
into equation (<ref>) to find the orthonormality relation in (<ref>).

We work out the explicit example for $\lambda = (3,2)$, which is a sufficiently small partition that we can still apply (<ref>) for comparison. Indeed, applying (<ref>) to $\avg{s_{(3,2)} s_{(3,2)} }_c= \sum_\nu s_{(3,2)/\nu} s_{(3,2)/\nu}   $ gives the following diagrams $(3,2)/\nu$.

*(white)     *(white)
(white)      *(white)

*(white)     *(white)

*(white)      *(white)

*(white)     *(white)

,  2

,   $\emptyset$

Note that 1 appears with multiplicity 2 as it arises from $\nu = (3,1)$ and $\nu = (2^2)$. By applying (<ref>) for $n=1$ (or, equivalently, (<ref>) or (<ref>)) to the leftmost diagram, we find the following.

*(white)     *(white)
(white)      *(white)

*(white)     *(white)
(white)      *(white)

*(white)     *(white)     *(white)

This leads to

s_(3,2) s_(3,2) _c  = (s_(2,2) + s_(3,1) )^2 + ( s_(3)+  s_(2,1))^2 + s_(2,1)^2  + ( s_(2)+  s_(1^2))^2 + s_(2)^2 + 2 s_(1)^2 +1   .

Note that the above equation is strictly speaking only correct when $x=y$. For $x\neq y$, we have e.g. $\left(s_{(2,2)}(x) + s_{(3,1)}(x) \right)\left(s_{(2,2)}(y) + s_{(3,1)}(y) \right)$ instead of $\left(s_{(2,2)} + s_{(3,1)} \right)^2$, but we write it as above to avoid clutter.

We now apply (<ref>) to compute $\avg{s_{(3,2)} s_{(3,2)}}_c$. To do so, we successively remove border strips from $(3,2)$ to find the various partitions $(3,2) \setminus \{\eta\}$ in (<ref>). This is illustrated below, where two diagrams connected by an arrow as $\overset{p_j}{\longrightarrow}$ again indicates that partitions $\lambda$ and $\mu$ are related by the removal of a border strip of size $j$. The graph below contains all information about the removal of border strips from $(3,2)$, except for those cases where the removal of a border strip leads to the empty diagram. All diagrams in figure <ref> except for $(3,2)$ and $(2,2)$ are hook shapes and therefore border strips, so that they are related to $\emptyset$ by the removal of a single border strip.

[on top/.style=preaction=draw=white,-,line width=#1,
on top/.default=7pt]

(A) at (0,0) $\ydiagram{3,2}$;
(B) at (3.5,.83) $\ydiagram{3,1}$;
(C) at (3.2,-1.15) $\ydiagram{2,2}$;
(D) at (7.5,1.5) $\ydiagram{3}$;
(E) at (7.2,-2.5) $\ydiagram{2,1}$;
(F) at (10.5,.65) $\ydiagram{2}$;
(G) at (10.5,-1.4) $\ydiagram{1,1}$;
(H) at (14,0) $\ydiagram{1}$;

every node/.style=fill=white,circle,
every edge/.style=draw=black,very thick]
[->] (A) edge node $p_1$ (B);
[->] (A) edge node $p_1$ (C);
[->] (B)[bend left=5] edge node $p_1$ (D);
[->] (C) edge node $p_1$ (E);

[->] (C) edge[bend left=5]
node $p_2$ (F);
[->] (D) edge node $p_1$ (F);
[->] (E) edge node $p_1$ (G);
[->] (F) edge node $p_1$ (H);
[->] (E) edge node $p_1$ (F);
[->] (G) edge node $p_1$ (H);
[->] (A) edge[bend left=30] node $p_2$ (D);
[->] (D) edge[bend left=17] node $p_2$ (H);

every node/.style=fill=white,circle,
every edge/.style=draw=black,ultra thick,dashed]
[->] (A) edge[bend left=50] node $p_4$ (H);
[->] (C) edge[bend right=58] node $p_3$ (H);
[->] (A) edge[on top, bend right=58] node $p_3$ (G);
[->] (C) edge[on top]
node $p_2$ (G);

every node/.style=fill=white,circle,
every edge/.style=draw=black,very thick]
[->] (B) edge[on top, bend left=5] node $p_1$ (E);
[->] (B) edge[on top, bend left=25] node $p_2$ (G);

The diagrams for $\lambda = (3,2)$ and those obtained by removal of border strips of size $j$, which is indicated by $p_j$. The solid and dashed lines again indicate the removal of border strips of even and odd heights, respectively. This graph indicates all ways to remove border strips from $(3,2)$ and the resulting diagrams, except for those cases where the removal of a border strip leads to the empty diagram.

The number of ways to arrive at a diagram by following the arrows in figure <ref> gives the multiplicity of that diagram in (<ref>). For example, for $~{\tiny \ydiagram{3,2}}$ , the diagram $~{\tiny \ydiagram{1}}$ has multiplicity 5 for $\alpha = (1^5)$, as there are 5 distinct ways to arrive at $~{\tiny \ydiagram{1}}$ following arrows indicated by $p_1$ in figure <ref>. For $\alpha$ containing elements of different sizes, a single ordering should be fixed to find the correct multiplicity i.e. we do not sum over different compositions of the same cycle type. Applying equation (<ref>) then leads to (<ref>), where we indicate the compositions $\alpha$ of the power sum $p_\alpha$ over which we contract. That is, to arrive at a term indicated by some $\alpha$, one should start from  3,2 and follow arrows indicated by the $\alpha_j$ in any order, keeping in mind the sign given by $(-1)^{\hg(\eta_j)}$ with $\abs{\eta_j} = \alpha_j$. This gives the following.

\begin{align}
\label{sps32}
\avg{s_{(3,2)} s_{(3,2)} }_c =  &  \overset{(1)}{\left(s_{(2,2)} + s_{(3,1)} \right)^2} + \overset{~~(2)}{\frac{1}{2} s_{(3)}^2}  +  \overset{(1^2)}{\frac{1}{2} \left( 2 s_{(2,1)} + s_{(3)} \right)^2   } + \overset{~~~~(3)}{+ \frac{1}{3} s_{(1^2)}^2 } + \overset{~~~(2,1)}{\frac{1}{2} s_{(2)}^2 } + \notag \\
&  + \underset{(1^3)}{\frac{1}{3!} \left(3 s_{(2)} + 2 s_{(1^2)}\right)^2 }
+  % \notag \\      &   +
\left(\frac{1}{3}+ \frac{1}{8} + \frac{1}{4} + \frac{1}{4} + \frac{5^2}{4!}   \right) s_{(1)} + 1 ~.
\end{align}

The last two terms on the bottom line of the expression above arise from $\abs{\alpha} = 4,5$. In the latter case, following arrows indicated by $p_{\alpha_j}$ leads to $\emptyset$.  One can see that there is only a single way to start at $~{\tiny \ydiagram{3,2}}$ and arrive at $\emptyset $ for $\alpha = (4,1),(3,2),(3,1^2),(2^2,1), (2,1^3)$ by following the corresponding arrows in the graph above. However, we already noted that there are 5 distinct ways $~{\tiny \ydiagram{3,2}}$ and arrive at $~{\tiny \ydiagram{1}}$ following arrows with $p_1$. This implies there are also 5 ways to arrive at $\emptyset$ by removing single elements, simply by taking the additional step $~{\tiny \ydiagram{1}} \overset{p_1}{\longrightarrow} \emptyset$. This gives

1/z_(4,1) + 1/z_(3,2) +  1/z_(3,1^2) + 1/z_(2,1^3) + 1/z_(2^2,1) +  5^2/z_(1^5) = 1/4 + 1/6 + 1/6 + 1/12 +1/8  + 5^2/5! = 1 ,

leading to the unit contribution in equation (<ref>). Via this reasoning, one may check that expression (<ref>) and (<ref>) are identical. The above example may not appear to give a very convincing argument in favor of equation (<ref>) over (<ref>), as the application of (<ref>) appears to be more complicated than (<ref>) for the case of $\lambda = (3,2)$. Indeed, for partitions containing few cells, such as $\lambda = (3,2)$, it is convenient to use (<ref>), as the skew partitions $\lambda/\nu$ can easily be related to non-skew partitions, as in (<ref>). However, for larger partitions, this is no longer the case, as (<ref>), (<ref>), and (<ref>) can no longer be applied. When considering larger partitions, then, equation (<ref>) can still be used to express general objects $\avg{s_\lambda s_\nu }_c$ in terms of non-skew Schur polynomials. This will allow us to express complicated correlation functions in terms of simpler ones in section <ref>, but can be used more generally in situations where an expression for $\avg{s_\lambda s_\nu}$ in terms of non-skew Schur polynomials is desirable.

§.§.§ Expansion in power sum symmetric polynomials

For certain applications, such as the LRRW models we will be considering in the next section, expanding $\avg{s_\lambda(U) s_\nu (U^{-1})}$ in terms of power sums $p_k(x)$ and $p_k(y)$ may be particularly useful. One way to do so is to use the expansion in (<ref>), calculate all $\chi^\lambda_{~\alpha},\chi^\nu_{~\alpha}$, and contract power sums using (<ref>). However, this is rather inconvenient as the computation of all the symmetric group characters is rapidly becomes more complicated for larger $\lambda,\nu$. Instead, it would be more effective to once more use equation (<ref>) to find a recursive expansion in terms of power sum polynomials. We will do so here, ultimately leading to equation (<ref>). For comparison, we will also consider the more complicated method that involves the calculation of all $\chi^\lambda_{~\alpha},\chi^\nu_{~\alpha}$ at the end of this subsection to demonstrate its inconvenience compared to equation (<ref>).

The expression that we will be deriving provides an iterative method for expanding in $p_\gamma (x)$ and $p_\omega(y)$ that does not require us to find $\chi^\lambda_{~\alpha}$ and $\chi^\nu_{~\alpha}$ for all $\alpha$ and then contract over all combinations of power sum polynomials. Essentially, we apply equations (<ref>) and (<ref>) to $\avg{s_\lambda(U) s_\nu(U^{-1})}$ and make use of the orthogonality properties of the symmetric group characters. In particular, we will revert the order of expansion in section <ref> and start from the term with all $p_\mu$ contracted, see equation (<ref>), then consider the term where a single $p_j$ is not contracted, then two uncontracted power sums, and so on. For simplicity, we start by considering autocorrelation function up to the subleading term, which gives

s_λ(U) s_λ(U^-1) =  ∑_μ,ρ  χ_ μ^λ/z_μχ_ ρ^λ/z_ρ p_μ(U) p_ρ(U^-1)
= 1+  ∑_j ≥1  ∑_μ(χ_ μ^λ)^2 /z_μ  m_j(μ)/jp_j(x)p_j(y) + …where sum is over all $\mu$ containing a row of size $j$. We have $\mu =\rho$ since this is the only way to contract all of $p_\mu$ with $p_\rho$ except for two copies of $p_j$ (one from $p_\mu$ and the other from $p_\rho$). Using the recursive formula for $\chi^\lambda_{~\mu}$ in (<ref>) and the orthogonality of symmetric group characters in (<ref>), the rightmost term in equation (<ref>) then gives

∑_j 1/j^2 ∑_μ ( ∑_η(-1)^(η) χ^λ∖η_ μ- (j) )  ( ∑_ζ(-1)^(ζ) χ^λ∖ζ_ μ- (j) )/z_μ-(j)    p_j(x)p_j(y) = ∑_j≥1 p_j(x)p_j(y)/j^2 ∑_η1 ,
where $\eta $ and $\zeta $ are border strips of size $j$. Equation (<ref>) then tells us that the coefficient $p_j(x)p_j(y)$ in the power sum expansion of $\avg{s_\lambda (U) s_\lambda(U^{-1})}$ is given by $\frac{1}{j^2}$ times the number of ways to remove a border strip of size $j$ from $\lambda$.  Consider another term in this expansion, where $\mu = (j,\alpha)$, $\rho=(k^2 ,\alpha)$ with $j=2k$, and contract over $p_\alpha$ leading to the term proportional to $p_j(x)p_k(y)^2$. Via the same reasoning as above, this is given by

p_j (x) p_k(y)^2/2 jk^2 ∑_η,ζ,ξ (-1)^(η)+  (ζ)+(ξ) δ_λ∖η,(λ∖ζ)∖ξ ,
where the sums are over all $\eta$, border strips of size $j$, and $\zeta$ and $ \xi$, border strips of size $k$. We see that the terms appearing in the power sum expansion of $\avg{s_\lambda (U) s_\lambda(U^{-1})}$ are found by removing border strips $\{\eta\}$ and $\{\zeta\}$ from $\lambda$ such that $\lambda \setminus \{\eta\} =\lambda \setminus \{\zeta\}$. Continuing this procedure gives the following. Consider not contracting $p_\omega (U)$ and $p_\gamma(U^{-1})$ for some $\omega$ and $\gamma$, leading to the term proportional to $p_\omega(y)p_\gamma(x)$. In general this leads to a sum over $\mu$ and $\rho$ (in the middle expression in (<ref>)) of the form

