# Self-Repelling Elastic Manifolds with Low Dimensional Range

Carl Mueller  Carl Mueller
Department of Mathematics
University of Rochester
Rochester, NY 14627<EMAIL_ADDRESS>and  Eyal Neuman  Eyal
Neuman
Department of Mathematics
Imperial College
London, UK<EMAIL_ADDRESS>
###### Abstract.

We consider self-repelling elastic manifolds with a domain
$[-N,N]^{d}\cap\mathbb{Z}^{d}$, that take values in $\mathbb{R}^{D}$. Our main
result states that when the dimension of the domain is $d=2$ and the dimension
of the range is $D=1$, the effective radius $R_{N}$ of the manifold is
approximately $N^{4/3}$. This verifies the conjecture of Kantor, Kardar and
Nelson [7]. Our results for the case where $d\geq 3$ and $D<d$ give a lower
bound on $R_{N}$ of order $N^{\frac{1}{D}\left(d-\frac{2(d-D)}{D+2}\right)}$
and an upper bound proportional to $N^{\frac{d}{2}+\frac{d-D}{D+2}}$. These
results imply that self-repelling elastic manifolds with a low dimensional
range undergo a significantly stronger stretching than in the case where
$d=D$, which was studied in [10].

###### Key words and phrases:

Gaussian free field, self-avoiding, elastic manifold

###### 2020 Mathematics Subject Classification:

Primary, 60G60; Secondary, 60G15.

CM was partially supported by Simons Foundation grant 513424.

##  1\. Introduction

Self-repelling elastic manifolds were first introduced by Kantor, Kardar and
Nelson in [5] as generalizations of polymer models to higher dimensions, in
order to capture the behaviour of sheets of covalently bonded atoms and of
polymerized lipid surfaces, among others. See [5, 7, 6, 11] and references
therein for additional details. In the mathematical literature this model was
first studied in [10] as a random surface with free boundary conditions is
modeled by $\mathbb{R}^{D}$-valued discrete Gaussian free field (DGFF) over
$[-N,N]^{d}\cap\mathbb{Z}^{d}$, with Neumann boundary conditions. A
penalization term for self-intersections, which reflects the fact that
different parts of the manifold cannot occupy the same position, is then added
to the Hamiltonian of the DGFF. If the domain of DGFF is one dimensional, then
we recover the well-known model of a random polymer. A typical object of study
is the end-to-end distance of such a polymer, or the closely related concept
of effective radius. There is a vast literature on such problems, see
Bauerschmidt, Duminil-Copin, Goodman, and Slade [2] and the included
citations.

Bounds on the effective radius $R_{N}$ of the self-repelling manifold were
derived in [10] for the case where $d=D$. It was proved that for the two
dimensional case, that is $d=D=2$, $R_{N}$ is proportional to $N$ in the upper
and lower bounds, up to a logarithmic correction. The bounds on $R_{N}$ in
higher dimensions are not as sharp, with a lower bound proportional to $N$,
but with an upper bound of order $N^{d/2}$. The results of [10] proved however
that self-repelling elastic manifolds experience a substantial stretching in
any dimension.

We should also mention the related paper [9] on the effective radius for
moving polymers. Most methods available in the polymer literature fail in
these more general settings, but we did take inspiration from work of
Bolthausen [4].

In this paper, we deal with the case where $D<d$. We first prove that when the
dimension of the domain is $d=2$ and the dimension of the range is $D=1$, the
effective radius $R_{N}$ of the manifold is approximately $N^{4/3}$. Our
results for the case $d\geq 3$ and $D<d$ give a lower bound on $R_{N}$ of
order $N^{\frac{1}{D}\left(d-\frac{2(d-D)}{D+2}\right)}$ and an upper bound
proportional to $N^{\frac{d}{2}+\frac{d-D}{D+2}}$. These results imply that
self-repelling elastic manifolds with a low dimensional range undergo a
significantly stronger stretching than in the case $d=D$, which was studied in
[10].

The remaining case, $D>d$, looks to be much harder. For example, consider the
case where the domain of the self-repelling DGFF is $\\{0,\dots,N\\}$ and it
takes values in $\mathbb{R}^{D}$. For $D=2,3,4$ the behavior of the effective
radius of the self-repelling polymer as $N\to\infty$ is still unknown,
although we have good information for $D=1$ and for $D>4$. See page 400 of
Bauerschmidt, Duminil-Copin, Goodman, and Slade [2] and also Bauerschmidt,
Slade, and Tomberg, and Wallace [3]. If $D$ is large enough, then for self-
avoiding walks, the lace expansion can be used. For DGFF however, there
appears to be no analogue of the lace expansion.

##  2\. Model Setup and Main Results

###  2.1. Setup

We briefly review some of the definitions and notation from Section 1 of [10]
which are essential for our context. In the following, ordinary letters such
as $x,u$ take values in $\mathbb{R}$ or $\mathbb{Z}$, while boldface letters
such as $\mathbf{x},\mathbf{u}$ take values in $\mathbb{R}^{d}$ for $d\geq 2$.

Fix $d\geq 2,N\geq 1$ and define our parameter set as follows:

$S^{d}_{N}:=[-N,N]^{d}\cap\mathbb{Z}^{d}.$

Note that

$S^{1}_{N}:=\\{-N,\dots,N\\}.$

Thus $S^{d}_{N}$ is a cube in $\mathbb{Z}^{d}$ centered at the origin.

We denote by $\Delta=\Delta_{N,d,D}$ the discrete Neumann Laplacian on
$S^{d}_{N}$ (see Section 1 of [10] for the precise definition). In the case
$D=1$, since $\Delta$ is a self-adjoint operator on a finite-dimensional
space, there exists a finite index set $\mathbb{I}=\mathbb{I}_{N,d}$ to be
defined later, and an orthonormal basis of eigenfunctions
$(\varphi_{\mathbf{k}})_{\mathbf{k}\in\mathbb{I}}$ with corresponding
eigenvalues $(\lambda_{\mathbf{k}})_{\mathbf{k}\in\mathbb{I}}$. We can assume
without loss of generality that there is a distinguished index $\mathbf{0}$
such that $\varphi_{\mathbf{0}}$ is constant and that
$\lambda_{\mathbf{0}}=0$.

Throughout, we fix a parameter $\beta>0$, which has a physical interpretation
as the inverse temperature. Let
$(X^{(i)}_{\mathbf{k}})_{\mathbf{k}\in\mathbb{I}\setminus\\{\mathbf{0}\\},i=1,\dots,D}$
be a collection of i.i.d. random variables defined on a probability space
$(\Omega,\mathcal{F},\mathbb{P})$ such that

$X^{(i)}_{\mathbf{k}}\sim N(0,(2\beta\lambda_{\mathbf{k}})^{-1}).$

For each $i=1,\dots,D$ define

(2.1)
$u^{(i)}=\sum_{\mathbf{k}\in\mathbb{I}\setminus\mathbf{0}}X^{(i)}_{\mathbf{k}}\varphi_{\mathbf{k}},\\\
$

and define DGFF as

(2.2) $\mathbf{u}=(u^{(1)},\dots,u^{(D)}).$

As explained in [10], this corresponds to the Gibbs measure with Hamiltonian
$H(\mathbf{u})=\sum_{\mathbf{x}\sim\mathbf{y}}|\mathbf{u}(\mathbf{x})-\mathbf{u}(\mathbf{y})|^{2}$,
where $\mathbf{x}\sim\mathbf{y}$ means that $\mathbf{x},\mathbf{y}$ should be
nearest neighbors on $S_{N}^{d}$. In other words, the energy $H(\mathbf{u})$
depends on the stretching $|\mathbf{u}(\mathbf{x})-\mathbf{u}(\mathbf{y})|$.

We recall the definition the local time of the field $\mathbf{u}$ at level
$\mathbf{z}\in\mathbb{R}^{D}$ as

(2.3) $\begin{split}\ell_{N}(\mathbf{z})&=\\#\\{\mathbf{x}\in
S^{d}_{N}:\mathbf{u}(\mathbf{x})\in[\mathbf{z}-\mathbf{1/2},\mathbf{z}+\mathbf{1/2}]\\}\\\
&=\sum_{\mathbf{x}\in
S_{N}^{d}}\mathbf{1}_{\\{\mathbf{u}(\mathbf{x})\in[\mathbf{z}-\mathbf{1/2},\mathbf{z}+\mathbf{1/2}]\\}},\end{split}$

where $\mathbf{\frac{1}{2}}=(1/2,\dots,1/2)\in\mathbb{R}^{D}$, and
$[\mathbf{x},\mathbf{y}]=\prod_{i}[x_{i},y_{i}]$ for
$\mathbf{x},\mathbf{y}\in\mathbb{R}^{D}$.

Now we define a weakly self-avoiding Gaussian free field. Throughout, we fix a
parameter $\gamma>0$. If $P_{N,d,D,\beta}$ denotes the original probability
measure of $(\mathbf{u}(\mathbf{x}))_{\mathbf{x}\in S_{N}^{d}}$, we define the
probability $Q_{N,d,D,\beta,\gamma}$ as follows. For ease of notation, we
write $E$ for the expectation with respect to $P_{N,d,D,\beta}$. Let

(2.4) $\displaystyle\mathcal{E}_{N,d,D,\gamma}$
$\displaystyle=\exp\left(-\gamma\int_{\mathbb{R}^{D}}\ell_{N}(\mathbf{y})^{2}d\mathbf{y}\right),$
$\displaystyle Z_{N,d,D,\beta,\gamma}$
$\displaystyle=E[\mathcal{E}_{N,d,D,\gamma}]=E^{P_{N,d,D,\beta}}[\mathcal{E}_{N,d,D,\gamma}].$

Then we define for any set $A\in\mathcal{F}$,

(2.5)
$Q_{N,d,D,\beta,\gamma}(A)=\frac{1}{Z_{N,d,D,\beta,\gamma}}E\big{[}\mathcal{E}_{N,d,D,\gamma}\mathbf{1}_{A}\big{]}.$

For ease of notation, we will often drop the subscripts except for $N$ and
write

$P_{N}=P_{N,d,D,\beta},\quad
Q_{N}=Q_{N,d,D,\beta,\gamma},\quad\mathcal{E}_{N}=\mathcal{E}_{N,d,D,\gamma}.$

For $Z_{N,d,D,\beta,\gamma}$ in (2.4) we often write,

$Z_{N,d,D}=Z_{N,d,D,\beta,\gamma}.$

Finally, we define the effective radius of the field $\mathbf{u}$ as

$R_{N,d,D}=\max_{\mathbf{w},\mathbf{z}\in
S^{d}_{N}}\|\mathbf{u}(\mathbf{z})-\mathbf{u}(\mathbf{w})\|,$

where $\|\cdot\|$ denotes the Euclidian norm.