μ= (ω,α)  ,    ρ= (γ,α) ,

so that $m_j(\mu)-m_j(\omega) = m_j(\alpha ) = m_j(\rho) - m_j(\gamma)$ for all $j$, and $\abs{\omega}=\abs{\eta}$. Since we sum over $\mu$ and $\rho$ for a  fixed choice of $\omega$ and $\gamma$, this effectively leads to a sum over $\alpha$. Since there are $ \frac{1}{m_j(\alpha)!}  \frac{m_j(\mu)!}{m_j(\omega)!} \frac{m_j(\rho)!}{m_j(\gamma)!}   $ ways to perform $m_j(\alpha)$ contractions of $p_j$ appearing in $p_\mu$ and $p_\rho$, we have

p_μp_ρ = …+  p_ω(y)p_γ(x) ∏_j≥1 1/m_j(α)!  m_j(μ)!/(m_j(μ)-m_j(α))! m_j(ρ)!/(m_j(ρ)-m_j(α))! j^m_j(α)  +…where we show only the term proportional to $p_\omega(y)p_\gamma(x)$. Using the fact that,

z_μ= z_α∏_j≥1 j^m_j(μ) - m_j(α) m_j(μ)!/ m_j(α)! ,
and again applying the orthogonality of symmetric group characters, we find that the term proportional to $p_\omega(y)p_\gamma(x)$ in the expansion of
$\avg{s_\lambda(U)  s_\lambda(U^{-1})}$ is given by

p_ω(y)p_γ(x)/z_ωz_γ ∑_{η} , { ξ}  (-1)^(T_η) +(T_ξ) δ_λ∖{η}, λ∖{ ξ} ,

where the sum is over border strips $\eta_1, \eta_2,\dots$ satisfying $\abs{\eta_j} = \omega_j$, as well as border strips $\xi_j$ satisfying $\abs{\xi_j} = \gamma_j$. Further, $T_\eta$ and $T_\xi$ are the (generally disconnected) skew diagrams consisting of the unions of the border strips in $\eta$ and $\xi$, respectively, and $\hg(T)$ is defined in (<ref>). It is clear that the examples in (<ref>) and (<ref>) arise as special cases of (<ref>). The autocorrelation function can then be expanded as a sum $\omega $ and $\gamma$, each contributing a term of the form appearing in (<ref>). The same reasoning can evidently be applied to more general correlation functions, which are then given by

s_λ(U) s_ν(U^-1) = ∑_ω, γ p_ω(y)p_γ(x)/z_ωz_γ ∑_{η} , { ξ}  (-1)^(T_η) +(T_ξ) δ_λ∖{η}, ν∖{ ξ} .

The above expression can be straightforwardly applied by considering the different ways to remove border strips from $\lambda$ and $\nu$ such that the resulting diagrams are identical, as expressed by the presence of $\delta_{\lambda\setminus \{\eta\}, \nu \setminus \{ \xi\}}$. This provides a recursive expression which significantly simplifies various computations, especially when $\lambda, \nu$ are large diagrams for which the computation of the symmetric group characters rapidly grows more difficult. Equation (<ref>) is particularly useful for applications where one has access to $p_k(x)$ and $p_k(y)$. This includes LRRW models, where $p_k(x)$ and $p_k(y)$  are proportional to the time parameter and the hopping parameters, as we will see in the next section.

We now apply (<ref>) to $\avg{s_\lambda(U) s_\lambda(U^{-1})}$ for $\lambda =(3,2)$. Consider all distinct ways to remove a border strip of size one, i.e. a single cell, from $~{\tiny \ydiagram{3,2}}$ . As can be seen in figure <ref>, this results in the following diagrams.

*(white)     *(white)     *(black)
(white)      *(white)

*(white)     *(white)        *(white)
(white)    *(black)

Removing a single cell twice results in the following diagrams.

*(white)     *(white)        *(white)
(black)    *(black)

*(white)     *(white)     *(black)
(white)     *(black)

We get multiplicity two for $~{\tiny \ydiagram{2,1}}$ on the right as one can remove the single cells indicated in black in either order, corresponding to the fact that there are two distinct ways to arrive at $~{\tiny \ydiagram{2,1}}$ from $~{\tiny \ydiagram{3,2}}$ by following $p_1$ in figure <ref>. Removing border strips of sizes $2,3,4$ results in the diagrams below.

*(white)     *(white)        *(white)
(black)    *(black)

*(white)     *(black)       *(black)
(white)    *(black)

*(white)     *(black)       *(black)
(black)     *(black)

Note that the height $\hg$ of the border strips of sizes $3$ and $4$ is given by 1, so that $(-1)^\hg = -1$. It is easy to see that no border strips of size $\geq 5 $ can be removed from $~{\tiny \ydiagram{3,2}}$ , since it is a partition of 5 that is not itself a border strip. From the fact that there are two distinct ways to remove a single cell from $\lambda$, and only a single way to remove border strips of sizes $2,3,4$, we see that the the corresponding contributions are given by

2 p_1(x) p_1(y) + p_2(x) p_2(y) /4 + p_3(x) p_3(y) /9+p_4(x) p_4(y) /16 .

Consider now the term proportional to $p_1(x)^2 p_1(y)^2$. The two diagrams which arise from consecutively removing two single cells are $~{\tiny \ydiagram{2,1}}$ and $~{\tiny \ydiagram{3}}$ , where the former appears with multiplicity two, as demonstrated above. We thus see that there are four ways to take two copies of $~{\tiny \ydiagram{3,2}}$ , consecutively remove two single cells, and end up with $~{\tiny \ydiagram{2,1}}$ . Conversely, there is only a single way to end up with $~{\tiny \ydiagram{3}}$ via this procedure. Combined with $(z_{(1^2)})^2 = 4$, this demonstrates that we have

5 p_1(x)^2 p_1(y)^2/4

appearing in the expansion of $\avg{s_{(3,2)}s_{(3,2)}}$. Lastly, we consider $p_2(x) p_1(y)^2 + p_1(x)^2 p_2(y)$. One may remove either a border strip of size two or two single cells and end up in $~{\tiny \ydiagram{3}}$ , see the leftmost diagrams in (<ref>) and (<ref>). Combined with $z_{(1^2)} = 2 = z_{(2)}$, this results in

p_2(x) p_1(y)^2 + p_1(x)^2 p_2(y)/4 .

Although the effectiveness of (<ref>) is already clear for $~{\tiny \ydiagram{3,2}}$ , this is still a
small partition. For larger $\lambda$, it becomes progressively harder to compute $\chi^\lambda_{~\alpha}$, increasing the advantage of (<ref>). We will work out more complicated examples in section <ref>, where we apply the above results to correlation functions of LRRW models.

We now proceed to compute $\avg{s_{(3,2)}(U) s_{(3,2)} (U^{-1})}$ via the more laborious method briefly outlined at the start of this subsection, where we compute $\chi^{(3,2)}_{~\alpha}$ for all $\alpha$, apply (<ref>), and perform all possible contractions between the power sum polynomials appearing in (<ref>). We will see in the following section, below equation (<ref>), that one may calculate $\chi^\lambda_{~\alpha}$ by using a relation with fermionic particles hopping on a one-dimensional lattice. One may then apply Wick's theorem to the resulting expression to find an expansion of LRRW correlation funtions in terms of power sums. Although this may provide a convenient method for calculating $\chi^\lambda_{~\alpha}$, we will see here that its application to the computation of $\avg{s_\lambda (U) s_\nu (U^{-1})}$ is much less convenient than simply applying equation (<ref>). Using (<ref>) and looking at figure <ref>, we see that the non-zero characters $\chi^\lambda_{~\alpha} $ are given by

χ^(3,2)_ (1^5)      =  5  ,     χ^(3,2)_ (2,1^3)        =  1         ,      χ^(3,2)_ (2^2,1)        =   1 ,

χ^(3,2)_ (3,1^2)      = 1  ,     χ^(3,2)_ (3,2)        =  -1         ,     χ^(3,2)_ (4,1)        =   1 .

Applying (<ref>) gives

\begin{align}
\label{y32pexp}
s_{(3,2)} = & \frac{p_1^5}{24} + \frac{ p_1^3 p_2}{12} + \frac{ p_1^2 p_3}{6}+\frac{p_2^2p_1}{8} - \frac{ p_2 p_3}{6} + \frac{p_1p_4}{4}\notag \\
= &  \frac{p_{(1^5)}}{24} + \frac{ p_{(2,1^3)} }{12} + \frac{ p_{(3,1^2)} }{6}+\frac{p_{(2^2,1)}}{8} - \frac{ p_{(3,2)} }{6} + \frac{p_{(4,1)}}{4} ~.
\end{align}

We then apply Wick's theorem to $\avg{s_{(3,2)}(U)s_{(3,2)}(U^{-1}}_c$. We give three examples below, again indicating the compositions $\alpha$ corresponding to the $p_\alpha$ which are contracted.

\begin{align}
\overset{(4)}{\frac{1}{4} p_1(x) p_1(y) } + \overset{(2^2)}{ \frac{1}{8^2} p_1(x)p_1(y) }  & + \overset{(3)}{\frac{3}{~6^2}\left( p_2(x)p_2(y) -   p_2(y)p_1(x)^2 \right)} + %\dots
\notag \\
& + \frac{3}{~6^2}\left( p_1(x)^2 p_1(y)^2 - p_2(x)p_1(y)^2  \right) +\dots
\end{align}

We will use the above method to find the prefactors of $p_\omega(y) p_\gamma(x)$ for some examples of $\omega,\gamma$. Consider first $\gamma = \omega = (4)$, leading to a term proportional to $p_4(x)p_4(y)$. This term is rather simple to find as we only get a contribution from the rightmost term in (<ref>). In particular, we have

1/4^2 p_(4,1)(U) p_(4,1)(U^-1) _c = p_4(x)p_4(y) /4^2 + p_1(x)p_1(y)/4 + 1/4 ,
so that $p_4(x)p_4(y)$ appears with a prefactor $\frac{1}{4^2}$ in the expansion of $\avg{s_{(3,2)}(U) s_{(3,2)}(U^{-1})}$. We consider now those terms proportional to $p_3(x)p_3(y)$ arising from (<ref>), where the dots below refer to terms not proportional to $p_3(x)p_3(y)$ and where we omit writing $(U^{\pm 1})$ explicitly henceforth.

\begin{align}
\label{mnxmpf2}
\frac{1}{6^2} \avg{ p_{(3,1^2)} p_{(3,1^2)}  }  +\frac{1}{6^2} \avg{ p_{(3,2)}  p_{(3,2)}  } & =   \frac{p_3(x)p_3(y)}{36} ( 2 +2)+ \dots \notag \\
& = \frac{p_3(x)p_3(y)}{9} + \dots
\end{align}

For $p_2(x)p_2 ( y)$, we find

\begin{align}
\label{mnxmpf3}
\frac{\avg{p_{(2,1^3)} p_{(2,1^3)}} }{(12)^2}  +  \frac{\avg{p_{(2^2,1)} p_{(2^2,1)}  }}{8^2}  +\frac{\avg{ p_{(3,2)}  p_{(3,2)} }}{6^2}  & =   p_2(x)p_2(y) \left( \frac{3!}{12}  + \frac{8}{8^2} + \frac{3}{6^2} + \right)+ \dots \notag \\
& = \frac{p_2(x)p_2(y)}{4} + \dots
\end{align}

We see a pattern emerge as we get $\frac{p_j(x)p_j(y)}{j^2}$ for all $j$ considered thus far. However, this pattern does not continue down to $j=1$. In particular, the following expression

p_(1^5)p_(1^5)  /(24)^2  + p_(2,1^3) p_(2,1^3)/(12)^2  + p_(3,1^2) p_(3,1^2) /6^2  +  p_(2^2,1) p_(2^2,1)  /8^2  + p_(4,1)  p_(4,1) /4^2  ,

is given by

p_1(x)p_1(y) ( (5!)^2/(4!)^3  + 6^2/(12)^2 + 12/6^2 + 8/8^2 +4/4^2 )+ …=   2 p_1(x)p_1(y) + …
Combining equations (<ref>), (<ref>), (<ref>), and (<ref>) then leads to equation (<ref>), albeit via a much less convenient method.

Consider now the term proportional to $p_1(x)^2 p_1(y)^2$, which receives contributions from

p_(1^5)p_(1^5) /(24)^2   + p_(2,1^3) p_(2,1^3)/(12)^2  + p_(3,1^2) p_(3,1^2) /6^2 .

In particular, this contributes

p_1(x)^2 p_1(y)^2 ( (5!)^2/(4!)^2 (2!)^2 3! + (12)^2/12 + 3/6^2 )  = 5  p_1(x)^2 p_1(y)^2 /4 ,

thus confirming equation (<ref>). Lastly, we consider a mixed term, namely, the term proportional to $p_2(x) p_1(y)^2 + p_1(x)^2 p_2(y) $. This is given by

\begin{align}
\label{mnxmpl}
\frac{ \avg{p_{(1^5)} p_{(2,1^3)} }}{12\cdot 24 }  - \frac{ \avg{ p_{(3,1^2)} p_{(3,2)}}}{6^2}   + \frac{\avg{p_{(2,1^3)} p_{(2^2,1)} }}{12\cdot 8}
= & p_2(x) p_1(y)^2 \left( \frac{5}{24} - \frac{1}{12} +\frac{1}{8} \right) +\dots \notag \\
= & \frac{p_2(x) p_1(y)^2}{4}+\dots
\end{align}

Of course, inverting the order of $p_\mu(U)$ and $p_\rho(U^{-1})$ in $\avg{ p_\mu(U) p_\rho(U^{-1})}$ above gives the same contribution with $x \leftrightarrow y$. The end result is therefore $\frac{ p_2(x) p_1(y)^2 + p_1(x)^2 p_2(y) }{4}$, thereby confirming (<ref>). It is clear from this simple example that the method applied in the second part of this subsection is much less powerful than equation (<ref>), and this becomes more acute when we consider larger partitions than $\lambda= (3,2)$ as the $\chi^\lambda_{~\alpha}$ are then a lot harder to compute.

§ APPLYING SYMMETRIC POLYNOMIAL THEORY TO LONG-RANGE RANDOM WALKS

We will now apply some of the results above to LRRW correlation functions by using their relation to weighted $U(N)$ integrals over Schur polynomials in equation (<ref>) and identities from symmetric polynomial theory, as well as results we derived in section <ref>. We first consider identities relating to elementary and complete homogeneous symmetric polynomials, before moving on to power sum polynomials and border strips. We will generally take $N\to \infty$ here, although the presentation in sections <ref>, <ref>, and <ref> is valid for finite $N$ as well.

§.§ Elementary and complete homogeneous symmetric polynomials

We repeat here the expression for the weight function in (<ref>),
f(e^iθ ;τ) = exp(τ∑_k∈Z a_ke^iθ )   ,
where $a_k$ are the hopping parameters of the hamiltonian in (<ref>). As noted before, the hamiltonian is a Toeplitz matrix, and $a_k$ is the number on its $k^{\text{th}}$ diagonal. By comparing with  $\eqref{genfuncsymp}$, we see that we can write the weight function as
f(z) =H(x;z)H(y;z^-1) ,
with the following identification, for $k\geq 1$,

\begin{align}
\label{hpid}
\tau a_k   & = \frac{p_k(x)}{k} ~, \notag \\
\tau a_{-k} & = \frac{p_k(y)}{k} ~.
%\frac{p_k(x)}{k}  = \tau a_k =\tau a_{-k}
\end{align}

Alternatively, we can write
f(z) =E(x;z)E(y;z^-1) ,
by identifying, for $k\geq 1$,

\begin{align}
\label{epid}
\tau a_k   & = \frac{(-1)^{k+1} p_k(x)}{k} ~, \notag \\
\tau a_{-k} & = \frac{(-1)^{k+1} p_k(y)}{k} ~,
%\frac{p_k(x)}{k}  = \tau a_k =\tau a_{-k}
\end{align}

and by transposing the diagrams as in (<ref>). When $\tau \to 0$, i.e. the CUE limit, we have

F_λ;μ (0) =  s_λ(U) s_μ(U^-1)_CUE =
δ_λ,μ  ,
which is again simply the orthonormality of Schur polynomials as the irreducible characters of $U(N)$.
By using the strong Szegő limit theorem, we can compute $F_{\emptyset;\emptyset}$ [ This has been noted before, see e.g [45].]. Assuming $a_0=0$, i.e. zero on-site energy, we have
F_∅;∅(τ)= exp( τ^2 ∑_k=1^∞ka_k^2) .
If we have $a_0\neq 0$, we get an additional multiplicative term $e^{-N \tau a_0}$ on the right, where one should remember that we take $N\to \infty$. Considering $a_1 =-1 = a_1$ and $a_k=0$ otherwise, i.e. the XX0-model, and choosing $\tau = it$, we recover the result of [46] and [45], see also [47],
F_∅;∅ (it) = e^-t^2 .

Remember that equations (<ref>) and (<ref>) state that
F_λ;μ(τ) = F_∅;∅(τ)s_λ(U) s_μ(U^-1)_τ ,
where $\avg{\dots}_\tau$ is given in equation (<ref>) with weight function given by $f(z;\tau)$ in (<ref>).
We therefore define

G_λ;μ (τ)  F_λ;μ(τ)/F_∅;∅(τ) = s_λ(U) s_μ(U^-1)_τ ,

i.e. we express correlations in terms of their proportionality to $F_{\emptyset;\emptyset}(\tau)$.
We will also write this as
G_λ;μ(τ) λμ_τ .

If $\mu=\emptyset$ (or $\lambda=\emptyset$), we will simply write
G_λ; ∅(τ) = λ∅_τλ_τ .
We also define the following - connected - correlation function,
G^c_λ;μ(τ) λμ_τ- λ_τμ^*_τ .

We consider now some explicit examples. Using (<ref>), we can express

e_n_τ=e_n(y)=G_(1^n); ∅  ,   h_n_τ=h_n(y)=G_(k); ∅  ,

and their complex conjugates $G_{\emptyset;(1^n)}$ and $ G_{\emptyset;(n) }$, in terms of $p_k(y)$ with $k\leq n$. For $h_n$ and $e_n$ with $n=4$, the diagrams and corresponding configurations are given below on the left and right, respectively.

That is, $h_n$ correspond to taking only the single rightmost particle and moving it $n$ sites to the right, whereas $e_n$ corresponds to taking the $n$ rightmost particles and moving them all a single site to the right. Note that with $x= (x_1,\dots ,x_K)$, we have $e_j(x) =0$ for $j>K$ (and likewise for $y$). This means that, for any $n,m >K$,

F_(1^m);(1^n)  =∑_j=0^K e_j e_j

That is, when we move $n>K$ adjacent particles by a single site, the effect is the same as to move $K$ adjacent particles a single site.

Not only can we read off $G_{(n);\emptyset}$ and $G_{(1^n) ; \emptyset}$ (and their complex conjugates). From the corollary of the Pieri formula in (<ref>), we have that, for any $\lambda$,

G^c_λ; (n) = s_λ(U) h_n(U^-1)_c =∑_j=1^n h_n-j s_λ/(j)=∑_j=1^n h_n-j ∑_ν^j s_ν^j  ,

where the rightmost sum is over all $\nu^j$ such that $\lambda/\nu^j$ is a horizontal strip of length $j$. Take, for example, $\lambda= (3,2)$,

which corresponds to

We remind the reader that the dots on the left (right) refer to an infinite string of particles (holes). We take $\lambda = (3,2)$ and $n=2$. For this choice of $\lambda$, the diagrams contributing to the sum over $\nu^1 $ and $\nu^2$ in (<ref>) are given in equation (<ref>).

We assign particles and holes to these diagrams and remind ourselves that

G_(1);∅ (τ) =

We then see that equation (<ref>) is given by the following expression. For each configuration, the dots on the left and right again indicate an infinite sequence of particles and holes, of which we show only the left- and rightmost, respectively.

\begin{align}
G_{(3,2);(2)} (\tau)  = & ~ \langle
\cdots
\vcenterbox{\unitlength=\cellsize\begin{picture}(7,2)
\linethickness{0.7pt}
\put(0,1){\line(1,0){7}}
\put(.5,1){\circle*{0.3}}
\put(3.5,1){\circle*{0.3}}
\put(5.5,1){\circle*{0.3}}
\put(1.5,1){\circle*{0.3}}
\put(2.5,1){\circle*{0.3}}
\put(4.5,1){\circle*{0.3}}
\put(6.5,1){\circle*{0.3}}
\put(1.5,1){\circle{0.3}}
\put(2.5,1){\circle{0.3}}
\put(4.5,1){\circle{0.3}}
\put(6.5,1){\circle{0.3}}
\end{picture}}
\cdots
\lvert
\cdots
\vcenterbox{\unitlength=\cellsize\begin{picture}(5,2)
\linethickness{0.7pt}
\put(0,1){\line(1,0){5}}
\put(.5,1){\circle*{0.3}}
\put(3.5,1){\circle*{0.3}}
\put(1.5,1){\circle*{0.3}}
\put(2.5,1){\circle*{0.3}}
\put(4.5,1){\circle*{0.3}}
\put(1.5,1){\circle{0.3}}
\put(2.5,1){\circle{0.3}}
\put(4.5,1){\circle{0.3}}
\end{picture}}
\cdots
\rangle_\tau \notag \\
= & \left( \langle  \cdots
\vcenterbox{\unitlength=\cellsize\begin{picture}(6,2)
\linethickness{0.7pt}
\put(0,1){\line(1,0){6}}
\put(.5,1){\circle*{0.3}}
\put(3.5,1){\circle*{0.3}}
\put(4.5,1){\circle*{0.3}}
\put(1.5,1){\circle*{0.3}}
\put(2.5,1){\circle*{0.3}}
\put(5.5,1){\circle*{0.3}}
\put(1.5,1){\circle{0.3}}
\put(2.5,1){\circle{0.3}}
\put(5.5,1){\circle{0.3}}
\end{picture}}
\cdots
\rangle_\tau
\langle  \cdots
\vcenterbox{\unitlength=\cellsize\begin{picture}(7,2)
\linethickness{0.7pt}
\put(0,1){\line(1,0){7}}
\put(.5,1){\circle*{0.3}}
\put(2.5,1){\circle*{0.3}}
\put(5.5,1){\circle*{0.3}}
\put(1.5,1){\circle*{0.3}}
\put(3.5,1){\circle*{0.3}}
\put(4.5,1){\circle*{0.3}}
\put(6.5,1){\circle*{0.3}}
\put(1.5,1){\circle{0.3}}
\put(3.5,1){\circle{0.3}}
\put(4.5,1){\circle{0.3}}
\put(6.5,1){\circle{0.3}}
\end{picture}}
\cdots
\rangle_\tau
\right) \times G_{(1);\emptyset}(\tau) \notag\\
& +
\langle  \cdots
\vcenterbox{\unitlength=\cellsize\begin{picture}(6,2)
\linethickness{0.7pt}
\put(0,1){\line(1,0){6}}
\put(.5,1){\circle*{0.3}}
\put(2.5,1){\circle*{0.3}}
\put(4.5,1){\circle*{0.3}}
\put(1.5,1){\circle*{0.3}}
\put(3.5,1){\circle*{0.3}}
\put(5.5,1){\circle*{0.3}}
\put(1.5,1){\circle{0.3}}
\put(3.5,1){\circle{0.3}}
\put(5.5,1){\circle{0.3}}
\end{picture}}
\cdots
\rangle_\tau
\langle  \cdots
\vcenterbox{\unitlength=\cellsize\begin{picture}(6,2)
\linethickness{0.7pt}
\put(0,1){\line(1,0){6}}
\put(0.5,1){\circle*{0.3}}
\put(4.5,1){\circle*{0.3}}
\put(1.5,1){\circle*{0.3}}
\put(2.5,1){\circle*{0.3}}
\put(3.5,1){\circle*{0.3}}
\put(5.5,1){\circle*{0.3}}
\put(1.5,1){\circle{0.3}}
\put(2.5,1){\circle{0.3}}
\put(3.5,1){\circle{0.3}}
\put(5.5,1){\circle{0.3}}
\end{picture}}
\cdots
\rangle_\tau
\end{align}

The second line above corresponds to $\nu^1$ in (<ref>), whereas the third line corresponds to $\nu^2$. By comparing with equation (<ref>), we see that the sum over $\nu^1$ contains all diagrams related to 3,2 by moving a single particle a single site to the left. The sum over $\nu^2$ consists of all ways to move one or two particles by a total of two sites, with the restriction that we cannot move two adjacent particles. This is the interpretation of the corollary of the Pieri formula in (<ref>), and it generalizes to $h_n $ for any $n\in\mathds{Z}^+$. In particular, we can assign a particle-hole interpretation to $s_{\lambda/(n)}$ by noting that it is given by summing over all distinct ways to move $\leq n$ particles a total of $n$ sites to the left without moving any two adjacent particles.