###  2.2. Statement of the main result

Note that in our main theorem below, we assume that $D\leq d$.

######  Theorem 2.1.

Let $\mathbf{u}$ be the weakly self-avoiding DGFF on $S_{N}^{d}$ taking values
in $\mathbb{R}^{D}$. There are constants $\varepsilon_{0},K_{0}>0$ not
depending on $\beta,\gamma,N$ such that

  * (i)

For $d=2$ and $D=1$,

$\displaystyle\lim_{N\to\infty}Q_{N}\Big{[}\varepsilon_{0}\frac{\gamma}{\beta+\gamma}N^{4/3}(\log
N)$ ${}^{-2/3}\leq R_{N,d,D}$ $\displaystyle\leq
K_{0}\Big{(}\frac{\beta+\gamma}{\beta}\Big{)}^{1/2}N^{4/3}(\log
N)^{4/3}\Big{]}=1.$

  * (ii)

For $d\geq 3$ and $1\leq D\leq d$,

$\displaystyle\lim_{N\to\infty}Q_{N}\Big{[}\varepsilon_{0}\Big{(}\frac{\gamma}{\beta+\gamma}\Big{)}^{1/D}N^{\frac{1}{D}\left(d-\frac{2(d-D)}{D+2}\right)}$
$\displaystyle\leq R_{N,d,D}$ $\displaystyle\leq
K_{0}\Big{(}\frac{\beta+\gamma}{\beta}\Big{)}^{1/2}N^{\frac{d}{2}+\frac{d-D}{D+2}}\Big{]}=1.$

######  Remark 2.2.

We compare the result of Theorem 2.1(i) to the result of Theorem 1.1(i) in
[10], where it was proved that $R_{N,2,2}\approx N$. Note that by reducing the
dimension of the range by 1, the manifold stretches, with a substantially
larger radius of $R_{N,2,1}\approx N^{4/3}$. By comparing Theorem 2.1(ii) and
Theorem 1.1(ii) in [10] we observe a similar phenomenon for $d\geq 3$, as the
lower bound on the radius increases from $R_{N,d,d}\gtrsim N$ to
$R_{N,d,D}\gtrsim N^{\frac{1}{D}\left(d-\frac{2(d-D)}{D+2}\right)}$ for $D<d$.
This additional stretching for manifolds of lower dimensional range can be
predicted by considering the local time expression in (2.3). Indeed for a
fixed value of $d$ and when the range has dimension $D<d$,
$\ell_{N}(\mathbf{z})$ counts the same number of vertices $(2N+1)^{d}$, but if
the radius $R_{N}$ remains the same, there is less space to fit these
vertices. Hence we would expect $\mathcal{E}_{N}$ to be larger. This in turn
gives the repelling term in (2.4) a stronger influence on the configuration of
the manifold (see e.g. (7.4) in Section 7).

######  Remark 2.3.

Theorem 2.1 verifies the conjecture by Kantor, Kardar and Nelson in [7] for
the case where $d=2$ and $D=1$. Although in the model that was presented in
[7] the DGFF is defined on the triangular lattice, the heuristics that yields
their result is based on Flory’s argument which also applies for the
rectangular lattice.

###  2.3. Outline of the proof

We describe the outline for the case $d=2$, $D=1$, as the proof for $d\geq 3$,
$D<d$ follows similar lines. Define the following events.

(2.6) $\displaystyle A^{(1)}_{N,d,D}$
$\displaystyle=\left\\{R_{N,d,D}>K_{0}\Big{(}\frac{\beta+\gamma}{\beta}\Big{)}^{1/2}N^{4/3}(\log
N)^{4/3}\right\\},$ $\displaystyle A^{(2)}_{N,d,D}$
$\displaystyle=\left\\{R_{N,d,D}<\varepsilon_{0}\frac{\gamma}{\beta+\gamma}N^{4/3}(\log
N)^{-2/3}\right\\}.$

It suffices to show that for $i=1,2$ we have

$\lim_{N\to\infty}Q_{N}\big{(}A^{(i)}_{N,d,D}\big{)}=0.$

From (2.5) we see that it is enough to find:

  1. (1)

a lower bound on $Z_{N,d,D}$, derived in Section 3,

  2. (2)

and an upper bound on
$E\big{[}\mathcal{E}_{N,d,D}\mathbf{1}_{A^{(i)}_{N,d,D}}\big{]}$ for $i=1,2$,
obtained in Sections 6 and 7, respectively.

Finally, the upper bounds divided by the lower bound should vanish as
$N\to\infty$.

##  3\.  Lower Bound on the Partition Function

In this section we derive the following lower bound on $Z_{N,d,D}$.

######  Proposition 3.1.

Let $\beta>0$. Then there exists a constant $C>0$ not depending on $N$,
$\beta$ and $\gamma$ such that

  * (i)

for $d=2$ and $D=1$,

$\log Z_{N,d,D}\geq-C(\beta+\gamma)N^{8/3}(\log N)^{2/3}.$

  * (ii)

for $d\geq 3$ and $D\leq d$,

$\log Z_{N,d,D}\geq-C(\beta+\gamma)N^{d+\frac{2(d-D)}{d-D+2}}.$

In order to prove Proposition 3.1 we will introduce some additional
definitions and auxiliary lemmas.

###  3.1. The orthonormal function basis

We first recall the orthonormal basis $\\{\varphi_{\mathbf{k}}\\}$ in (2.1) of
eigenfunctions of $\Delta_{N,d,1}$ on $[-N,N]^{d}\cap\mathbb{Z}^{d}$ taking
values in $\mathbb{R}$. Each basis function $\varphi_{\mathbf{k}}$ can be
represented as a product of $d$ functions
$\phi_{j}:\mathbb{R}^{d}\rightarrow\mathbb{R}$, as follows

(3.1)
$\varphi_{\mathbf{k}}(\mathbf{x})=\phi_{k_{1}}(x_{1})\dots\phi_{k_{d}}(x_{d}),$

where $\mathbf{x}=(x_{1},\dots,x_{d})$ and $\mathbf{k}=(k_{1},\dots,k_{d})$,
$-N\leq k_{i}\leq N$, and $1\leq i\leq d$. Here $\\{\phi_{j}\\}_{j=-N}^{N}$ is
an orthonormal basis of eigenfunctions of $\Delta_{N,1,1}$, the Laplacian with
Neumann boundary conditions on $S_{N}^{1}=\\{-N,\dots,N\\}$. Note that if
$\lambda_{\mathbf{k}}$ is the eigenvalue of $\varphi_{\mathbf{k}}$ and
$\lambda_{k}$ is the eigenvalue corresponding to $\phi_{k}$, then satisfies

(3.2) $\lambda_{\mathbf{k}}=\sum_{i=1}^{d}\lambda_{k_{i}}.$

We can explicitly compute these eigenfunctions and eigenvalues (see Section
3.1 of [10]).

Our basis comprises all such combinations as in (3.1), excluding the constant
eigenfunction

$\varphi_{(0,\dots,0)}(\mathbf{x})=\phi_{0}(x_{1})\dots\phi_{0}(x_{d}).$

We denote by $N(d)$ the number of function in our basis,

(3.3) $N(d)=|S_{N}^{d}|-1=(2N+1)^{d}-1.$

###  3.2. Incorporating drift

Next we incorporate a linear drift into each of the components $u^{(i)}$ of
$\mathbf{u}$, calling the resulting component $u^{(i)}_{a}$. This drift should
stretch out the values of $\mathbf{u}(\cdot)$, so that $\mathbf{u}$ looks more
like what we believe it would be under $Q_{N}$. Since our goal is to use
Jensen’s inequality, this drift should make the inequality more exact. Let

$u^{(i)}_{a}(\mathbf{x})=\sum_{k=1}^{N(d)}X^{(i)}_{\mathbf{k}}\varphi_{\mathbf{k}}(\mathbf{x})+ax_{i},\quad
i=1,\dots,D,$

where $a>0$ is a constant to be determined later.

Using (2.1) and (3.1), we get

(3.4) $u_{a}^{(i)}(\mathbf{x})=\sum_{(k_{1},\dots,k_{d})\in
S_{N}^{d}\setminus\\{\mathbf{0}\\}}X^{(i)}_{k_{1},\dots,k_{d}}\phi_{k_{1}}(x_{1})\dots.\phi_{k_{d}}(x_{d})+ax_{i},$

where $\mathbf{0}=(0,\dots,0)\in\mathbb{Z}^{d}$.

Regarding $x_{i}$ as a function on $S_{N}^{1}=\\{-N,\dots,N\\}$ and expanding
it in terms of our eigenfunctions, we see that there are coefficients
$\alpha_{j}^{(i)}$ such that

(3.5) $x_{i}=(\phi_{0})^{d-1}\sum_{j\in
S_{N}^{1}\setminus\\{0\\}}\phi_{j}(x_{i})\alpha_{j}^{(i)},$

where we have included $(\phi_{0})^{(1-d)}$ for convenience in later
calculations. Recall that

$\phi_{0}=\phi_{0}(x)=(2N+1)^{-1/2}.$

In (3.5), we do not include $j=0$ because $x_{i}$ is orthogonal to the
constant function $\phi_{0}$.