We can apply the same reasoning to $\avg{s_\lambda (U) e_n(U^{-1}}$ by using (<ref>). Similar to (<ref>), we have

G_λ; (1^n) =∑_j=1^n e_n-j s_λ/(1^j) =∑_j=1^n e_n-j ∑_ν^j s_ν ,
where $\sum_{\nu^j}$ is now a sum over all $\nu$ such that $\lambda/\nu$ is a vertical strip of length $j$. The skew Schur polynomial $s_{\lambda/(1^n)}$ can be interpreted as taking the configuration corresponding to $\lambda$ and summing over all distinct ways to move $ n$ particles a single site to the left.

Consider the case where $x=y$ (e.g. $\tau = \beta \in \mathds{R}$ and $a_k = a_{-k}$). We can then apply the Pieri formula once again, in this case to the products $h_{n-j}s_{\lambda/(j)}$ in (<ref>), as
h_n-js_λ/(j) = ∑_νs_ν ,
where the sum is over all $\nu$ obtained from $\lambda$ by removing a horizontal strip of size $j$ and then adding to the resuting diagram a horizontal strip of size $n-j$. We consider again $G_{(3,2);(2)} = s_{(3,2)/(1)} h_1 +  s_{(3,2)/(2)}$, where $(3,2)/(2) = (2,1) + (3)$. Applying the Pieri formula to $s_{(3,2)/(1)} h_1 = s_{(3,2)/(1)} s_{(1)} $ gives the following diagrams, where the cell that is added again indicated in gray. It is clear that $\lambda = (3,2)$ appears twice, as there are two ways to remove (and then add again) a single cell from $\lambda$.

*(white)     *(white)     *(gray)
(white)      *(white)

*(white)     *(white)
(white)      *(white)

*(white)     *(white)     *(white)      *(gray)

*(white)     *(white)     *(white)
(white)       *(gray)

*(white)     *(white)     *(white)

We now briefly consider the single particle correlation functions $F_{j;l}$. The generating function $f(z) = \sum_j F_{j;l}z^{j-l}$ for one-particle correlations is written in (<ref>). Writing again $f(z) =H(x;z)H(x;z^{-1})$ for some $x$, we have

f(z) = ∏_j(1-x_jz)^-1(1-y_j z^-1)^-1 =∑_j,k=0^∞h_j(x) h_k(y) z^j-k ∑_m=0^∞d_m(z^m+z^-m) ,

where $h_j$ are complete homogeneous symmetric polynomials. Then,

d_m=∑_j=m^∞h_j(y)h_j-m(x) = h_M(U) h_M-m(U^-1) = G_j;l(τ)  ,    j-l=m   .
where $M$ is the largest number $k$ such that $h_k\neq 0$, which is typically infinite. Remember that the $G_{j;l}(\tau)$ are the single-particle wavefunctions at site $j$ and time $\tau$ for a particle released from site $l$. The last two equalities in (<ref>) relate $F_{j;l}(\tau)$ to single-particle wavefunctions at site $M$ and time $\tau$ for a particle released from site $M-m$. Since $M$ is infinite for most choices of hopping parameters $a_k\sim p_k$, this correspond to releasing a particle at infinity and finding its wavefunction $m$ lattice sites away from where it was released at some Euclidean time $\tau$. The fact that $h_{M-m}(U^{-1}) = G_{j;l}(\tau)$ for $M\to \infty$ agrees with the intuition that a single particle at infinity does not feel the presence of the other remaining particles, which are infinitely far away from it.

Lastly, we note that one can apply the Jacobi-Trudi identity (<ref>) to $h_n$ and $e_n$ to find any skew Schur polynomials on the right hand side of (<ref>). This, in effect, provides a way to compute any correlation function. Although the resulting expressions generally grow quickly as one considers large partitions, the application of this method itself is rather simple as it only requires the computation of a determinant. In <ref>, we will outline various other, closely interrelated, methods to calculate any $\avg{s_\lambda  s_\mu }$, based on power sum polynomials rather than elementary and complete homogeneous symmetric polynomials.

§.§.§ Quasi-local particle-hole duality

Remember that one can replace the generating function $f_1(z) =  H(x;z) H(y;z^{-1})$ by $f_2(z) =  E(x;z) E(y;z^{-1})$ at the cost of transposing the partitions appearing in the correlation functions. This leads to a duality between models related by $a_n \to (-1)^{n+1}a_n$, as we demonstrate here. Though mathematically trivial, the physical interpretation of this duality is quite surprising. As mentioned at the start of this section, this duality holds for finite as well as infinite $N$. From (<ref>), it follows that switching between $f_1(z) $ and $f_2(z)$ corresponds to taking
p_k(x) →(-1)^k+1 p_k(x) .
From equations (<ref>) and (<ref>), it is clear that replacing as $p_k(x) \to (-1)^{k+1} p_k(x)$ corresponds to $a_k \to (-1)^{k+1} a_k$. In particular, we consider two hamiltonians for the same $a_n$,

\begin{align}
\label{invham}
\hat{H}_1 & =- \sum_{m=0}^{\infty}\sum_{n}a_n \left(\sigma^-_m \sigma_{m+n}^+ + \sigma_m^-\sigma_{m-n}^+  \right)~, \notag \\
\hat{H}_2 & =- \sum_{m=0}^{\infty}\sum_{n} (-1)^{n+1} a_n \left(\sigma^-_m \sigma_{m+n}^+ + \sigma_m^-\sigma_{m-n}^+  \right)~.
\end{align}

Correlation functions for $\hat{H}_1$ correspond to weight function $f_1(z) $, whereas those for $\hat{H}_2$ correspond to $f_2(z) $. Let us compare correlation functions for $\hat{H}_1$ and $\hat{H}_2$, which we will write as $
F^{(1)}_{\lambda;\mu} (\tau) $ and $
F^{(2)}_{\lambda;\mu} (\tau) $, respectively. From Szegő's theorem, we know that $F_{\emptyset;\emptyset}$ only depends on $p_k(x) p_k(y) $, so taking $p_k \to (-1)^{k+1}p_k$, has no effect on $F_{\emptyset;\emptyset}$. Therefore,
F^(1)_∅;∅ (τ) = F^(2)_∅;∅ (τ)   ,
that is, the return probability for $N$ adjacent particles for $\hat{H}_1$ is identical to that of $\hat{H}_2$ for any $\tau$. Moreover, by comparing (<ref>) and (<ref>), we see that, for general $\lambda$ and $\mu$,

F^(1)_λ;μ (τ)  = F^(2)_λ^t;μ^t (τ)   .

This establishes the following duality between correlations functions of $\hat{H}_1$ and $\hat{H}_2$. It is well known that transposition of diagrams induces a particle-hole and parity transformation on the corresponding particle-hole configuration. This simply follows from the fact that transposition exchanges left and right, and that it maps vertical edges to horizontal ones (and vice versa). However, we are not implementing particle-hole symmetry in (<ref>) but instead we establish an equality between different correlation functions corresponding to different models. Let us consider the configuration associated to $\lambda = (5,4,2,1^3)$ and $\lambda^t=(6,3,2^2,1)$, given on the left and right below, respectively.

These correspond to the following configurations,

\begin{align}
\label{transpdiag2}
\ket{\lambda} = &  \cdots
\vcenterbox{\unitlength=\cellsize\begin{picture}(13,2)
\linethickness{0.7pt}
\drawline(1,.3)(1,1.7)
\drawline(12,.3)(12,1.7)
\put(0,1){\line(1,0){13}}
\put(.5,1){\circle*{0.3}}
\put(2.5,1){\circle*{0.3}}
\put(3.5,1){\circle*{0.3}}
\put(4.5,1){\circle*{0.3}}
\put(6.5,1){\circle*{0.3}}
\put(9.5,1){\circle*{0.3}}
\put(11.5,1){\circle*{0.3}}
\put(1.5,1){\circle*{0.3}}
\put(5.5,1){\circle*{0.3}}
\put(7.5,1){\circle*{0.3}}
\put(8.5,1){\circle*{0.3}}
\put(10.5,1){\circle*{0.3}}
\put(12.5,1){\circle*{0.3}}
\put(1.5,1){\circle{0.3}}
\put(3.5,1){\circle{0.3}}
\put(5.5,1){\circle{0.3}}
\put(7.5,1){\circle{0.3}}
\put(8.5,1){\circle{0.3}}
\put(10.5,1){\circle{0.3}}
\put(12.5,1){\circle{0.3}}
\end{picture}}
\cdots  ~, \notag\\
\ket{\lambda^t} = &   \cdots
\vcenterbox{\unitlength=\cellsize\begin{picture}(13,2)
\linethickness{0.7pt}
\drawline(1,.3)(1,1.7)
\drawline(12,.3)(12,1.7)
\put(0,1){\line(1,0){13}}
\put(.5,1){\circle*{0.3}}
\put(2.5,1){\circle*{0.3}}
\put(4.5,1){\circle*{0.3}}
\put(5.5,1){\circle*{0.3}}
\put(7.5,1){\circle*{0.3}}
\put(11.5,1){\circle*{0.3}}
\put(1.5,1){\circle*{0.3}}
\put(3.5,1){\circle*{0.3}}
\put(6.5,1){\circle*{0.3}}
\put(8.5,1){\circle*{0.3}}
\put(9.5,1){\circle*{0.3}}
\put(10.5,1){\circle*{0.3}}
\put(12.5,1){\circle*{0.3}}
\put(1.5,1){\circle{0.3}}
\put(3.5,1){\circle{0.3}}
\put(6.5,1){\circle{0.3}}
\put(8.5,1){\circle{0.3}}
\put(9.5,1){\circle{0.3}}
\put(10.5,1){\circle{0.3}}
\put(12.5,1){\circle{0.3}}
\end{picture}}
\cdots  ~.
\end{align}

where the vertical lines on the left and right correspond to the diagonal lines at the lower left and upper right corners of $\lambda$ and $\lambda^t$ in (<ref>), respectively. We see that the regions in between the vertical lines for $\ket{\lambda} $ and $\ket{\lambda^t}$ are related by performing a parity transformation and exchanging between particles and holes. Only a finite interval $\ell(\lambda) \leq r \leq \lambda_1$ is affected non-trivially by this combination of particle-hole and parity transformations. We thus establish a bijection between the correlation functions corresponding to $\hat{H}_1$ and $\hat{H}_2$, given by the transposition of diagrams, which we refer to as quasi-local particle-hole duality. It follows trivially from the above treatment that if $
a_{2k} =0~~,~~~\forall~ k \in \mathds{Z}^+~$, then $H(x;z)  = E(x;z)$, as can be seen immediately from (<ref>). It follows that when all even hopping parameters $a_{2k}$ are zero, we have $F^{(1)}_{\lambda;\mu} = F^{(2)}_{\lambda;\mu}  $, so we can suppress the superscript. In this case, it follow from (<ref>) that
F_λ;μ (τ)  =F_λ^t;μ^t (τ)   .
That is, any system with $a_{2k}=0$ satisfies quasi-local particle-hole duality with itself.

§.§ Power sum symmetric polynomials and border strips

We consider now the particle-hole configurations corresponding to power sum polynomials $p_k$, which are a very natural basis for the application to LRRW models due to their proportionality to $\tau a_k$. Consider specifically the case where $\tau = it$. Then, from (<ref>), we have

p_k(y) = -it k a_k^*  = (-p_k(x))^* = e^i ϕ p_k(x)  ,   ϕπ-2(p_k(x)) .
Since $p_k(x)$ only differs from $p_k(y)$ by a phase, we can apply the usual multiplication rules to objects of the form $p_\rho(x) p_\mu(y)$, which is an advantage over the usage of $h_j$, $e_j$.

Remember that power sum polynomials are expanded as an alternating sum over hook-shaped Schur polynomials, see equation (<ref>). Note that
U^n_τ=  p_n =  ∑_r=0^n-1(-1)^r G_(n-r,1^r);∅ ,
can be read off from the hamiltonian by the identification in either (<ref>) or (<ref>). As an example, the configuration corresponding to a hook shape $\lambda= (4,1^2)$ is as follows.

That is, a hook-shaped diagram corresponds to taking the particle at site $-b$ and moving it $a+b$ sites to the right. More generally, consider

p_ns_λ =
∑_r=0^n-1(-1)^r  G^c_λ;(n-r,1^r) =
∑_r=0^n-1(-1)^r  (n-1,1^r)λ  .

Using equation (<ref>), this is given by

∑_r=0^n-1(-1)^r s_λ/(n-r,1^r)=∑_η(-1)^(η)s_λ∖η ,

where the sum is over all $\eta$ which are border strips of size $n$, see e.g. the example in (<ref>) for $n=4$. Note that there is only a single Schur polynomial appearing on the right hand side, as opposed to (<ref>) and (<ref>), which contain factors of $e_{n-j}$ and $h_{n-j}$, respectively. We consider a few specific examples of (<ref>), already noted in [40], before treating it in generality. In particular, consider $ \avg{s_\lambda p_n}_c$ with $(n-r,1^r) \nsubseteq \lambda ~\forall~ r \in \{ 0,\dots, n-1\}$, such that $\lambda_1+\lambda_1^t-1 < n$. This gives

U^-n _λU_c =0 .
Writing $\lambda $ in Frobenius notation as $\lambda = (a_1,\dots ,a_k| b_1,\dots,b_k)$ with $a_j$ satisfying $a_1> \dots >a_k$ and similar for $b_j$. In this case, $a_1+b_1+1$ equals the hook-length of the top left cell of $\lambda$. Equation (<ref>) then states that $\avg{ \tra U^{-n} \tra_\lambda U}_c =0 $ if $ a_1+b_1 +1<n$. In terms of particle-hole configurations, (<ref>) states that $\sum_{r=0}^{n-1}(-1)^r  G^c_{\lambda ;(n-r,1^r) } = 0$ if $n$ is greater than the distance (in units of lattice spacing) between the leftmost hole and rightmost particle in the configuration corresponding to $\lambda$ .

Further, write $m = a_1+b_1+1-n $ and consider $\lambda=(a|b)=(a_1,\dots,a_k|b_1,\dots b_k)$ such that $m \leq a_1-a_2-1 $ and $m \leq b_1 -b_2-1 $, respectively.
Take $\mu = (a_2,\dots,a_k|b_2,\dots,b_k) $, obtained by removing the first row and column from $\lambda$.
Any $(n-r,1^r) \subseteq \lambda$ then satisfies
λ/(n-r,1^r) = (a_1+1,1^b_1) /(n-r,1^r) ⊗μ= (a_1+1-n+r)⊗(1^b_1-r) ⊗μ .

Consider the example of $(6,4,3,1^2)/(5,1^3)$, which is given by the diagram below.

*(black)      *(black)      *(black)      *(black)      *(black)      *(white)
(black)       *(white)     *(white)       *(white)
(black)       *(white)     *(white)

We then apply (<ref>) to find

s_λ(U) U^-n _c =    ∑_r=n-a_1-1^b_1 (-1)^r s_λ/(n-r,1^r)  =±s_μ∑_k=0^m (-1)^k  h_m-ke_k =0 .

Assigning particle-hole configurations to the diagrams above, we see that the correlation function corresponding to $	\avg{s_\lambda (U) \tra U^{-n} }_c$ vanishes when the distance between the leftmost hole and rightmost particle, as well as the distance between the rightmost particle and the second-to-rightmost particle (and vice versa for holes), are sufficiently small compared to $n$.

We now consider (<ref>) more generally. In this expression, $\nu$ is related to $\lambda$ by the removal of a border strip, i.e. a connected skew diagram not containing a subdiagram that is a 2 by 2 block. In terms of particle-hole configurations, a 2 by 2 block corresponds to moving two adjacent particles by two sites to the right, see below.

In equation (<ref>), the fact that a border strip has no 2 by 2 subdiagram therefore states that (the configuration corresponding to) $\nu$ is related to $\lambda$ by moving (a) particle(s) left by $n$ sites without moving two or more adjacent particles by two or more sites. The number of rows that $\lambda/\nu$ occupies (which is $\hg(\lambda/\nu)+1$) equals the number of particles that are involved in this process, which follows immediately from the fact that vertical edges on the boundary of a diagram correspond to particles. From the fact that the border strip is connected, it follows that it occupies only consecutive rows. For example, the skew diagram on the left is a border strip, whereas the one on the right clearly is not.

This means that only consecutive (but not necessarily adjacent) particles are affected. However, connectedness is a stronger condition, and in the case of border strips this leads to the following observation. For a skew diagram $\lambda/\nu$, we call the outer rim the (horizontal and vertical) edges on the bottom right of the diagram of $\lambda/\nu$, as in e.g. [41]. Conversely, the inner rim consists of the edges on the top left of $\lambda/\nu$. For a border strip containing $n$ cells, we number the edges of the inner and outer rims by $j=\{1,\dots,n+1 \}$. We will refer to the $j^\text{th}$ edge on the outer rim as the $j^\text{th}$ outer edge, and likewise for the inner rim. We consider an explicit example, where $\lambda = (8,6^2,4,1)$ and $\nu = (5^2,3,2,1)$, leading to a border strip of size $n=9$ below. On the right hand side, we number the vertical edges of the inner and outer rims of $\lambda/\nu$.

*(black)      *(black)      *(black)      *(black)      *(black)      *(white)      *(white)      *(white)
(black)      *(black)      *(black)      *(black)      *(black)      *(white)
(black)      *(black)      *(black)      *(white)       *(white)       *(white)
(black)      *(black)      *(white)       *(white)

[     7]                  [  10]

[     6]          [  7]

[     3]                  [  6]
[     1]               [  3]

One can see in the above example that the first edge on the inner rim and the ledge edge on the outer rim are both vertical, which is clearly true for any (skew) diagram. Further, we see that all other vertical edges occupy the same positions on the inner and outer rims, namely $\{3,6,7\}$. If e.g. the third edge on the inner rim were horizontal, the resulting skew diagram would be disconnected and therefore not a valid border strip, shown below.

It is clear that this holds generally for border strips. In particular, the inner and outer edges of any border strip are identical for $j=2,3,\dots,n$, and the outer (inner) edge for $j=1$ is horizontal (vertical), whereas the outer (inner) edge for $j=n+1$ is vertical (horizontal).

The configurations corresponding to $\lambda$ and $\nu$ are related by taking the configuration on the outer rim of $\lambda/\nu$ (as a subset of the outer rim of $\lambda$) and replacing it by the configuration corresponding to the inner rim of $\lambda/\nu$. What we effectively get is the following. We take a particle at some site $k$ and move it to site $k-n$, and we get a factor $(-1)^{\hg(\lambda/\nu)}$. Here, $\hg(\lambda/\nu)$ equals the number of particles the affected particle jumps over, that is, the number of particles occupying sites $\{ k-n+1,\dots,k-1\}$. We thus see that $(-1)^{\hg(\lambda/\nu)}$ simply implements fermionic statistics. Below, we show the border strip $\lambda/\nu$ with empty cells as a subset of $\lambda$, where we indicate particles and holes. The inner edge $j$ and outer edge $j$ for $j=2,\dots ,n $ are connected by diagonal lines.

The configurations corresponding to $\lambda$ and $\nu$ are given by

\begin{align}
\label{transpdiag}
\ket{\lambda} = &  \cdots
\vcenterbox{\unitlength=\cellsize\begin{picture}(15,2)
\linethickness{0.7pt}
\put(0,1){\line(1,0){15}}
\put(.5,1){\circle*{0.3}}
\put(2.5,1){\circle*{0.3}}
\put(6.5,1){\circle*{0.3}}
\put(9.5,1){\circle*{0.3}}
\put(10.5,1){\circle*{0.3}}
\put(13.5,1){\circle*{0.3}}
\put(1.5,1){\circle*{0.3}}
\put(3.5,1){\circle*{0.3}}
\put(4.5,1){\circle*{0.3}}
\put(5.5,1){\circle*{0.3}}
\put(7.5,1){\circle*{0.3}}
\put(8.5,1){\circle*{0.3}}
\put(11.5,1){\circle*{0.3}}
\put(12.5,1){\circle*{0.3}}
\put(14.5,1){\circle*{0.3}}
\put(1.5,1){\circle{0.3}}
\put(3.5,1){\circle{0.3}}
\put(4.5,1){\circle{0.3}}
\put(5.5,1){\circle{0.3}}
\put(7.5,1){\circle{0.3}}
\put(8.5,1){\circle{0.3}}
\put(11.5,1){\circle{0.3}}
\put(12.5,1){\circle{0.3}}
\put(14.5,1){\circle{0.3}}
\end{picture}}
\cdots  ~, \notag\\
\ket{\nu} = &  \cdots
\vcenterbox{\unitlength=\cellsize\begin{picture}(15,2)
\linethickness{0.7pt}
\put(0,1){\line(1,0){15}}
\put(.5,1){\circle*{0.3}}
\put(2.5,1){\circle*{0.3}}
\put(4.5,1){\circle*{0.3}}
\put(6.5,1){\circle*{0.3}}
\put(9.5,1){\circle*{0.3}}
\put(10.5,1){\circle*{0.3}}
\put(13.5,1){\circle*{0.3}}
\put(1.5,1){\circle*{0.3}}
\put(3.5,1){\circle*{0.3}}
\put(5.5,1){\circle*{0.3}}
\put(7.5,1){\circle*{0.3}}
\put(8.5,1){\circle*{0.3}}
\put(11.5,1){\circle*{0.3}}
\put(12.5,1){\circle*{0.3}}
\put(13.5,1){\circle*{0.3}}
\put(14.5,1){\circle*{0.3}}
\put(1.5,1){\circle{0.3}}
\put(3.5,1){\circle{0.3}}
\put(4.5,1){\circle{0.3}}
\put(5.5,1){\circle{0.3}}
\put(7.5,1){\circle{0.3}}
\put(8.5,1){\circle{0.3}}
\put(11.5,1){\circle{0.3}}
\put(12.5,1){\circle{0.3}}
\put(13.5,1){\circle{0.3}}
\put(14.5,1){\circle{0.3}}
\end{picture}}
\cdots  ~.
\end{align}

It is clear that they are identical except for a single particle which has moved $n=9$ sites to the left, whereby it jumps over 3 other particles, leading to a factor $(-1)^3$. Summarizing the above, we have

∑_r=0^n-1(-1)^r G^c_λ;(n-r,1^r)  = ∑(-1)^P s_λ∖η(y) {
Distinct ways to move a particle in $\lambda$
to the left by $n$ sites, thereby hopping

over $P \leq n-1$ other particles.

}  ,

where $\lambda\setminus \eta $ again represents a diagram obtained from $\lambda$ by the removal of a border strip, in this case of size $n$. We can apply the reasoning presented above to the calculation of $\chi^{\lambda/\mu}_{~\alpha}$, by considering $ p_\alpha s_\mu$
in terms of fermionic particles hopping on a 1D lattice. In particular, starting from some partition $\mu$ and fixing a choice of $\alpha$, we consider all way to consecutively take a particle in the configuration corresponding to $\mu$ and move it $\alpha_j$ sites to the right, summing over $j\geq 1$, and adding a multiplicative factor $-1$ for each other particle which it hops over. We then add the resulting numbers $(\pm 1)$ for all cases where the end result of this process is the configuration $\lambda$. The outcome of this computation is precisely $\chi^{\lambda/\mu}_{~\alpha} $. This might provide a convenient method for computing $\chi^{\lambda/\mu}_{~\alpha}$, as it involves moving particles around on a line instead of a border strip tiling problem. Although these two problems evidently identical, the former might be simpler to implement practically.

§.§.§ The action of the hamiltonian in terms of Young diagrams

When we consider the action of the hamiltonian in (<ref>) in terms of diagrams, the following picture arises. The presentation given here does not depend on taking $N\to \infty$ and holds for finite $N$ as well. We first consider the XX0-model, in which case the action of $H^n$ on some state with $k$ particles can be described in terms of $k$ non-intersecting (vicious) random walkers that are allowed to take a single site to the left or right at each time step, for $n$ time steps [25]. For the XX0-model, considering $H\ket{\lambda}$ in terms of diagrams then corresponds to summing over all ways to add one cell to and to remove one cell from $\lambda$, as this gives vicious random walkers that can move a single site to the right or left. For a general long-range hamiltonian such as in (<ref>), the action of $H$ is given by vicious random walkers which can move $n$ sites right or left, weighted by $a_{\pm n}$ and summed over $n$. We saw above that removing a border strip of size $n$ corresponds to taking a single particle and moving it by $n$ sites to the left, so it appears that

H |λ⟩ = ∑_n a_n  (∑_η |λ∖η⟩ + ∑_ν  |ν⟩ ) ,

where $\eta$ is a border strip of size $n$ and $\nu$ is related to $\lambda$ by the addition of a border strip of size $n$, and we sum over all such $\eta$ and $\nu$. Note that we do not get the minus sign that we get when considering $\avg{p_n(U) s_\lambda(U^{-1}}_c$ as in (<ref>). For $n=1$, when we consider all ways to start with the empty partition and look at all ways to successively add single cells, we simply get Young's lattice. The action of the XX0-model on some state $\ket{\lambda}$ then corresponds to moving from $\lambda$ along all edges in Young's lattice. For a general hamiltonian as in (<ref>), the picture is clearly more involved. Take, for example, the case where $a_4\neq 0$ and $a_n=0$ for $n\neq 4$. If we have  $\lambda=(5,3,1)$, given below,

which corresponds to

|λ⟩ =      ⋯=(10,2)

⋯ .
It is easy to see that there are two ways to move a single particle four sites to the left, and there are six ways to move a particle four sites to the right. Correspondingly, the action of the hamiltonian $H\ket{\lambda}$ produces the following eight diagrams.