Next we find $\alpha_{j}^{(i)}$ in (3.5). Since $\alpha_{j}^{(i)}$ are used to
expand the function $f(x)=x$ for each coordinate $i$, we can omit the
superscript $i$ and write just $\alpha_{j}$ in what follows. Since
$\\{\phi_{j}\\}_{j=-N}^{N}$ forms an orthonormal basis, we get

(3.6) $\alpha_{j}=\phi_{0}^{(1-d)}\sum_{n=-N}^{N}n\phi_{j}(n),\quad
j\not=0,\quad\textrm{and }\quad\alpha_{0}=0.$

From (3.4) and (3.5) we have

$\displaystyle u_{a}^{(i)}(\mathbf{x})$
$\displaystyle=\sum_{(k_{1},\dots,k_{d})\in
S_{N}^{d}\setminus\\{\textbf{0}\\}}X^{(i)}_{k_{1},\dots,k_{d}}\phi_{k_{1}}(x_{1})\dots.\phi_{k_{d}}(x_{d})$
$\displaystyle\quad+a\phi_{0}^{d-1}\sum_{j\in
S_{N}^{1}\setminus\\{0\\}}\phi_{j}(x_{i})\alpha_{j}.$

We can represent $u_{a}^{(i)}$ as follows:

(3.7) $\displaystyle u_{a}^{(i)}(x)$
$\displaystyle=\sum_{(k_{1},\dots,k_{d})\in
S_{N}^{d}\setminus\\{j\mathbf{e}_{i}:\,j\in
S_{N}^{1}\\}}X^{(i)}_{k_{1},\dots,k_{d}}\phi_{k_{1}}(x_{1})\cdots\phi_{k_{d}}(x_{d})$
$\displaystyle\quad+\phi_{0}^{d-1}\sum_{j\in
S_{N}^{1}\setminus\\{0\\}}(X_{j\mathbf{e}_{i}}^{(i)}+a\alpha_{j}^{(i)})\phi_{j}(x_{i}),$

where $\\{\mathbf{e}_{i}\\}_{i=1}^{d}$ is the standard basis of
$\mathbb{R}^{d}$.

For $i=1,\dots,D$ let $\mathbf{x}^{(i)}\in\mathbf{V}$ and define

$F(\mathbb{x}^{(1)},\dots,\mathbb{x}^{(D)})=\sum_{i=1}^{D}\sum_{(k_{1},\dots,k_{d})\in
S_{N}^{d}\setminus\\{\mathbf{0}\\}}\frac{(x^{(i)}_{k_{1},\dots,k_{d}})^{2}}{2(2\beta\lambda_{k_{1},\dots,k_{d}})^{-1}}.$

We rewrite $Z_{N,d,D}$ in (2.4) as follows. We should emphasize that the local
time $\ell_{N}$ is random and hence a function of the random variables
$(X^{(i)}_{\mathbf{k}})$, so we can write

$\ell_{N}(\mathbf{y})=\ell_{N}\left((X_{\mathbf{k}}^{(i)}),\mathbf{y}\right),$

where for readability we have omitted the specification that $i=1,\dots,D$ and
$\mathbf{k}\in S_{N}^{d}\setminus\\{\mathbf{0}\\}$. Then we have

(3.8) $\displaystyle Z_{N,d,D}$ $\displaystyle=\int_{\mathbb{R}^{D\cdot
N(d)}}\exp\left(-F(\mathbb{x}^{(1)},\dots,\mathbb{x}^{(D)})-\gamma\int_{\mathbb{R}^{D}}\ell_{N}^{2}\big{(}(\mathbf{x}^{(i)}),\mathbf{y}\big{)}d\mathbf{y}\right)$
$\displaystyle\hskip
56.9055pt\times\prod_{i=1}^{D}\prod_{(k_{1},\dots,k_{d})\in
S_{N}^{d}\setminus\\{\textbf{0}\\}}dx_{k_{1},\dots,k_{d}}^{(i)}.$

In order to find the Radon-Nikodym derivative corresponding to the drift in
(3.7) we note that,

(3.9) $\displaystyle\sum_{i=1}^{D}\sum_{(k_{1},\dots,k_{d})\in
S_{N}^{d}\setminus\\{\textbf{0}\\}}\frac{(x^{(i)}_{k_{1},\dots,k_{d}})^{2}}{2(2\beta\lambda_{k_{1},\dots,k_{d}})^{-1}}$
$\displaystyle=\sum_{i=1}^{D}\bigg{(}\sum_{(k_{1},\dots,k_{d})\in
S_{N}^{d}\setminus\\{j\textbf{e}_{i}:\,j\in
S_{N}^{1}\\}}\frac{(x^{(i)}_{k_{1},\dots,k_{d}})^{2}}{2(2\beta\lambda_{k_{1},\dots,k_{d}})^{-1}}$
$\displaystyle\qquad+\sum_{j\in
S_{N}^{1}\setminus\\{0\\}}\frac{(x^{(i)}_{j\mathbf{e}_{i}}+a\alpha_{j})^{2}}{2(2\beta\lambda_{j\mathbf{e}_{i}})^{-1}}-\sum_{j\in
S_{N}^{1}\setminus\\{0\\}}\frac{2a\alpha_{j}x^{(i)}_{j\mathbf{e}_{i}}+(a\alpha_{j})^{2}}{2(2\beta\lambda_{j\mathbf{e}_{i}})^{-1}}\bigg{)}.$

We therefore define $\hat{P}^{(a)}$ (resp. $\hat{E}^{(a)}$) to be the
probability measure (expectation) under which $u$ is shifted as in (3.7). Then
(3.8) and (3.9) imply

(3.10) $\frac{d\hat{P}^{(a)}}{dP}=\exp\bigg{(}-\sum_{i=1}^{D}\sum_{j\in
S_{N}^{1}\setminus\\{0\\}}\frac{2a\alpha_{j}x^{(i)}_{j\mathbf{e}_{i}}+(a\alpha_{j})^{2}}{2(2\beta\lambda_{j\mathbf{e}_{i}})^{-1}}\bigg{)}.$

We can therefore rewrite $Z_{N,d,D}$ in (3.8) as follows,

(3.11)
$\begin{split}Z_{N,d,D}&=\hat{E}^{(a)}\Bigg{[}\exp\bigg{(}\sum_{i=1}^{D}\sum_{j\in
S_{N}^{1}\setminus\\{0\\}}\frac{2a\alpha_{j}X^{(i)}_{j\mathbf{e}_{i}}+(a\alpha_{j})^{2}}{2(2\beta\lambda_{j\mathbf{e}_{i}})^{-1}}\\\
&\hskip
113.81102pt-\gamma\int_{\mathbb{R}^{D}}\ell_{N}^{2}(\mathbf{y})d\mathbf{y}\bigg{)}\Bigg{]}.\end{split}$

We define

(3.12)
$Y_{j\mathbf{e}_{i}}^{(i)}=\frac{2a\alpha_{j}X^{(i)}_{j\mathbf{e}_{i}}+(a\alpha_{j})^{2}}{2(2\beta\lambda_{j\mathbf{e}_{i}})^{-1}}.$

Using Jensen’s inequality, we get that

(3.13) $\displaystyle\log Z_{N,d,D}$
$\displaystyle\geq\hat{E}^{(a)}\left[-\gamma\int_{\mathbb{R}^{D}}\ell_{N}^{2}(\mathbf{y})d\mathbf{y}\right]-\hat{E}^{(a)}\left[-\sum_{i=1}^{D}\sum_{j\in
S_{N}^{1}\setminus\\{0\\}}Y_{j\mathbf{e}_{i}}^{(i)}\right]$
$\displaystyle=:-(I_{1,d,D}+I_{2,d,D}).$

The following proposition gives some essential bounds on $I_{i,d,D}$, $i=1,2$.

######  Proposition 3.2.

Let $\beta,\gamma>0$. Then there exists a constant $C>0$ not depending on
$N,\beta,\gamma$ such that

  * (i)

for $d=2$ and $D=1$,

$I_{1,2,1}\leq C\gamma N^{2}\big{(}(\beta^{-1/2}a^{-1}N\log N)\vee 1\big{)},$

  * (ii)

for $d\geq 3$ and $D\leq d$,

$I_{1,d,D}\leq C\gamma N^{d}\big{(}1\vee(\beta^{-D/2}N^{d-D}a^{-D})\big{)}.$

  * (iii)

for any $d\geq 2$,

$I_{2,d,D}\leq C\beta a^{2}N^{d}.$

The proof of Proposition (3.2)(i) and (ii) is postponed to Section 4. The
proof of Proposition (3.2)(iii) is given in Section 5.

###  3.3. Proof of Proposition 3.1

###### Proof of Proposition 3.1.

From (3.13) and Proposition 3.2(i) and (iii) it follows that for $d=2$ and
$D=1$,

(3.14) $\displaystyle\log\hat{Z}_{N,2,1}$
$\displaystyle\geq-(I_{1,2,1}+I_{2,2,1})$ $\displaystyle\geq-C\left[\gamma
N^{2}\big{(}(\beta^{-1/2}a^{-1}N\log N)\vee 1\big{)}+\beta N^{2}a^{2}\right].$

Taking $a=\beta^{-1/2}(N\log N)^{1/3}$ in (3.14) gives,

$\log\hat{Z}_{N,2,1}\geq-C(\beta+\gamma)N^{8/3}(\log N)^{2/3}.$

The proof for $d\geq 3$ and $1\leq D<d$ follows the same lines with the only
modification that we are using Proposition 3.2(ii) and choosing
$a=\beta^{-1/2}N^{\frac{d-D}{D+2}}$ to get

$\log\hat{Z}_{N,d,D}\geq-C(\beta+\gamma)N^{d+\frac{2(d-D)}{D+2}}.$

∎

##  4\. Proof of Proposition (3.2)(i) and (ii)

###### Proof of Proposition 3.2(i) and (ii).

We can write

(4.1) $\displaystyle\tilde{I}_{1,d,D}$
$\displaystyle:=\hat{E}^{(a)}\left[\int_{\mathbb{R}^{D}}\ell_{N}(\mathbf{y})^{2}d\mathbf{y}\right]$
$\displaystyle=\hat{E}^{(a)}\left[\int_{\mathbb{R}^{D}}\Big{(}\sum_{\mathbf{z}\in
S^{d}_{N}}\mathbf{1}_{[\mathbf{y}-\mathbf{1/2},\mathbf{y}+\mathbf{1/2}]}(\mathbf{u}(\mathbf{z}))\Big{)}^{2}d\mathbf{y}\right]$
$\displaystyle=\sum_{\mathbf{z}\in
S^{d}_{N}}\hat{E}^{(a)}\left[\int_{\mathbb{R}^{D}}\mathbf{1}_{[\mathbf{y}-\mathbf{1/2},\mathbf{y}+\mathbf{1/2}]}(\mathbf{u}(\mathbf{z}))d\mathbf{y}\right]$
$\displaystyle\quad+\sum_{\mathbf{z},\mathbf{w}\in
S^{d}_{N},\,\mathbf{z}\not=\mathbf{w}}\hat{E}^{(a)}\left[\int_{\mathbb{R}^{D}}\mathbf{1}_{\mathbf{u}(\mathbf{z}),\mathbf{u}(\mathbf{w})\in[\mathbf{y}-\mathbf{1/2},\mathbf{y}+\mathbf{1/2}]}d\mathbf{y}\right]$
$\displaystyle=(2N+1)^{d}+\sum_{\mathbf{z},\mathbf{w}\in
S^{d}_{N},\,\mathbf{z}\not=\mathbf{w}}\int_{\|\mathbf{y}\|\leq
1}\hat{p}_{\mathbf{z},\mathbf{w}}(\mathbf{y})d\mathbf{y},$

where $\hat{p}_{\mathbf{z},\mathbf{w}}$ is the density of
$\mathbf{u}(\mathbf{z})-\mathbf{u}(\mathbf{w})$ under $\hat{P}^{(a)}$.