*(white)     *(white)     *(black)      *(black)      *(black)
(white)      *(white)         *(black)

*(white)     *(white)     *(white)     *(white)     *(white)
(black)      *(black)       *(black)

*(white)     *(white)     *(white)     *(white)     *(white)     *(gray)     *(gray)      *(gray)     *(gray)
(white)     *(white)     *(white)

*(white)     *(white)     *(white)     *(white)     *(white)     *(gray)
(white)     *(white)     *(white)     *(gray)      *(gray)      *(gray)

*(white)     *(white)     *(white)     *(white)     *(white)
(white)     *(white)     *(white)     *(gray)
(white)     *(gray)      *(gray)      *(gray)

*(white)     *(white)     *(white)     *(white)     *(white)
(white)     *(white)     *(white)
(white)     *(gray)      *(gray)
(gray)      *(gray)

*(white)     *(white)     *(white)     *(white)     *(white)
(white)     *(white)     *(white)
(white)     *(gray)
(gray)      *(gray)

*(white)     *(white)     *(white)     *(white)     *(white)
(white)     *(white)     *(white)

Consider a generalization of the Young's lattice in the form of a graph where each edge connects two diagrams that are related by addition or removal of some size $n$ border strip, for all $n$. The action $H^n \ket{\lambda}$ of a general long-range hamiltonian as in (<ref>) corresponds to an $n$-site random walk on this graph, where edges corresponding to a border strip of size $n$ carry weight $a_n$. This provides a general description of the action of a hamiltonian in (<ref>) in terms of (addition and removal of) border strips. We know from equation (<ref>) how many diagrams there are with some $n$-core $\mu$ and $n$-weight $w$, but due to the fact that the graph described above has varying connectivity, the $n$-site random walk is more likely to end up in some diagrams than others.

§.§.§ Border strip tableaux and fermionic models

As in section <ref>, the present treatment up to equation (<ref>) does not require $N\to \infty$. We saw that adding or removing a border strip $\eta$ and multiplying with $(-1)^{\hg(\eta)}$ implements fermionic statistics, and that this can be applied to the calculation of $\chi^{\lambda/\mu}_{~\alpha}$. Inverting this line of reasoning, we will now derive two relations for fermionic models using the properties of $\chi^{\lambda/\mu}_{~\alpha}$. The first of these involves the fact, noted at equation (<ref>) and below, that the order in which one adds border strips in the construction of a border strip tableau $\text{BST}(\lambda/\mu,\alpha)$ is irrelevant to its outcome. That is, $\chi^{\lambda/\mu}_{~\alpha}$ does not depend on the order of the entries of $\alpha$. The same is true for the removal of border strips, as can be seen in equation (<ref>) and the example in (<ref>) and figure <ref>. From the relation between removing or adding border strips in the construction of $\chi^{\lambda/\mu}_{~\alpha}$ and fermions hopping on a line, we can make the following observation. Consider a one-dimensional fermion configuration corresponding to some diagram $\mu$, and consider all ways of moving not necessarily distinct fermions to the right by $\alpha_1,\alpha_2,\dots$ sites, where $\alpha_j$ are unordered non-negative integers. We then see that the order of the step sizes $\alpha_j$ by which we move fermions has no effect on the outcome of this process. That is, starting from a fermionic configuration and consecutively moving fermions to the right by various step sizes, the outcome depends only on the distribution of the step sizes and not the order in which the steps are taken. It is clear from applying (<ref>) and the removal of border strips in the computation of $\chi^{\lambda/\mu}_{~\alpha}$ that the same statement holds when we consider fermions that can only hop to the left instead of the right.

Consider a simple example which is partly given in figure <ref> and in its entirety in figure <ref>. Namely, consider again $\lambda = (3,2) = {\tiny \ydiagram{3,2}}~$ and now remove twice a single cell and once a border strip of size 2 from $\lambda$.
We can see this gives ${\tiny ~ \ydiagram{1}}$ , regardless of the order of the border strip sizes.

\begin{align}
\label{y32p211}
{\small   \ydiagram{3,2} } & ~ \overset{p_2}{\longrightarrow}~  {\small   \ydiagram{3} } ~ \overset{p_1}{\longrightarrow} ~ {\small   \ydiagram{2} } ~\overset{p_1}{\longrightarrow}  ~{\small   \ydiagram{1} } \notag\\[5pt]
{\small   \ydiagram{3,2} } & ~  \overset{p_1}{\longrightarrow}  ~{\small   \ydiagram{3,1} } + {\small   \ydiagram{2,2} }  ~\overset{p_2}{\longrightarrow} ~ {\small   \ydiagram{2} } + ( 1-1) {\small \times ~ } {\small   \ydiagram{1,1} }~ \overset{p_1}{\longrightarrow} ~ {\small   \ydiagram{1} } \notag\\[5pt]
{\small   \ydiagram{3,2} } & ~  \overset{p_1}{\longrightarrow}  ~{\small   \ydiagram{3,1} } + {\small   \ydiagram{2,2} }  ~\overset{p_1}{\longrightarrow} ~ 2 {\small \times ~ } {\small   \ydiagram{2,1} } +  {\small   \ydiagram{3} }~ \overset{p_2}{\longrightarrow} ~ {\small   \ydiagram{1} }
\end{align}

It is interesting to see that the fact that we arrive at ${\tiny ~ \ydiagram{1}~}$ follows either from the cancellation between two different ways to arrive at ${\tiny ~ \ydiagram{1,1}}$ , in the second line of (<ref>), or the fact that one cannot remove a size 2 border strip from ${\tiny ~ \ydiagram{2,1}}$ , in the third line of (<ref>). Assigning particle-hole configurations to the diagrams above, given in (<ref>) for ${\tiny ~ \ydiagram{3,2}}$ , and treating the particles as fermions gives an example of the irrelevance of the order of step sizes.

Our second result for fermionic systems arises from equations (<ref>), (<ref>). These state that $\chi^{\lambda/\mu}_{~(n^k)}$ is cancellation-free, where $\mu$ is the $n$-core of $\lambda$ [42], [41]. This leads to the following observation. Take a fermionic hamiltonian $\hat{H}_f$, where fermions are only allowed to hop $n$ sites, and denote a fermionic state corresponding to the $n$-core $\mu$ of some $\lambda$ as $\lVert \mu \rrangle$. The result of $\hat{H}_f \lVert \mu \rrangle $ can then be expressed in terms of symmetric functions as

s_μp_n + ∑_r=0^n-1 (-1)^r  s_μ/(n-r,1^r) =  ∑_ν  (-1)^(ν/μ)  s_ν + ∑_ν_n  (-1)^(η)   s_μ∖η  ,

where, as before, $\nu$ and $\mu \setminus \eta$ are related to $\mu$ by the addition and removal of a border strip of size $n$, respectively. Compare with equation (<ref>), especially the factors $(-1)^\hg$. We can keep iterating this step by adding and removing border strips of size $n$ to and from the resulting diagrams. From (<ref>) and the comments below, it then follows that all diagrams appearing in the expansion of $\left(\hat{H}_f\right)^k \lVert \mu \rrangle $ have the same sign for any $k$. Further, we saw that adding or removing border strips $\mu/\nu$ and multiplying by $(-1)^{\hg(\mu/\nu)}$ corresponds to letting fermions hop over a distance of $\abs{\mu/\nu}$ lattice sites. It follows that, for a fermionic model where particles can only hop $n$ sites, all distinct ways to go from some configuration $\mu$ to another configuration $\lambda$ appear with the same sign, with the sign depending only only the choice of $\mu$ and $\lambda$. In other words, all different ways to go from any configuration $\mu$ to any other configuration $\lambda$ involves fermions hopping over either an even or an odd number of other fermions. Physically, this means that there is no interference between various ways to arrive at some fermionic state. Fermionic states will spread through Hilbert space rapidly upon time evolution, as they are not restricted by destructive interference.

The same reasoning can be applied to various expectation values of the LRRW models with hamiltonian (<ref>). In particular, we will consider

$\avg{  (\tra U^n)^k s_\lambda(U^{-1}) }_c$  .

Repeatedly applying (<ref>), which we have used at various points above, we get

(U^n)^k s_λ(U^-1) _c  = ∑_j=1^k ∑_ν_n,j (-1)^(T_j) s_ν_n,j (x) p_n(y)^k-j .
where we use the notation $\nu_{n,j} $ for diagrams related to $\lambda$ by the consecutive removal of $j$ border strips of size $n$. For example, consider again $\lambda =  (6,5,2^2,1)$ and $(n^k) = (4^4)$. Then, the diagrams $\nu_{4,2}$ are given by removing the yellow and blue regions from the diagrams in (<ref>), for $\nu_{4,3}$ the orange regions are also removed, and after removing the red regions we end up with $\nu_{4,4} = \emptyset$. Note that $\chi^{\lambda}_{~(n^k)} =0$ for most $\lambda$, one sufficient but far from necessary condition for this being that $nk \neq \abs{\lambda}$. In general, consecutively removing border strips of some size $n$ leads eventually to the aforementioned $n$-core of $\lambda$ [44]. From (<ref>), we have
s_ν_n,j(x)p_n^k-j(x) = ∑_μd^μ_n,j,k s_μ(x)  ,

for some coefficients $d^\mu_{n,j,k} $. From (<ref>), we know that $s_\lambda p_n^k $ is expanded as a sum over all diagrams related to $\lambda$ by subsequently adding a $k$ border strips of size $n$. On the other hand, $\nu_{n,j}$ is related to $\lambda$ by removal of $j$ border strips of size $j$. Therefore,

(U^n)^k s_λ(U^-1) _c  =  ∑_j=1^k ∑_ν̃_n,j^k e^iϕ(k-j) (-1)^(T_1) + (T_2) s_ν̃_n,j^k-j (x) .

In the above expression, the sum is over all $\tilde{\nu}_{n,j}^{k-j}$ constructed by first removing $j$ border strips of size $n$ from $\lambda$ (which results in $\nu_{n,j}$) and then adding $k-j$ border strips of size $n$, including multiplicities. Further, $T_1$ is given by the BST constructed from of the union of the $j$ border strips that are removed from $\lambda$, and $T_2$ is the tableau that is the union of the $k-j$ border strips that are added to $\nu_{n,j}$ to construct $\tilde{\nu}_{n,j}^{k-j}$. For $\lambda =  (6,5,2^2,1)$, we consider

(U^4)^2 s_λ(U^-1) _c  .
As mentioned above, contracting both copies of $\tra U^4$ with $s_\lambda$ gives rise to the diagrams in (<ref>) after removing the yellow and blue border strips. The $\tilde{\nu}_{4,1}^{1} $ are given by the ways to remove from and then add to $\lambda$ a border strip of size four. We see from (<ref>) that $\lambda$ appears in the expansion of (<ref>) with a multiplicity two, as there are two distinct border strips of size four that one can remove from $\lambda$, namely, the yellow and blue border strips on the leftmost diagram in (<ref>). To find the remaining diagrams appearing in (<ref>), one should add border strips of size four to the diagrams obtained after removing the yellow and blue border strips on the leftmost diagram in (<ref>). In general, we get the following picture.

(U^n)^k s_λ(U^-1) _c  =  ±∑_j=1^k e^iϕ(k-j)  s_ν̃_n,j^k-j (x) {
Distinct ways to consecutively
move $j$ particles in $\lambda$ to
the left
by $n$ sites and then move $k-j$
particles to the right by $n$ sites .

Note that the particles that are moved by $n$ sites are not necessarily distinct. Further, the right hand side appears with a positive or negative sign depending only on the final configuration. This again follows from the fact that $\chi^{\lambda/\mu}_{~\alpha}$ is cancellation-free.