We recall Proposition 3.1 from [10].

######  Proposition 4.1.

There exist constants $C_{1},C_{2}>0$ such that,

  * (i)

for $d=2$, for all $\mathbf{w},\mathbf{z}\in S_{N}^{2}$ with
$\mathbf{w}\neq\mathbf{z}$, and for $i=1,2$ we have

$C_{1}\beta^{-1}\leq\textrm{Var}\left(u^{(i)}(\mathbf{z})-u^{(i)}(\mathbf{w})\right)\leq
C_{2}\beta^{-1}(\log N)^{2},$

  * (ii)

for $d\geq 3$, for all $\mathbf{w},\mathbf{z}\in S_{N}^{d}$ with
$\mathbf{w}\neq\mathbf{z}$, and for $i=1,\dots,D$ we have

$C_{1}\beta^{-1}\leq\textrm{Var}\left(u^{(i)}(\mathbf{z})-u^{(i)}(\mathbf{w})\right)\leq
C_{2}\beta^{-1}.$

Note that from (3.4) we have

$\hat{E}^{(a)}[u^{(i)}(\mathbf{z})-u^{(i)}(\mathbf{w})]=a(z_{i}-w_{i}),\quad\text{
for }i=1,\dots,D.$

Since $(u^{(i)})_{i=1,\dots,D}$ are independent, we have for any
$\mathbb{y}\in\mathbb{R}^{D}$

(4.2)
$\hat{p}_{\mathbf{z},\mathbf{w}}(\mathbb{y}):=\prod_{i=1}^{D}\hat{p}^{(i)}_{\mathbf{z},\mathbf{w}}(y_{i})$

and therefore

(4.3) $\int_{\|\mathbf{y}\|\leq
1}\hat{p}_{\mathbf{z},\mathbf{w}}(\mathbf{y})d\mathbf{y}\leq\prod_{i=1}^{D}\int_{-1}^{1}\hat{p}^{(i)}_{\mathbf{z},\mathbf{w}}(y_{i})dy_{i},$

where $\hat{p}^{(i)}_{\mathbf{z},\mathbf{w}}$ is the density of
$u^{(i)}(\mathbf{z})-u^{(i)}(\mathbf{w})$ under $\hat{P}^{(a)}$.

We distinguish between the following two cases.

Case 1: $d\geq 3$ and $D\leq d$. From Proposition 4.1(ii) we have

(4.4) $\hat{p}^{(i)}_{\mathbf{z},\mathbf{w}}(y_{i})\leq\frac{1}{\sqrt{2\pi
C_{1}\beta^{-1}}}\exp\Big{(}-\frac{a^{2}(z_{i}-w_{i}-y_{i})^{2}}{2C_{2}\beta^{-1}}\Big{)},\quad\textrm{i=1,...,D}.$

From (4.3) and (4.4) we therefore get

$\displaystyle\sum_{\mathbf{z},\mathbf{w}\in
S_{N}^{d},\,\mathbf{z}\not=\mathbf{w}}\int_{\|\mathbf{y}\|\leq
1}\hat{p}_{\mathbf{z},\mathbf{w}}(\mathbf{y})$ $\displaystyle\leq
C\sum_{\mathbf{z},\mathbf{w}\in
S_{N}^{d},\,\mathbf{z}\not=\mathbf{w}}\int_{\mathbf{y}\in[-1,1]^{D}}\frac{1}{(2\pi
C_{1}\beta^{-1})^{d/2}}$
$\displaystyle\qquad\times\exp\Big{(}-\frac{a^{2}\sum_{i=1}^{D}(z_{i}-w_{i}-y_{i})^{2}}{2C_{2}\beta^{-1}}\Big{)}d\mathbf{y}$
$\displaystyle\leq\int_{\mathbf{y}\in[-1,1]}J(\mathbf{y})d\mathbf{y}.$

where

$J(\mathbf{y}):=\sum_{\mathbf{z},\mathbf{w}\in S_{N}^{d}}\frac{1}{2\pi
C_{1}\beta^{-1}}\exp\Big{(}-\frac{a^{2}\sum_{i=1}^{D}(z_{i}-w_{i}-y_{i})^{2}}{2C_{2}\beta^{-1}}\Big{)}.$

The following lemma follows immediately from the proof of Lemma 3.2 from [10].

######  Lemma 4.2.

Let $\kappa>0$. Then for all $y\in[-1,1]$ and $w\in S^{1}_{N}$ we have

$\displaystyle\sum_{z\in S^{1}_{N}}\exp\Big{(}-\kappa(z-w-y)^{2}\Big{)}$
$\displaystyle\leq\sum_{z=w-1}^{w+1}\exp\Big{(}-\kappa(z-w-y)^{2}\Big{)}$
$\displaystyle\quad+\int_{-\infty}^{\infty}\exp\Big{(}-\kappa(z-w-y)^{2}\Big{)}dz.$

Using Lemma 4.2 and integrating over the Gaussian density gives,

(4.5) $\displaystyle J(\mathbf{y})$ $\displaystyle=\frac{(2N+1)^{2(d-D)}}{2\pi
C_{1}\beta^{-1}}\prod_{i=1}^{D}\left(\sum_{z_{i}\in S_{N}^{1}}\sum_{w_{i}\in
S_{N}^{1}}\exp\Big{(}-\frac{a^{2}(z_{i}-w_{i}-y_{i})^{2}}{2C_{2}\beta^{-1}}\Big{)}\right)$
$\displaystyle\leq CN^{2(d-D)}\prod_{i=1}^{D}\sum_{w_{i}\in
S_{N}^{1}}\Bigg{(}\sum_{z_{i}=w_{i}-1}^{w_{i}+1}\exp\Big{(}-\frac{a^{2}(z_{i}-w_{i}-y_{i})^{2}}{2C_{2}\beta^{-1}}\Big{)}$
$\displaystyle\quad+\beta^{-1/2}a^{-1}\frac{1}{\sqrt{2\pi
C_{2}\beta^{-1}a^{-2}}}\int_{-\infty}^{\infty}\exp\Big{(}-\frac{(z_{i}-w_{i}-y_{i})^{2}}{2C_{2}\beta^{-1}a^{-2}}\Big{)}dz_{i}\Bigg{)}$
$\displaystyle\leq CN^{2(d-D)}\prod_{i=1}^{D}\sum_{w_{i}\in
S_{N}^{1}}\left(\sum_{z_{i}=w_{i}-1}^{w_{i}+1}\exp\Big{(}-\frac{a^{2}(z_{i}-w_{i}-y_{i})^{2}}{2C_{2}\beta^{-1}}\Big{)}+\beta^{-1/2}a^{-1}\right).$

It follows that

(4.6) $\displaystyle\int_{\mathbf{y}\in[-1,1]}J(\mathbf{y})d\mathbf{y}$
$\displaystyle\leq CN^{2(d-D)}\left(\int_{-1}^{1}\sum_{w\in
S_{N}^{1}}\left(\sum_{z=w-1}^{w+1}\exp\Big{(}-\frac{a^{2}(z-w-y)^{2}}{2C_{2}\beta^{-1}}\Big{)}+\beta^{-1/2}a^{-1}\right)dy\right)^{D}$
$\displaystyle\leq
CN^{2(d-D)}N^{D}\left(\int_{-1}^{1}\left(\sum_{k=-1}^{1}\exp\Big{(}-\frac{a^{2}(k-y)^{2}}{2C_{2}\beta^{-1}}\Big{)}+\beta^{-1/2}a^{-1}\right)dy\right)^{D}.$

Using again integration over the Gaussian density gives for any $M>0$ and
$k=-1,0,1$,

$\int_{-1}^{1}e^{-M(k+x)^{2}}dx\leq\int_{-\infty}^{\infty}e^{-M(k+x)^{2}}dx\leq
CM^{-1/2}.$

Plugging in these bounds to (4.6) gives

$\int_{\mathbf{y}\in[-1,1]}J(\mathbf{y})d\mathbf{y}\leq
CN^{2(d-D)}N^{D}(\beta^{-1/2}a^{-1})^{D}.$

Together with (4.1) this leads to,

$\tilde{I}_{1}\leq\tilde{C}N^{d}\big{(}1\vee(\beta^{-D/2}N^{d-D}a^{-D})\big{)}.$

Case 2: $d=2$ and $D=1$. Then from Proposition 4.1(i) we have

(4.7) $\hat{p}^{(1)}_{\mathbf{z},\mathbf{w}}(y)\leq\frac{1}{\sqrt{2\pi
C_{1}\beta^{-1}}}\exp\Big{(}-\frac{a^{2}(y-(z_{1}-w_{1}))^{2}}{2\beta^{-1}C_{2}(\log
N)^{2}}\Big{)}.$

Then following similar steps as in Case 1 we arrive to

(4.8) $\displaystyle\tilde{I}_{1,2,1}$
$\displaystyle\leq(2N+1)^{2}+C\beta^{-1/2}a^{-1}N^{3}\log N$
$\displaystyle\leq\tilde{C}\big{(}(\beta^{-1/2}a^{-1}N\log N)\vee
1\big{)}N^{2}.$

Since from (3.13) and (4.1) we have that

$I_{1,d,D}=\gamma\tilde{I}_{1,d,D},$

this completes the proof of Proposition 3.2 parts (i) and (ii). ∎

##  5\. Proof of Proposition (3.2) (iii)

######  Proof of Proposition (3.2)(iii).

Recall that $I_{2,d,D}$ was defined in (3.13),

(5.1) $\displaystyle I_{2,d,D}$ $\displaystyle=-\sum_{i=1}^{D}\sum_{j\in
S_{N}^{1}\setminus\\{0\\}}\hat{E}^{(a)}\left[Y_{j\mathbf{e}_{i}}^{(i)}\right],$

where $Y_{j\mathbf{e}_{i}}^{(i)}$ was defined in (3.12). Recall also that
$N(d)$ was defined in (3.3). Since the expectation on the right-hand side of
(5.1) is taken over a Gaussian measure, we define the following normalizing
constant

(5.2) $\displaystyle C_{N,\beta,d,D}$ $\displaystyle=\int_{\mathbb{R}^{D\cdot
N(d)}}\exp\left(-\sum_{i=1}^{D}\sum_{k=1}^{N(d)}\frac{(x^{(i)}_{k})^{2}}{2(2\beta\lambda_{k})^{-1}}\right)\prod_{i=1}^{D}\prod_{k=1}^{N(d)}dx^{(i)}_{k}$
$\displaystyle=\frac{1}{(2\beta)^{DN(d)/2}}\prod_{k=1}^{N(d)}\frac{1}{\lambda_{k}^{D/2}}.$

We refer to Section 1.2 in [10] for additional in formation about the setup of
the GFF measure.