§.§.§ Schur polynomial expansions of correlation functions

We consider now the expansions for general correlation functions which were derived in section <ref>. We first consider the application of (<ref>) here, before moving on to (<ref>) in section <ref>. Equation (<ref>) expresses $\avg{s_\lambda(U) s_\nu(U^{-1})}$ as a sum over diagrams obtained form $\lambda $ and $\nu$ by removing border strips of size $\alpha_1$, $\alpha_2 ,\dots$ for a composition $\alpha$, summed over $\alpha$ up to permutations of the entries $\alpha_j$. In particular, it establishes a relation between $\avg{s_\lambda s_\nu}_c $ and the correlation functions $\avg{s_{\lambda\setminus \{\eta\} }}= \bket{\emptyset}{s_{\lambda\setminus \{\eta\} }}$, involving only a single non-trivial configuration $\lambda\setminus \{\eta\}$ (and likewise for $\nu\setminus \{\eta\}$). Fixing some $\alpha$, we sum over all ways to start from $\lambda$ and $\nu$ and move particles $\alpha_j$ sites to the left with fermionic statistics (in the form of $(-1)^{\hg (T_\eta)}$). Consider the autocorrelation for $\nu=\lambda  = (8,6^2,4,1)$, which we show below.

Equation (<ref>) gives an expression in terms of diagrams obtained by removing border strips from $\lambda$. Below, we show from left to right all border strips of size $1,2,3$ that can be removed from $\lambda=\nu$.

*(white)      *(white)      *(white)     *(white)      *(white)      *(white)     *(white)      *(black)
(white)      *(white)      *(white)      *(white)      *(white)      *(white)
(white)      *(white)      *(white)      *(white)      *(white)     *(black)
(white)      *(white)      *(white)     *(black)

*(white)      *(white)      *(white)     *(white)      *(white)      *(white)     *(black)      *(black)
(white)      *(white)      *(white)      *(white)      *(white)       *(black)
(white)      *(white)      *(white)      *(white)      *(black)     *(green)
(white)      *(white)      *(black)      *(black)

*(white)      *(white)      *(white)     *(white)      *(white)      *(white)     *(white)      *(white)
(white)      *(white)      *(white)      *(white)      *(white)    *(black)
(white)      *(white)      *(white)      *(white)    *(black)       *(black)
(white)     *(black)     *(black)     *(black)

In the middle diagram above, the green cell is shared between both a horizontal and a vertical strip of size two. We can thus remove four border strips of both size 1 and 2, and two of size 3. It is easy to see that there is a single border strip of size 4 that can be removed from $\lambda$, three border strips for both sizes 5 and 6, and so on, up to a single border strip of size $\lambda_1 + \ell(\lambda) -1 = 12$. Applying (<ref>) then gives

\begin{align}
\avg{s_\lambda (U) s_\lambda(U^{-1})}_c = & \left( s_{(7,6^2,4,1)} +  s_{(8,6,5,4,1)}+s_{(8,6^2,3,1)} + s_{(8,6^2,4)}  \right)^2+ \notag \\
& + \frac{1}{2} \left( s_{(6^3,4,1)} - s_{(8,5^2,4,1)} + s_{(8,6,4^2,1)} \right)^2 + \frac{1}{3} \left( - s_{(8,5,4,1^2 )} + s_{(8,6^2,1^2)} \right)^2+ \dots
\end{align}

In the above expression, the first, second, and third terms correspond to the removal of border strips of size one, two, and three, respectively. Further, we again write $\left( s_\mu  + s_\rho +\dots \right)^2 $ instead of
$\left( s_\mu(x)  + s_\rho(x) +\dots \right)\left( s_\mu(y)  + s_\rho(y) +\dots \right)$.

Consider now $\avg{s_\lambda (U) s_\nu (U^{-1})}$ with $\lambda$ as above and $\nu = (8^3,4,1)$, the latter of which is shown below. We have here $\lambda \subset \nu$, and we indicate $\nu/\lambda$ in gray.

*(white)      *(white)      *(white)     *(white)      *(white)      *(white)     *(white)      *(white)
(white)      *(white)      *(white)      *(white)      *(white)      *(white)      *(gray)      *(gray)
(white)      *(white)      *(white)      *(white)      *(white)      *(white)      *(gray)      *(gray)
(white)      *(white)      *(white)      *(white)

Consider again all ways to remove border strips from $\nu$, and apply (<ref>). This then leads to the following terms appearing in the expansion of $\avg{s_\lambda(U) s_\nu(U^{-1})}$,

\begin{align}
\label{slsnxmp}
& \left( s_{(7,6^2,4,1)} +  s_{(8,6,5,4,1)}+s_{(8,6^2,3,1)} + s_{(8,6^2,4)}  \right)\left( s_{(8^2,7,4,1)} +s_{(8^3,3,1)} + s_{(8^3,4)} \right) + \notag \\
& +  \frac{1}{2} \left( s_{(6^3,4,1)} - s_{(8,5^2,4,1)} + s_{(8,6,4^2,1)} \right)\left( s_{(8,7^2,4,1)} + s_{(8^2,6,4,1)} + s_{(8^3,2,1)} \right) +  \dots
\end{align}

The diagrams appearing on the left (right) in the top and bottom lines of (<ref>) are found by removing border strips of sizes 1 and 2 from $\lambda$ ($\nu$), respectively. This expansion can easily be continued by considering more or larger border strips. From the relation between removal or addition of border strips as in (<ref>) and fermionic particles hopping on a line, we can interpret (<ref>) in the following manner.

s_λs_ν = ∑_α1/z_α ( (-1)^P s_λ∖{η} (y) {
Distinct ways to take particles in $\lambda$
and move them $\alpha_1, \alpha_2,\dots$ sites to the

left, thereby hopping over $P$ particles

} )×(  y →x λ→ν )

We remind the reader that $\{ \eta \} = \{ \eta_1 ,\eta_2,\dots\}$ are border strips of sizes $\abs{\eta_j}  = \alpha_j$. Note that the particles mentioned above are not required to be distinct.

§.§.§ Power sum expansions of correlation functions

From the fact that, for $\tau = it$, $p_k(x)$ and $p_k(y)$ are related by a phase, it is useful to express LRRW correlation functions as expansions in terms of $p_k(x)$ and $p_k(y)$ in order to reveal these phases. To do so, one may use the relation between Schur and power sum polynomials in equation (<ref>) and apply the Murnaghan-Nakayama rule, as done for the relatively simple example of $\avg{s_\lambda(U) s_\lambda(U^{-1})}$ for $\lambda = (3,2)$ in section <ref>. However, as noted there, even this relatively simple example is already somewhat non-trivial, as it requires the computation of $\chi^\lambda_{~\alpha}$ for all $\alpha$. For these purposes, it is more convenient to employ (<ref>), which provides an expansion of $\avg{s_\lambda(U) s_\nu(U^{-1})}$ in terms of $p_k(x)$ and $p_k(y)$. The prefactors appearing in this expansion depend on the number of ways to remove border strips from $\lambda$ and $\nu$ in such a way that the resulting diagram is the same for both $\lambda$ and $\nu$. Taking again $\nu=\lambda  = (8,6^2,4,1)$, for which the diagrams resulting from removal of border strips were treated in section <ref> above. Applying (<ref>) then gives

\begin{align}
\label{scpe}
\avg{s_\lambda(U)s_\lambda(U^{-1})}_c = &  ~ 4 p_1(x)p_1(y)+ p_2(x)p_2(y) + \frac{2}{9} p_3(x)p_3(y) +\frac{1}{16} p_4(x)p_4(y)+ \notag \\
& + 3 \left( \frac{1}{25} p_5(x) p_5(y) + \frac{1}{36} p_6(x) p_6(y) \right) + \notag \\
& + \frac{1}{2} \left(p^2_1(x)p_2(y)  + p_2(x) p_1^2(y)  \right)+ \notag \\    & +
\frac{1}{6} \left(p_3(x)p_1(y)p_2(y) + p_3(y)p_1(x)p_2(x)\right)  + \dots
\end{align}

In the above expression, the first two lines on the right hand side are given by (<ref>), which is a special case of (<ref>). As we saw previously, the terms $\sim \frac{p_j(x)p_j(y)}{j^2}$ in the first two lines of (<ref>) arise as follows. The denominator is given by the inverse of $z_{(j)}z_{(j)} = j^2$, and the numerator equals the number of distinct ways to remove a border strip of size $j$ from $\lambda =(3,2)$. The third and fourth line give mixed power sums obtained from not contracting a single $p_j(U^\pm)$ and two copies of $p_k(U^\mp)$, $p_m(U^\mp)$. From the diagrams in (<ref>), one can see that there are four ways to remove two border strips of unit size and arrive at a diagram that can alternatively be obtained by removing a single border strip of size 2. This follows from the fact that there are four ways to remove a border strip of size two, which can alternatively be achieved by removing the two cells of such a border strip successively. However, due to the factors $(-1)^{\hg(T)}$ in (<ref>), one of those four contributes with a negative sign, leading to a prefactor of $\frac{2}{z_{(1^2)}z_{(2)}} = \frac{1}{2}$ multiplying $\sim p_1^2 p_2 $ in the third line of (<ref>). There is a similar cancellation occurring from the term proportional to $p_3p_1p_2$. This expansion can then be continued by removing further border strips.
Consider again $\avg{s_\lambda (U) s_\nu(U^{-1})}$ with $\lambda=(8,6^2,4,1) $ and $\nu = (8^3,4,1)$, as at the end of section <ref>. applying (<ref>) leads to

s_λ(U) s_ν(U^-1) = -2/3p_1(x)p_3(x) + 1/4 p_2^2(x) + 1/12p_1(x)^4 + 1/2 p_2^2(x) p_1(x)p_1(y) +…
This can be seen by considering diagram corresponding to $\nu$ in (<ref>), and covering the gray 2 by 2 square with border strips of sizes from 1 to 3. We get no contribution proportional to $p_2(x)p_1(x)^2$ since the two ways to cover a 2 by 2 diagram with a single border strip of size 2 and 2 border strips of size 1 appear with opposite sign. This expansion can be continued by removing more or larger border strips from $\lambda$ and $\nu$, which can be conveniently done by using the relation to fermionic configurations.

The above examples show that (<ref>) can be conveniently applied to $\avg{s_\lambda (U) s_\nu(U^{-1})}$ for larger $\lambda$, $\nu$ as well, for which $\chi^\lambda_{~\alpha}$ and $\chi^\nu_{~\alpha}$ would be very hard to compute. Indeed, as hinted at above, it is particularly useful for the following three reasons.

* Equation (<ref>) provides a controlled expansion of general correlation functions $\avg{s_\lambda(U) s_\nu(U^{-1})}$ in terms of power sums. These power sums can be directly read off from the hamiltonian, see (<ref>) and (<ref>).

* This expansion can be straightforwardly applied, including to correlation functions involving large diagrams $\lambda$, $\nu$. Using the comments below (<ref>), the removal of border strips is related to fermionic particles hopping on a line, which could further simplify the application of this method.

* The power sums $p_k(x)$ and $p_k(y)$ are proportional to $\tau$, so that (<ref>) provides an expansion in terms of powers of $\tau$. Depending on the application and the range of $\tau$ one would like to consider, this expansion can be truncated at any desirable order that provides sufficient precision. For $\tau = it$, this $p_k(x)$ and $p_k(y)$ are related by a complex phase, leading to various simplifications that are hard to reveal otherwise.

From the treatment above, it follows that the expression for $\avg{ s_\lambda (U) s_\nu(U^{-1})} $ in equation (<ref>) has the following particle-hole interpretation.

s_λ(U) s_ν(U^-1) =  ∑_ω, γ   p_ω(y)p_γ(x)/z_ωz_γ (-1)^P

Distinct ways to move particles in $\lambda$ and $\nu$
to the left by $\gamma_1, \gamma_2,\dots$  and $\omega_1, \omega_2,\dots$ sites,

respectively, hopping over $P$ other particles

and ending up in the same configuration.