We further introduce the following notation:

$z_{k_{1},\dots,k_{d}}^{(i)}=\frac{(x^{(i)}_{k_{1},\dots,k_{d}})^{2}}{2(2\beta\lambda_{k_{1},\dots,k_{d}})^{-1}},\quad
w^{(i)}_{j\mathbf{e}_{i}}=\frac{(x^{(i)}_{j\mathbf{e}_{i}}+a\alpha_{j})^{2}}{2(2\beta\lambda_{j\mathbf{e}_{i}})^{-1}},$

and

$y_{j\mathbf{e}_{i}}^{(i)}=\frac{2a\alpha_{j}x^{(i)}_{j\mathbf{e}_{i}}+(a\alpha_{j})^{2}}{2(2\beta\lambda_{j\mathbf{e}_{i}})^{-1}}.$

Then from (3.9) and (3.10) we have

(5.3) $\displaystyle\hat{E}^{(a)}\left[Y_{j\mathbf{e}_{i}}^{(i)}\right]$
$\displaystyle=\frac{1}{C_{N,\beta,d,D}}\int
y_{j\mathbf{e}_{i}}^{(i)}\exp\bigg{(}-\sum_{i=1}^{D}\Big{(}\sum_{(k_{1},\dots,k_{d})\in
S_{N}^{d}\setminus\\{j\mathbf{e}_{i}:\,j\in
S_{N}^{1}\\}}z_{k_{1},\dots,k_{d}}^{(i)}$ $\displaystyle\qquad+\sum_{l\in
S_{N}^{1}\setminus\\{0\\}}w^{(i)}_{l\mathbf{e}_{i}}\Big{)}\bigg{)}\prod_{i=1}^{D}\prod_{l=1}^{N(d)}dx^{(i)}_{l},$

where $C_{N,\beta,d,D}$ was defined in (5.2).

Since the expected value in (5.3) is symmetric with respect to $i$, we can use
$i=1$ in what follows in order to ease the notation. We therefore consider

(5.4)
$\begin{split}&\hat{E}^{(a)}\left[Y_{j\mathbf{e}_{1}}^{(1)}\right]=\frac{1}{C_{N,\beta,d,D}}\int
y_{j\mathbf{e}_{1}}^{(1)}\exp(-w^{(1)}_{j\mathbf{e}_{1}})\\\
&\quad\times\int\exp\bigg{(}-\Big{(}\sum_{i=1}^{D}\sum_{(k_{1},\dots,k_{d})\in
S_{N}^{d}\setminus\\{l\mathbf{e}_{i}:\,l\in
S_{N}^{1}\\}}z_{k_{1},\dots,k_{d}}^{(i)}\\\
&\quad+\sum_{i=2}^{D}\sum_{l\in\\{-N,\dots,N\\}\setminus\\{0\\}}w^{(i)}_{l\mathbf{e}_{i}}+\sum_{l\in
S_{N}^{1}\setminus\\{0,j\\}}w^{(1)}_{l\mathbf{e}_{1}}\Big{)}\bigg{)}\prod_{i=1}^{D}\prod_{l=1}^{N(d)}dx^{(i)}_{l}.\end{split}$

We notice that we have three types of integrals above, which can be evaluated
as follows. We have $D((2N+1)^{d}-(2N+1))$ integrals of the form

$\displaystyle\int_{\mathbb{R}}\exp\big{(}-z_{k_{1},\dots,k_{d}}^{(i)}\big{)}dx^{(i)}_{k_{1},\dots,k_{d}}$
$\displaystyle=\int_{\mathbb{R}}\exp\left(-\frac{(x^{(i)}_{k_{1},\dots,k_{d}})^{2}}{2(2\beta\lambda_{k_{1},\dots,k_{d}})^{-1}}\right)dx^{(i)}_{k_{1},\dots,k_{d}}$
$\displaystyle=\sqrt{2\pi}(2\beta\lambda_{k_{1},\dots,k_{d}})^{-1/2}.$

We have $2N(D-1)+2N-1$ integrals of the form

$\displaystyle\int_{\mathbb{R}}\exp\big{(}-w^{(i)}_{l\mathbf{e}_{i}}\big{)}dx^{(i)}_{l\mathbf{e}_{i}}$
$\displaystyle=\int_{\mathbb{R}}\exp\left(-\frac{(x^{(i)}_{\textbf{0}_{i}(l)}+a\alpha_{l})^{2}}{2(2\beta\lambda_{l\mathbf{e}_{i}})^{-1}}\right)dx^{(i)}_{l\mathbf{e}_{i}}$
$\displaystyle=\sqrt{2\pi}(2\beta\lambda_{l\mathbf{e}_{i}})^{-1/2},$

and one integral as follows

$\displaystyle\int
y_{j\mathbf{e}_{1}}\exp(-w^{(1)}_{j\mathbf{e}_{1}})dx_{j\mathbf{e}_{1}}^{(1)}$
$\displaystyle=\int_{\mathbb{R}}\frac{2a\alpha_{j}x^{(1)}_{j\mathbf{e}_{1}}+(a\alpha_{j})^{2}}{2(2\beta\lambda_{j\mathbf{e}_{1}})^{-1}}\exp\left(-\frac{(x^{(1)}_{j\mathbf{e}_{1}}+a\alpha_{j})^{2}}{2(2\beta\lambda_{j\mathbf{e}_{1}})^{-1}}\right)dx_{j\mathbf{e}_{1}}^{(1)}$
$\displaystyle=-\sqrt{2\pi}\frac{(a\alpha_{j})^{2}}{2(2\beta\lambda_{j\mathbf{e}_{1}})^{-1/2}}.$

Plugging in all the above integrals into (5.4) gives

$\displaystyle\hat{E}^{(a)}\left[Y_{j\mathbf{e}_{1}}^{(1)}\right]$
$\displaystyle=-\frac{1}{C_{N,\beta,d,D}}\sqrt{2\pi}\frac{(a\alpha_{j})^{2}}{2(2\beta\lambda_{j\mathbf{e}_{1}})^{-1/2}}$
$\displaystyle\qquad\times\prod_{i=1}^{D}\prod_{(k_{1},\dots,k_{d})\in
S_{N}^{d}\setminus\\{l\mathbf{e}_{i}:\,l\in
S_{N}^{1}\\}}\sqrt{2\pi}(2\beta\lambda_{k_{1},\dots,k_{d}})^{-1/2}$
$\displaystyle\qquad\times\prod_{i=2}^{D}\prod_{l\in
S_{N}^{1}\setminus\\{0\\}}\sqrt{2\pi}(2\beta\lambda_{{l\mathbf{e}_{i}}})^{-1/2}\prod_{l\in
S_{N}^{1}\setminus\\{0,k\\}}\sqrt{2\pi}(2\beta\lambda_{l\mathbf{e}_{1}})^{-1/2}$
$\displaystyle=-\frac{1}{C_{N,\beta,d,D}}\frac{(2\pi)^{((2N+1)^{d}-1)D/2}(a\alpha_{j})^{2}\lambda_{j\mathbf{e}_{1}}}{2(2\beta)^{((2N+1)^{d}-1)D/2-1}}\prod_{i=1}^{D}\prod_{(k_{1},\dots,k_{d})\in
S_{N}^{d}\setminus\\{\textbf{0}\\}}\frac{1}{\lambda_{k_{1},\dots,k_{d}}^{1/2}}.$

Together with (5.2) we get

(5.5) $\displaystyle\hat{E}^{(a)}\left[Y_{j\mathbf{e}_{1}}^{(1)}\right]$
$\displaystyle=-\beta(a\alpha_{j})^{2}\lambda_{j\mathbf{e}_{1}}.$

Plugging (5.5) into (5.1) gives

(5.6) $\displaystyle I_{2,d,D}$ $\displaystyle=\beta\sum_{i=1}^{D}\sum_{j\in
S_{N}^{1}\setminus\\{0\\}}(a\alpha_{j})^{2}\lambda_{j\mathbf{e}_{i}}$
$\displaystyle=D\beta a^{2}\sum_{j\in
S_{N}^{1}\setminus\\{0\\}}\alpha_{j}^{2}\lambda_{j\mathbf{e}_{1}},$

where we used the fact that
$\lambda_{j\mathbf{e}_{i}}=\lambda_{j\mathbf{e}_{1}}$, by the symmetry of the
eigenvalues (see (3.2)).

In order to complete the proof we recall Lemma 4.1 from [10].

######  Lemma 5.1.

There exists a constant $C>0$ not depending on $N$ and $\beta$ such that,

$\sum_{j\in
S_{N}^{1}\setminus\\{0\\}}\alpha_{j}^{2}\lambda_{j\mathbf{e}_{1}}\leq CN^{d}.$

From Lemma 5.1 and (5.6) we conclude that

(5.7) $I_{2,d,D}\leq C\beta a^{2}N^{d},$

which completes the proof of Proposition 3.2 part (iii). ∎

##  6\. Large distance tail estimates

Assume first that $d=2$ and $d=1$. Let $\alpha>0$, then from (2.5) we have

(6.1) $\displaystyle\log Q_{N}(R_{N,2,1}>\alpha N^{4/3}(\log N)^{4/3})$
$\displaystyle\leq\log P(R_{N,2,1}>\alpha N^{4/3}(\log N)^{4/3})-\log
Z_{N,2,1}.$

From the proof in Section 5 of [10], which uses standard Gaussian estimates,
it follows that for any constant $\kappa(N)>0$ we have

$P(R_{N,2,1}>\alpha\beta^{-1/2}\kappa(N))\leq
C_{1}\exp\left\\{-c_{2}\kappa(N)^{2}(\log N)^{-2}\right\\}.$

where $C_{1},c_{2}>0$ are constants not depending on $N$.

We therefore get,

$P(R_{N,2,1}>\alpha\beta^{-1/2}N^{4/3}(\log N)^{4/3})\leq
C_{1}\exp\left\\{-c_{2}\alpha^{2}N^{8/3}(\log N)^{2/3}\right\\}.$

Using this bound together with Proposition 3.1(i) and (6.1) we get for all
$\alpha\geq 1$,

$\displaystyle\log
Q_{N}(R_{N,2,1}>\alpha\beta^{-1/2}(\beta+\gamma)^{1/2}N^{4/3}(\log N)^{4/3})$
$\displaystyle\leq\log
P(R_{N,2,1}>\alpha\beta^{-1/2}(\beta+\gamma)^{1/2}N^{4/3}(\log N)^{4/3})-\log
Z_{N,2,1}$ $\displaystyle\leq-(\beta+\gamma)N^{8/3}(\log
N)^{2/3}\big{(}c_{3}\alpha^{2}-c_{4}\big{)}.$

We then can choose $\alpha$ to be large enough to get the large distance tail
estimate in Theorem 2.1.

The proof for $d\geq 3$ and $1\leq D\leq d$ follows similar lines, only now we
use the bound

$P(R_{N,d,D}>\alpha\beta^{-1/2}\kappa(N))\leq
C_{1}\exp\left\\{-c_{2}\kappa(N)^{2}\right\\}.$

Note that the $\log$-correction does not appear on the right-hand side due to
Proposition 4.1(ii).

Together with Proposition 3.1(ii) we get

$\displaystyle\log
Q_{N}(R_{N,d,D}>\alpha\beta^{-1/2}(\beta+\gamma)^{1/2}N^{\frac{d}{2}+\frac{d-D}{D+2}})$
$\displaystyle\leq\log
P(R_{N,d,D}>\alpha\beta^{-1/2}(\beta+\gamma)^{1/2}N^{\frac{d}{2}+\frac{d-D}{D+2}})-\log
Z_{N}$
$\displaystyle\leq-(\beta+\gamma)N^{d+\frac{2(d-D)}{D+2}}\big{(}c_{3}\alpha^{2}-c_{4}\big{)}.$

We then can choose $\alpha$ to be large enough to get the large distance tail
estimate in Theorem 2.1.

##  7\. Small distance tail estimates

Let $\varepsilon>0$, and $\kappa(N)>0$ to be specified later. Then from (2.5)
we have the following:

(7.1) $\begin{split}\log&Q_{N}(R_{N,d,D}<\varepsilon\kappa(N))\\\ &\leq\log
E\left[\exp\left\\{-\gamma\int_{\mathbb{R}^{D}}\ell_{N}(\mathbf{y})^{2}d\mathbf{y}\right\\}\mathbf{1}_{\\{R_{N,d,D}<\varepsilon\kappa(N)\\}}\right]-\log
Z_{N,d,D}.\end{split}$

Let

(7.2)
$\tilde{J}_{N,d,D}:=E\left[\exp\left\\{-\gamma\int_{\mathbb{R}^{D}}\ell_{N}(\mathbf{y})^{2}d\mathbf{y}\mathbf{1}_{\\{R_{N,d,D}<\varepsilon\kappa(N)\\}}\right\\}\right].$

Note that on $\\{R_{N,d,D}<\varepsilon\kappa(N)\\}$ we have

(7.3) $\displaystyle\int_{\mathbb{R}^{D}}\ell_{N}(\mathbf{y})^{2}d\mathbf{y}$
$\displaystyle=2^{d}\kappa(N)^{D}\varepsilon^{D}\int_{[-\varepsilon\kappa(N),\varepsilon\kappa(N)]^{D}}\ell_{N}(\mathbf{y})^{2}\frac{1}{2^{D}\kappa(N)^{D}\varepsilon^{D}}d\mathbf{y}$
$\displaystyle\geq
2^{D}\kappa(N)^{D}\varepsilon^{D}\left(\int_{[-\varepsilon\kappa(N),\varepsilon\kappa(N)]^{D}}\ell_{N}(\mathbf{y})\frac{1}{2^{D}\kappa(N)^{D}\varepsilon^{D}}d\mathbf{y}\right)^{2}$
$\displaystyle=\frac{1}{2^{D}\kappa(N)^{D}\varepsilon^{D}}\left(\int_{[-\varepsilon\kappa(N),\varepsilon\kappa(N)]^{D}}\ell_{N}(\mathbf{y})d\mathbf{y}\right)^{2},$

where we used Jensen’s inequality. Since on
$\\{R_{N,d,D}<\varepsilon\kappa(N)\\}$ we have

$\int_{[-\varepsilon\kappa(N),\varepsilon\kappa(N)]^{D}}\ell_{N}(\mathbf{y})d\mathbf{y}=|S_{N}|=(2N+1)^{d},$

together with (7.3) we get that

(7.4)
$\int_{\mathbb{R}^{D}}\ell_{N}(\mathbf{y})^{2}d\mathbf{y}\geq\frac{2^{2d-D}N^{2d}\kappa(N)^{-D}}{\varepsilon^{D}}.$

From (7.2) and (7.4) we have

(7.5)
$\tilde{J}\leq\exp\left(-\gamma\frac{2^{2d-D}N^{2d}\kappa(N)^{-D}}{\varepsilon^{D}}\right).$

From (7.1), (7.2), (7.5) and Proposition 3.1(i) we get for $d=2$ and D=1
(taking $\kappa(N)=\gamma(\beta+\gamma)^{-1}N^{4/3}(\log N)^{-2/3}$):

$\displaystyle\log$ $\displaystyle
Q_{N}(R_{N,2,1}<\varepsilon\gamma(\beta+\gamma)^{-1}N^{4/3}(\log N)^{-2/3})$
$\displaystyle\leq-(\beta+\gamma)\left(\frac{8N^{8/3}(\log
N)^{2/3}}{\varepsilon}-CN^{8/3}(\log N)^{2/3}\right).$

By choosing $\varepsilon>0$ small enough it follows that

$\lim_{N\rightarrow\infty}\log
Q_{N}(R_{N,2,1}<\varepsilon\gamma(\beta+\gamma)^{-1}N^{4/3}(\log
N)^{-2/3})=0.$

Repeating the same steps as in the case where $d=2$ and $D=1$, plugging in

$\kappa(N)=\gamma^{1/D}(\beta+\gamma)^{-1/D}N^{\frac{1}{D}\left(d-\frac{2(d-D)}{d-D+2}\right)}$

to (7.5) we get,

$\tilde{J}\leq
e^{-(\beta+\gamma)\frac{2^{2d-D}N^{d+\frac{2(d-D)}{d-D+2}}}{\varepsilon^{D}}}.$

Together with Proposition 3.1(ii) this give the following bound for $d\geq 3$
and $1\leq D\leq d$,

$\displaystyle\log
Q_{N}\left(R_{N,d,D}<\varepsilon\gamma^{1/D}(\beta+\gamma)^{-1/D}N^{\frac{1}{D}\left(d-\frac{2(d-D)}{D+2}\right)}\right)$
$\displaystyle\leq-(\beta+\gamma)\left(\frac{N^{d+\frac{2(d-D)}{D+2}}}{\varepsilon^{D}}-CN^{d+\frac{2(d-D)}{D+2}}\right).$

Then choosing $\varepsilon>0$ sufficiently small and taking the limit where
$N\rightarrow\infty$ completes the proof of Theorem 2.1.

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# Self-descriptive Sequences directed by two Periodic Sequences

Shigeki Akiyama Institute of Mathematics, University of Tsukuba
1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8571 Japan<EMAIL_ADDRESS>Univ. Lorraine, Loria, UMR 7503
Vandœuvre-lès-Nancy, F-54506, FranceUniv Rouen Normandie, CNRS,
Normandie Univ LMRS UMR 6085,
F-76000 Rouen, FranceÉcole Normale Supérieure de Lyon
15 parvis René Descartes, F-69342 Lyon, France Damien Jamet Univ. Lorraine,
Loria, UMR 7503
Vandœuvre-lès-Nancy, F-54506, France<EMAIL_ADDRESS>Univ Rouen
Normandie, CNRS,
Normandie Univ LMRS UMR 6085,
F-76000 Rouen, FranceÉcole Normale Supérieure de Lyon
15 parvis René Descartes, F-69342 Lyon, France Irène Marcovici Univ Rouen
Normandie, CNRS,
Normandie Univ LMRS UMR 6085,
F-76000 Rouen, France<EMAIL_ADDRESS>École Normale Supérieure
de Lyon
15 parvis René Descartes, F-69342 Lyon, France Mai-Linh Trân Công École
Normale Supérieure de Lyon
15 parvis René Descartes, F-69342 Lyon, France<EMAIL_ADDRESS>
##  1 Introduction

A self-descriptive sequence $(u_{n})_{n\in\mathbb{N}}$ is an infinite
concatenation of finite powers of a letter (usually called runs)
$(w_{n})_{n\in\mathbb{N}}$ such that $|w_{n}|=u_{n}$ where $|x|$ denotes the
length of the finite word $x$. The best known self-descriptive sequence is
certainly the Oldenburger word $\mathbb{O}_{1,2}=(k_{n})_{n\in\mathbb{N}}$
[10, 9] defined by $u_{0}=1$, $w_{2n}=1^{u_{2n}}$ and $w_{2n+1}=2^{u_{2n+1}}$
for all $n\in\mathbb{N}$. Until recently, the Oldenburger word was still
called the Kolakoski word in reference to [9], but actually, it first appeared
in [10].

The Oldenburger word is a special case of a self-descriptive sequence. Indeed,
the run $w_{n}$ is entirely determined by knowledge of its index $n$: its
length is equal to $u_{n}$ and its single letter is determined by the parity
of $n$.