§.§.§ Correlations for power sums and applications to experimental benchmarking

We give here some suggestions for the benchmarking of experimental setups using correlation functions involving power sum polynoials. In experimental contexts such as trapped ion systems, one could measure the correlation functions corresponding to $\avg{p_n(U^{\pm 1} )}=\avg{\tra U^{\pm n}} = p_n $ experimentally. Remember that $p_n$ can be read off from the hamiltonian as in (<ref>) or (<ref>) due to their direct proportionality to $a_{\pm n}$, and is given by a superposition of configurations corresponding to hook-shaped diagrams as in (<ref>). Therefore, one could measure $\avg{p_n(U^{\pm 1} )}$ and compare them with the intended values of $a_{\pm n}$ to benchmark experimental setups. Further, one can use equation (<ref>),
U^n^2 = n + p_n(x)p_n(y)  ,
and consider $\tra U^n$ as a superposition of states (when properly normalized), as before. Then, the correlation function corresponding $\avg{\abs{\tra U^n}^2}$ is proportional to $F_{\emptyset;\emptyset}$ with proportionality given by $n+p_n(x)p_n(y) $, up to normalization. When we have certain hopping parameters $a_n \neq 0$, the only way to get independence from time is to have no dependence on the power sums $p_n \sim \tau n a_n$, as they contain a factor of $\tau$. The $\tau$-independence of the connected part of the correlation function corresponding to (<ref>) therefore sets it apart from other correlation functions, and might offer an effective way to benchmark experimental setups. Remember that the $p_\lambda$ form a basis for all symmetric polynomials.  In case $\ell(\lambda) \geq 2$ (or $\ell(\nu) \geq 2$) so that $p_\lambda = p_{\lambda_1} p_{\lambda_2} \dots$, Wick's theorem tells us that $\avg{p_\lambda p_\mu}_c$ will contain terms where not all $p_{\lambda_j}$ and $p_{\mu_k}$ are contracted. These give contributions containing $p_k(x) \sim \tau$ (and/or $p_j( y)\sim \tau $). Therefore, $\frac{1}{\sqrt{n}}\avg{\abs{\tra U^n}^2}_c$  for any integer $n$ is the only non-zero connected correlation function that does not depend on $\tau$. Lastly, one could use $\avg{\tra U^n \tra U^{-k}}_c = n\delta_{n,k}$ (after proper normalization) to see if the system is truly translationally invariant, as its derivation is predicated on the assumption of translational invariance.

§ CONCLUSIONS

This work focused on weighted $U(N)$ integrals over symmetric polynomials and their relation with  correlation functions of LRRW models. The weighting is given by a weight function $f(z)$, which is required to satisfy Szegő's strong limit theorem, corresponding to hopping parameters $a_k$ whose modulus decays faster than $k^{-1}$ for $k\to \infty$. Writing the weight function in terms of generating functions of elementary or complete homogeneous symmetric polynomials as $f(z) = E(x;z)E(y;z^{-1})$ or $f(z) = H(x;z)H(y;z^{-1})$, respectively, general correlation functions can be expressed in terms of Schur polynomials with variables $x$ and $y$.
In particular, the power sum polynomials are related to the hopping parameters as $ p_k(x) =\pm \tau k a_k $ and $ p_k(y) = \pm \tau k a_{-k}$ for $k\geq 1$. By using our earlier result [40] that
$\avg{\tra U^n \tra U^{-k}}_c = n \delta_{n,k}$ and applying Wick's theorem, we derive various identities. In particular, we compute $\avg{p_\mu(U) p_\rho(U^{-1})}$, generalizing a
result due to Diaconis and Shahshahani, who computed said object in the CUE, corresponding to $f=1$. Further, we derive two expressions for correlation functions $\avg{s_\lambda(U) s_\nu (U^{-1})}$ for general $\lambda$ and $\nu$,
both of which are obtained by removing border strips from $\lambda$ and $\nu$. In particular, the first of these expressions is an expansion in terms of diagrams related to $\lambda$ and $\nu$ by the removal of border strips. The second is an expansion in terms of $p_\gamma(x)$ and $p_\omega(y)$, where the expansion coefficients are determined by the number of ways to remove border strips from $\lambda$ and $\nu$ and arrive at the same diagram, up to a sign determined by the height of the (generally disjoint) border strip tableau that is removed.

We applied these results to correlation functions of LRRW models, where we consider configurations starting with an infinite number of particles (down spins) and ending
with and infinite number of holes (up spins). Before applying our own results, we considered various standard relations in the theory of symmetric
polynomials, such as the Pieri formula, which lead to a simple ways to compute and interpret various correlation functions. The most striking of these results is the fact that the involution between elementary and complete
homogeneous symmetric polynomials, corresponding to the transposition of diagrams, leads to a duality between models related by switching the sign of the even hopping parameters, $a_k \to (-1)^{k+1}a_k$. In particular, we found that the correlation functions of these two models equal each other when we perform a particle-hole and parity transformation, which acts non-trivially on the
finite interval in between the infinite strings of particles and holes. We therefore refer to this duality as quasi-local particle-hole duality. Further, we noted that the Jacobi-Trudi
identity allows one to calculate any $\avg{s_\lambda(U) s_\nu (U^{-1})}$ directly. Although the number of terms appearing in this expansion grows quickly with the size of $\lambda$, $\nu$,
it is simple to implement as it only involves the calculation of a determinant of a matrix with known entries.

We then considered the basis of power sum polynomials $p_\gamma(x)$ and $p_\omega(y)$ and the closely related border strips, which is particularly useful in the application to LRRW models due to their aforementioned relation to hopping parameters $a_k$ and generalized time $\tau$. As mentioned above, we derived expressions for LRRW correlation functions involving addition or removal of a border strip $T$ of size $n$ and height $\hg(T)$, leading to a sign $(-1)^{\hg(T)}$. We found that this corresponds to moving a fermionic particle a distance $n$ to the right or left, respectively, thereby hopping
over $\hg (T)$ other particles. This provides an interpretation of various expansions of correlation functions in terms of particles hopping with fermionic exchange statistics,
even though the LRRW model consists of hard-core bosons rather than fermions. For example, the procedure of the second expansion of $\avg{s_\lambda  s_\nu}$, in (<ref>), which involves the removal of border strips, can be interpreted as replacing the (hard-core boson) particles in $\lambda$ and $\nu$ by fermions and considering all distinct ways to move them to the left such that the final configurations are identical. Due to the aforementioned relation between power sums and hopping parameters, this expression provides an expansion in powers of $\tau$ where the numbers appearing in the expansion coefficients can be read off from the hamiltonian.

The mathematical results derived in section <ref> may be generally applied to the case where the strong Szegő limit theorem applies. The expansion of general correlation functions in terms of power sum polynomials is particularly useful in such applications where these can be accessed directly, such as for LRRW models, where they are provided as input. In section <ref>, we saw that the addition or removal of border strips with sign $(-1)^{\hg}$ can be interpreted as fermionic particles hopping to the right or left, respectively. One may apply this `physical' interpretation of this process to the power sum expansion of $\avg{s_\lambda s_\nu}$, by considering all ways to move fermionic particles in $\lambda $ and $\nu$ to the left to arrive at the same configuration. Further, the same fermionic interpretation may be applied to the Murnaghan-Nakayama rule, which could provide a more convenient practical method for the calculation of general $\chi^{\lambda/\mu}_{~\alpha}$. Inverting this reasoning, we applied properties of these character to long-range fermionic models. First of all, the fact that $\chi^{\lambda/\mu}_{~\alpha}$ does not depend on the order of the entries of $\alpha$, means that taking a fermionic configuration and considering all ways to move particles by $\alpha_1,\alpha_2,\dots$ sites to the right gives the same outcome regardless of the ordering of the step sizes $\alpha_j$. Further, the fact that the symmetric group character are cancellation-free for $\alpha = (n^k)$ lead to the conclusion that all ways to go from any configuration to any other configuration by taking only $n$ sites involves either an even or an odd number of particles being hopped over, depending only on $n$ and the choice of initial and final configurations.

As mentioned in the introduction, LRRW models such as we consider here have seen increasing activity in both experimental and theoretical contexts due to the experimental accessibility of such systems and the surprising phenomena they exhibit. We believe our work can be applied along some of these lines of research, including the consideration of localization by addition of (diagonal) disorder [29], [30]. Adding such disorder will generally break translational invariance, and the expressions in this work will no longer apply. One may also consider translationally invariant disorder such as random hopping parameters, as in [48], [49], in which case the results derived here would still apply as long as the hopping parameters satisfy the aforementioned asymptotic fall-off conditions.
All results in section <ref> can, in principle, be checked experimentally, where the expressions in this work would be expected to hold with reasonable accuracy up to the time that finite size effects start to occur. Besides checking our results in experimental setups, correlation functions involving power sum polynomials may be used for experimental benchmarking.

§ ACKNOWLEDGEMENTS

We would like to thank Ivan Khaymovich, Vladimir Kravtsov, Jiří Minář, and Arghavan Safavi-Naini for useful discussions and suggestions.
This work is part of the DeltaITP
consortium, a program of the Netherlands Organization for Scientific Research (NWO) funded
by the Dutch Ministry of Education, Culture and Science (OCW).

§ SYMMETRIC POLYNOMIALS AND YOUNG DIAGRAMS

We review here some aspects of symmetric polynomials and Young diagrams, starting with some simple examples. Take a set of variables $x =( x_1,x_2,\dots )$. The elementary symmetric polynomials are then defined as

e_k (x) = ∑_i_1<…<i_k x_i_1…x_i_k .
Some examples include

\begin{align}
e_0& = 1 ~, \notag \\
e_1(x_1)& = x_1 ~, \notag\\
e_1(x_1, x_2 )& = x_1 +x_2~, \notag\\
e_2(x_1, x_2 )& = x_1 x_2~.
\end{align}

Closely related are the complete homogeneous symmetric polynomials, defined as

h_k (x) = ∑_i_1≤…≤i_k x_i_1…x_i_k ,
which contain all monomials of degree $k$. Note the difference in the summation bounds between (<ref>) and (<ref>). Some examples of $h_k$ include

\begin{align}
h_0 & = 1 ~,  \notag\\
h_1(x_1)& = x_1 ~, \notag\\
h_1(x_1, x_2 )& = x_1 +x_2~, \notag\\
h_2(x_1, x_2 )& = x_1 x_2 +  x_1^2+x_2^2~.
\end{align}

Another type of symmetric polynomial is the power sum symmetric polynomial, also simply referred to as power sum polynomial,

p_k(x)= ∑_j x_j^k =x_1^k+x_2^k+…
The generating functions of $e_k$ and $h_k$ and their relation with power sums are as follows [see e.g [50]],

\begin{align}
\label{genfuncsymp}
E(x;z) &= \sum_{k=0}^\infty e_k(x) z^k = \prod_{k=1}^\infty (1+x_k z) = \exp \left[ \sum_{k=1}^\infty \frac{(-1)^{k+1}}{k} p_k(x) z^k   \right]  ~,\notag \\
H(x;z) & = \sum_{k=0}^\infty h_k(x) z^k = \prod_{k=1}^\infty \frac{1}{1-x_k z} = \exp \left[ \sum_{k=1}^\infty \frac{1}{k} p_k(x) z^k   \right]  ~.
\end{align}

From the above expressions, it is clear that $H(x;z) E(x;-z) = 1$. Checking every order of $z$ then gives, for all $n\geq 1$ and any choice of $x$ [e.g. (2.6') from [50]],

∑_r=0^n(-1)^r h_n-r (x) e_r (x) =0 .

Partitions play an important role in the study of symmetric polynomials. A partition of $n\in \mathds{Z}^+$ is a sequence of non-negative integers $\lambda = ( \lambda_1 , \lambda_2,\dots,\lambda_{\ell(\lambda)} )$, which we will order as $\lambda_1 \geq \lambda_2 \geq \dots $, satisfying $\sum_j \lambda_j=n$. The size (or weight) of a partition is given by the sum of its terms $\abs{\lambda}= \sum_j\lambda_j$ and its length $\ell(\lambda) $ is the largest value of $j$ such that $\lambda_j \neq 0$. Closely related to partitions of $n$ are compositions of $n$, consisting also of a sequence of non-negative integers which sum to $n$, but where a different ordering in these integers defines a different composition. A weak composition of $n$ is a composition which may include zeroes as its entries, that is, a set of non-negative integers which sum up to $n$. A partition of $n$ corresponds to a Young (or Ferrers) diagram containing $n$ cells, or `boxes'. We will use these terms interchangeably. As an example, the diagram corresponding to a partition of 12 given by $\lambda = (6,4,2,1)$ is given below, where $\lambda_j$ equals the number of cells in the $j^\text{th}$ row.

We will denote a diagram consisting of $b$ rows of $a$ cells by $(a^b)$.

For a partition $\lambda$, we will write

e_λ= ∏_j≥1 e_λ_j ,    h_λ= ∏_j≥1 h_λ_j ,    p_λ= ∏_j≥1 p_λ_j .
Further, we write

z_λ=∏_j≥1j^m_jm_j! ,    m_j(λ)=Card {k:λ_k=j} ,
i.e.  $m_j(\lambda)$ is the number of rows in $\lambda$ of length $j$. We also write $\varepsilon_\lambda=(-1)^{\abs{\lambda} -\ell(\lambda)}$. Newton's identities then read

\begin{align}
h_n & = \sum_{\abs{\lambda}=n} z_\lambda^{-1} p_\lambda~, \notag \\
e_n & = \sum_{\abs{\lambda}=n} \varepsilon_\lambda z_\lambda^{-1} p_\lambda~.
\end{align}