In [4], the authors focus on a larger family of self-descriptive sequences
where the $w_{n}$’s are determined not only by their index $n$ but also by
another sequence, namely the directing sequence of $u$. In practice, given a
sequence $t=(t_{n})_{n\in\mathbb{N}}$ on the alphabet
$\mathcal{A}\in\\{1,2,\dots\\}$, the sequence directed by $t$ is the sequence
$u$ defined by: $u={t_{0}}^{u_{0}}{t_{1}}^{u_{1}}\cdots{t_{n}}^{u_{n}}\cdots$
For example, the Oldenburger word
$\mathbb{O}_{1,2}=1^{u_{0}}2^{u_{1}}1^{u_{2}}2^{u_{4}}\cdots$ is directed by
the sequence $t=(12)^{\omega}$.

One of the most fascinating questions about the sequence $\mathbb{O}_{1,2}$
concerns the existence and possible value of the frequencies of occurrences of
each of its letters [8]: Do the letters $1$ and $2$ have frequencies of
occurrences $f_{1}$ and $f_{2}$ in $\mathbb{O}_{1,2}$? If so, does
$f_{1}=f_{2}=\frac{1}{2}$? Recall that the frequency of occurrences of the
letter $a$ in the sequence $u$ is the limit, when $n$ tends to $+\infty$, of
the average number of $a$ in the prefix $u_{0}\cdots u_{n-1}$ of $u$.

The notion of self-descriptive sequence is related to that of differentiable
word and smooth word [3, 5]. A sequence over $\mathcal{A}$ is differentiable
over $\mathcal{A}$ if it is the infinite concatenation of runs whose lengths
have values in $\mathcal{A}$. More precisely, a sequence
$(u_{n})_{n\in\mathbb{N}}$ over a finite alphabet
$\mathcal{A}\subset\mathbb{N}$ is differentiable if there exist two sequences
$(x_{n})_{n\in\mathbb{N}}$ and $(\alpha_{n})_{n\in\mathbb{N}}$ over
$\mathcal{A}$, such that
$u=x_{0}^{\alpha_{0}}x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\cdots$ with
$x_{n}\neq x_{n+1}$ and $\alpha_{n}\neq 0$ for all $n\in\mathbb{N}$. The
sequence $(\alpha_{n})_{n\in\mathbb{N}}$ is the derivative sequence of $u$.
Finally, the sequence $u$ is smooth if it is infinitely differentiable.

The sequence $\mathbb{O}_{1,2}$ is a fixed point for differentiation. It is
self-descriptive, differentiable and smooth over $\\{1,2\\}$. As with the
sequence $\mathbb{O}_{1,2}$, the question of the existence of frequencies of
occurrences and their values in smooth words on the alphabet $\\{1,2\\}$ is
still open. The first significant result on this question is due to V. Chvátal
[7]: for $n$ large enough,

$0.49916\leq\dfrac{|k_{0}\cdots k_{n-1}|_{1}}{n}\leq 0.50084,$

where $|w|_{1}$ denotes the number of occurrences of the letter $1$ in $w$.
These bounds have been slightly improved by M. Rao [11] using a method quite
similar to Chvátal’s but with greater computing power:

$0.49992\leq\dfrac{|k_{0}\cdots k_{n-1}|_{1}}{n}\leq 0.50008.$

This is the best currently known bound for $\mathbb{O}_{1,2}$ and for the set
of smooth words on the alphabet $\\{1,2\\}$. In other words, neither the
existence nor the values of the frequencies of occurrences are known for any
smooth word on the $\\{1,2\\}$ alphabet. On the other hand, over the
$\\{a,b\\}$ alphabets where $a$ and $b$ have the same parity, it is possible
to determine the frequencies of certain smooth words. This is the case at
least for the Oldenburger word $\mathbb{O}_{1,3}$ (resp. $\mathbb{O}_{3,1}$)
defined on the alphabet $\\{1,3\\}$ directed by $(13)^{\omega}$ (resp. by
$(31)^{\omega}$) [2] and for the extreme smooth words (in the sense of
lexicographic order) [6].

Since the work of V. Chvátal [7] and M. Rao [11], it is reasonable to expect
that the frequencies of occurrences of each letter in $\mathbb{O}_{1,2}$ are
equal to $\frac{1}{2}$. In other words, it is reasonable to assume that the
frequencies of occurrence in $\mathbb{O}_{1,2}$ and in its directing sequence
are identical.

As far as we know, none of the works on smooth words or on the Oldenburger
word has shown the existence or the non-existence of frequencies of
occurrences in a non-trivial deterministic self-descriptive sequence over the
alphabet $\\{1,2\\}$.

Similarly, none of these works has proved the existence of a non-trivial
deterministic self-descriptive sequence that shares (resp. does not share) its
frequencies of occurrences with its directing sequence.

In the present work, we exhibit a class of self-descriptive sequences that can
be explicitly computed and whose frequencies are known. In particular, as a
corollary of our main result, we prove that the sequence introduced in [4] has
the expected frequencies of occurrences.

##  2 Definitions and basic notions

Let $\mathcal{A}$ be a finite alphabet. The set of finite words over
$\mathcal{A}$ is denoted by $\mathcal{A}^{\star}$. If $w=w_{0}\cdots
w_{k}\in\mathcal{A}^{\star}$ is a finite word over the alphabet $\mathcal{A}$
with $w_{i}\in\mathcal{A}$ for $i=0,1,\cdots,k$. Let $|w|$ stand for the
length of $w$, that is the number of letters occurring in $w$. If
$w=w_{0}\cdots w_{k}$, then $|w|=k+1$. In particular, $|\varepsilon|=0$. Let
$w\in\mathcal{A}^{\star}$ and let $a\in\mathcal{A}$. We set
$|w|_{a}=\\#\left\\{i\in\\{0,1,\dots,|w|-1\\}\,|\ w_{i}=a\right\\}$.

######  Definition 1 (Self-descriptive sequence).

Let $\mathcal{A}\subset\mathbb{N}^{\star}$ be a finite alphabet. The infinite
sequence $u=(u_{n})_{n\in\mathbb{N}}\in\mathcal{A}^{\mathbb{N}}$ is said to be
self-descriptive if there exists a sequence
$\delta=(\delta_{n})_{n\in\mathbb{N}}$ over $\mathcal{A}$ such that

$u=\delta_{0}^{u_{0}}\delta_{1}^{u_{1}}\delta_{2}^{u_{2}}\cdots\delta_{n}^{u_{n}}\cdots$
(1)

The sequence $\delta$ is called the directing sequence of $u$ and one says
that $u$ is directed by $x$.

In other words, the sequence $u$ is self-descriptive if it is the
concatenation of runs of size $u_{0}$, $u_{1}$, $u_{2}$, … respectively. Note
that in the definition of self-descriptive sequences, unlike that of
differentiable or smooth words, it is not necessary that $x_{n}\neq x_{n+1}$.
Furthermore, if $0\notin\mathcal{A}$, then the sequence
$(x_{n})_{n\in\mathbb{N}}$ is entirely determined by $u$. In other words,
there exists a canonical bijection between sequences over $\mathcal{A}$ and
self-descriptive sequences over $\mathcal{A}$.

Let
$u=\delta_{0}^{u_{0}}\delta_{1}^{u_{1}}\delta_{2}^{u_{2}}\cdots\delta_{n}^{u_{n}}\cdots$
be a self-descriptive sequence over the alphabet $\\{1,2\\}$. Let
$(m_{k})_{k\in\mathbb{N}}$ (resp. $(n_{k})_{k\in\mathbb{N}}$) be the
increasing sequence over $\mathbb{N}$ such that $u_{i}=1$ (resp. $u_{i}=2$) if
and only if there exists $k\in\mathbb{N}$ such that $i=m_{k}$ (resp.
$i=n_{k}$). In other words, $(m_{k})_{k\in\mathbb{N}}$ (resp.
$(n_{k})_{k\in\mathbb{N}}$) is exactly the ordered sequences of the indices
where $u$ is equal to $1$ (resp. $2$). Let
$T_{1}=\left(\delta_{m_{k}}\right)_{k\in\\{1,2\\}^{\mathbb{N}}}$ and
$T_{2}=\left(\delta_{n_{k}}\right)_{k\in\\{1,2\\}^{\mathbb{N}}}$.

In the present work, since we are only interested in frequencies of letters,
we assume, without loss of generality, that $u_{0}=u_{1}=2$. The sequence $u$
and its directing sequence $\delta$ are then computable from $T_{1}$ and
$T_{2}$ as follows:

⬇

def OK(T1, T2):

u = [2,2]

delta = [2]

k = 1

while len(T1) > 0 and len(T2) > 0:

if u[k] == 1:

c = T1.pop(0)

u += [c]

else:

c = T2.pop(0)

u += [c] * u[k]

delta += [l]

k += 1

return u,delta

Program 1: Python function computing $u$ and $\delta$ from $T_{1}$ and
$T_{2}$.

One then says that $u$ is also directed by sequences $T_{1}$ and $T_{2}$. For
instance, if $T_{1}={\color[rgb]{1,.5,0}121\cdots}$ and
$T_{2}={\color[rgb]{0,0.6,0}12\cdots}$, then

$\displaystyle u$ $\displaystyle=$
$\displaystyle\underset{2}{\leavevmode\hbox{\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt\vtop{\hbox{\kern
1.49994pt\hbox{$22$}\kern 1.49994pt}\kern 1.43515pt\hrule}\kern-0.4pt\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt}}\
\cdot\underset{2}{\leavevmode\hbox{\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt\vtop{\hbox{\kern
1.49994pt\hbox{${\color[rgb]{0,0.6,0}11}$}\kern 1.49994pt}\kern
1.43515pt\hrule}\kern-0.4pt\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt}}\
\cdot\underset{1}{\leavevmode\hbox{\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt\vtop{\hbox{\kern
1.49994pt\hbox{${\color[rgb]{1,.5,0}1}$}\kern 1.49994pt}\kern
1.43515pt\hrule}\kern-0.4pt\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt}}\
\underset{1}{\leavevmode\hbox{\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt\vtop{\hbox{\kern
1.49994pt\hbox{${\color[rgb]{1,.5,0}2}$}\kern 1.49994pt}\kern
1.43515pt\hrule}\kern-0.4pt\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt}}\
\cdot\underset{1}{\leavevmode\hbox{\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt\vtop{\hbox{\kern
1.49994pt\hbox{${\color[rgb]{1,.5,0}1}$}\kern 1.49994pt}\kern
1.43515pt\hrule}\kern-0.4pt\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt}}\
\underset{2}{\leavevmode\hbox{\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt\vtop{\hbox{\kern
1.49994pt\hbox{${\color[rgb]{0,0.6,0}22}$}\kern 1.49994pt}\kern
1.43515pt\hrule}\kern-0.4pt\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt}}\ \cdots$ (2)
$\displaystyle=$ $\displaystyle
2^{2}\cdot{\color[rgb]{0,0.6,0}1}^{2}\cdot{\color[rgb]{1,.5,0}1}^{1}{\color[rgb]{1,.5,0}2}^{1}\cdot{\color[rgb]{1,.5,0}1}^{1}{\color[rgb]{0,0.6,0}2}^{2}\cdots$
(3) $\displaystyle=$ $\displaystyle\underset{2}{\leavevmode\hbox{\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt\vtop{\hbox{\kern
1.49994pt\hbox{$22$}\kern 1.49994pt}\kern 1.43515pt\hrule}\kern-0.4pt\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt}}\
\cdot\underset{w_{0}}{\leavevmode\hbox{\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt\vtop{\hbox{\kern
1.49994pt\hbox{${\color[rgb]{0,0.6,0}11}$}\kern 1.49994pt}\kern
1.43515pt\hrule}\kern-0.4pt\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt}}\
\cdot\underset{w_{1}}{\leavevmode\hbox{\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt\vtop{\hbox{\kern
1.49994pt\hbox{${\color[rgb]{1,.5,0}1}{\color[rgb]{1,.5,0}2}$}\kern
1.49994pt}\kern 1.43515pt\hrule}\kern-0.4pt\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt}}\
\cdot\underset{w_{2}}{\leavevmode\hbox{\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt\vtop{\hbox{\kern
1.49994pt\hbox{${\color[rgb]{1,.5,0}1}{\color[rgb]{0,0.6,0}22}$}\kern
1.49994pt}\kern 1.43515pt\hrule}\kern-0.4pt\vrule
height=2.15277pt,width=0.4pt,depth=1.43515pt\kern-0.4pt}}\ \cdots=22\hskip
5.12128pt\cdot\hskip 1.42271ptw_{0}\hskip 2.98741pt\cdot\hskip
2.84544ptw_{1}\hskip 6.04604pt\cdot\hskip 4.26773ptw_{2}\hskip 8.5359pt\cdots$
(4)

with $w_{i}\in\\{a,b,c,d\\}^{*}$, for $i\in\mathbb{N}$,
$a={\color[rgb]{1,.5,0}1}$, $b={\color[rgb]{1,.5,0}2}$,
$c={\color[rgb]{0,0.6,0}1}$ and $d={\color[rgb]{0,0.6,0}1}$.

##  3 Main result

The main result of the present work is:

######  Theorem 1 (Main result).

Let $u\in\\{1,2\\}^{\mathbb{N}}$ be a sequence over $\\{1,2\\}$ directed by
two periodic sequences $T_{1}=(x_{1})^{\omega}\in\\{1,2\\}^{\mathbb{N}}$ and
$T_{2}=(x_{2})^{\omega}\in\\{1,2\\}^{\mathbb{N}}$, with
$x_{1},x_{2}\in\\{1,2\\}^{*}$. Let $p_{1}=\dfrac{|x_{1}|_{1}}{|x_{1}|}$ and
$q_{2}=\dfrac{|x_{2}|_{2}}{|x_{2}|}$. One has

$f_{1}:=\lim_{n\to\infty}\dfrac{|u_{0}\cdots
u_{n-1}|_{1}}{n}=\frac{(1-q_{2})(p_{1}+2q_{2}+\sqrt{\Delta})}{2+\sqrt{\Delta}-p_{1}}$

with $\Delta=(p_{1}+2q_{2})^{2}-8(p_{1}+q_{2}-1)$.

If $\delta=(\delta_{n})_{n\in\mathbb{N}}\in\\{1,2\\}$ is directing $u$, then

$\lim_{n\to\infty}\dfrac{|\delta_{0}\cdots\delta_{n-1}|_{1}}{n}=p_{1}f_{1}+p_{2}(1-f_{1}).$

###### Sketch of proof.

Let us recode $T_{1}$, $T_{2}$ and $u$ over $\\{a,b,c,d\\}$ as follows:
rewrite $T_{1}$ (resp. $T_{2}$) as the image of $T_{1}$ (resp. $T_{2}$) by the
morphism $1\mapsto a$, $2\mapsto b$ (resp. $1\mapsto c$, $2\mapsto d$). In
$u$, let us substitute $a$ (resp. $b$) for isolated $1$’s (resp. isolated
$2$’s) and $cc$ (resp. $dd$ ) for double $1$’s (resp. double $2$’s).

  1. 1.

Let $u=(u_{n})_{n\in\mathbb{N}}=22\cdot w_{0}\cdot w_{1}\cdot\cdots\cdot
w_{n}\cdot\cdots$, where $w_{n+1}$ is the image of $w_{n}$ by the recoding
rule along $T_{1}$ and $T_{2}$ (see (4) for an example).

  2. 2.

Let Let $p_{2}=1-p_{1}$ and $q_{1}=1-q_{2}$. Let

$A=\begin{pmatrix}p_{1}&0&p_{1}&0\\\ p_{2}&0&p_{2}&0\\\ 0&2q_{1}&0&2q_{1}\\\
0&2q_{2}&0&2q_{2}\end{pmatrix}\text{ and }v_{n}=\begin{pmatrix}|w_{n}|_{a}\\\
|w_{n}|_{b}\\\ |w_{n}|_{c}\\\ |w_{n}|_{d}\end{pmatrix}\qquad\implies
v_{n+1}=A\cdot v_{n}+e_{n},$

where $e_{n}$ is an "error" vector and is bounded.

  3. 3.

$A$ is primitive with exactly two eigenvalues $0<|\alpha_{2}|\leq
1<\alpha_{1}$. By the Perron-Frobenius theorem, there exists a right (resp.
left) eigenvector vectors $\mathbf{r}$ (resp. $\mathbf{\ell}$) of $A$ such
that: $\mathbf{\ell}\mathbf{r}=1$ and
$\lim_{n\to\infty}{\alpha^{-n}_{1}A^{n}}=\mathbf{r}\cdot{\leavevmode\nobreak\
}^{t}\mathbf{\ell}$.

  4. 4.

One cuts the sequence $u$ into words $(g_{n})_{n\in\mathbb{N}}$ following
Algorithm 1 and shows:

     * •

$|w_{0}\cdots w_{\ell_{n}}|=o(|g_{n}|)$ and
$\displaystyle\lim_{n\to\infty}|g_{n}|=+\infty$

     * •

$\dfrac{|u_{0}\cdots u_{n}|_{a}}{|u_{0}\cdots u_{n}|}=\dfrac{|w_{0}\cdots
w_{l_{n}-1}|_{a}+|g_{n}|_{a}}{|w_{0}\cdots
w_{l_{n}-1}|+|g_{n}|}=\dfrac{\frac{|g_{n}|_{a}}{|g_{n}|}+o(1)}{1+o(1)}\underset{n\to\infty}{\longrightarrow}r_{0}$

Input: $u=w_{0}w_{1}\cdots w_{n}\cdots=u_{0}u_{1}\cdots$

$\ell_{0}\leftarrow 0$

// initial left index

1 for  _each $n\in\mathbb{N}$_ do

$g_{n}\leftarrow u_{\ell_{n}}\cdots u_{n}$

// $|g_{n}|=(n+1)-|w_{0}\cdots w_{\ell_{n}-1}|$

2 if  _$|g_{n}|+1 >|w_{0}\cdots w_{\ell_{n}}|^{2}$ _ then

$\ell_{n+1}\leftarrow\ell_{n}+1$

// increment left index

3

4 else

$\ell_{n+1}\leftarrow\ell_{n}$

// keep left index

5

6

Algorithm 1 Cutting the sequence $\mathcal{O}$ into $(g_{n})_{n\in\mathbb{N}}$

∎

As a direct consequence of Theorem 1, we prove the existence of the
frequencies in the sequence introduced in Section 5 of [4]:

######  Definition 2 (BJM sequence [4]).

Let $U=(u_{n})_{n\in\mathbb{N}}$ be the self-descriptive sequence
$U=x_{0}^{u_{0}}x_{1}^{u_{1}}x_{2}^{u_{2}}\cdots$ defined by
$x_{0}=u_{0}=u_{1}=2$, and for all $n\in\mathbb{N}^{\star}$:

  1. i)

if $u_{n}=1$, then $x_{n}=1$ (resp. $x_{n}=2$) if $|u_{0}\cdots u_{n}|_{1}$ is
odd (resp. even),

  2. ii)

if $u_{n}=2$ then $x_{n}=1$.

In other words, the runs of size 2 (except the first one) are filled by 1, and
the runs of size 1 are filled alternatively by 1 and 2. The sequence
$X=(x_{n})_{n\in\mathbb{N}}$ is the directed sequence of $U$.

In [4], the authors showed that the frequencies of occurrence in $U$ cannot be
equal to those of its directing sequence $X$. However, the authors did not
prove the existence of the frequencies but only that they cannot be identical…
if they exist. Since $U$ is directed by $T_{1}=(12)^{\omega}$ and
$T_{2}=1^{\omega}$, it directly follows from Theorem 1 that:

######  Corollary 1.

Let $U$ be the sequence directed by $T_{1}=(12)^{\omega}$ and
$T_{2}=1^{\omega}$. Then

$\lim_{n\to\infty}\dfrac{|U_{0}\cdots
U_{n-1}|_{1}}{n}=\dfrac{7-\sqrt{17}}{4}\text{ and
}\lim_{n\to\infty}\dfrac{|X_{0}\cdots
X_{n-1}|_{1}}{n}=\dfrac{1+\sqrt{17}}{8}.$

###### Proof.

In that present case, $p_{1}=0.5$ and $q_{2}=0$. ∎

##  4 Conclusion and perspectives

In the present work, we have shown that self-descriptive sequences directed by
two periodic sequences have frequencies. We have also given an explicit
expression for these frequencies.

In future work, it will be interesting to extend this result to non-periodic
sequences. For example, Sturmian words, namely the aperiodic sequences with
the least number of finite factors, are good candidates for directing
sequences.

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