# Current fluctuations of a self-interacting diffusion on a ring

Francesco Coghi<EMAIL_ADDRESS>Nordita, KTH Royal Institute of
Technology and Stockholm University, Hannes Alfvéns väg 12, SE-106 91
Stockholm, Sweden

###### Abstract

We investigate fluctuations in the average speed or current of a self-
interacting diffusion (SID) on a ring, mimicking the non-Markovian behaviour
of an agent influenced by its own path. We derive the SID’s phase diagram,
showing a delocalisation-localisation phase transition from self-repelling to
self-attracting. Current fluctuations are analysed using: (i) an adiabatic
approximation, where the system reaches its stationary distribution before
developing current fluctuations, and (ii) an original extension of level 2.5
large deviations for Markov processes. The latter, combined with perturbation
theory, provides an upper bound to current fluctuations, which is tight in the
self-repelling region, while the former approximates current fluctuations well
in all regimes. Both methods accurately estimate the asymptotic variance and
suggests a phase transition at the onset of the localised regime.

###### Contents

  1. I Introduction
  2. II Model, typical behaviour and current observable
    1. II.1 Model formulation
    2. II.2 Typical behaviour
      1. II.2.1 Stationary Fokker–Planck equation
      2. II.2.2 Phase diagram for $\rho_{\text{inv}}$
    3. II.3 Current observable
  3. III Methods
    1. III.1 Adiabatic approximation
    2. III.2 Extended level 2.5 large deviations
      1. III.2.1 Markov processes
      2. III.2.2 Self-interacting diffusions
      3. III.2.3 Current contraction and the Euler–Lagrange equation
      4. III.2.4 Perturbation theory around $\rho_{\text{inv}}$
      5. III.2.5 Current rate function for small $\jmath$
  4. IV Results
    1. IV.1 Asymptotic variance
    2. IV.2 Comparison with Monte-Carlo simulations
  5. V Conclusion
  6. A Linear stability of $\rho_{\text{inv}}=(2\pi)^{-1}$
  7. B Connection to the mean-field XY model

##  I Introduction

We consider the Self-Interacting Diffusion (SID) on the ring $\mathcal{S}^{1}$
described by the process $\left\\{\theta_{t}\right\\}_{t\geq 0}$ solution of
the Stochastic Differential Equation (SDE)

$d\theta_{t}=\sqrt{D}dW_{t}+dt\int_{\mathcal{S}^{1}}2c\sin(\theta-\theta_{t})\rho_{t}(\theta)\,d\theta\,,$
(1)

where $D>0$ is the diffusion coefficient of the Brownian motion
$W_{t}=\int_{0}^{t}dW_{s}$ on the ring, $c\in\mathbb{R}^{+}$ a constant
determining the magnitude of the periodic drift and $\rho_{t}$ the empirical
occupation measure of the process, which is a scalar field marking the
fraction of time spent by the process on each and every point of the ring, up
to time $t$. Roughly, the process evolves stochastically on a ring and its
position at time $t$ is affected by the whole path of the process up to time
$t$ through $\rho_{t}$. We refer the reader to Sec. II for further details on
the process. We are interested in the study of the average speed, or current,
observable

$j_{t}=\frac{1}{t}\int_{0}^{t}d\theta_{s}\,,$ (2)

and aim to characterise its fluctuation behaviour in the long-time limit using
large deviation theory methods.

An SID is a type of non-Markov (memory-dependent) process that has been
introduced in the probability community in [1] as an example of a continuous-
time path-interaction (or reinforcement) process to model growing polymers,
extending [2, 3], and discrete-time reinforced random walks; we refer the
reader to [4] for a thorough historical account of self-interacting random
processes. Asymptotic properties of SIDs have been studied in [1, 5, 6, 7]
restricting to compact spaces and symmetric interaction potentials and
extended to $\mathbb{R}^{d}$ in [8, 9, 10] requiring confining potentials.
More recently, in [11, 12], refinements of Kramer’s law for SIDs have also
appeared (as well as in [13] for generalised overdamped Langevin systems).

In parallel, various forms of SIDs have been employed in statistical physics
to model autochemotaxis—the ability of organisms to communicate through local
secretions that form chemical trails, which would enter $\rho_{t}$.
Researchers have studied models where the signaling molecule remains in place
to mark the trail, as seen in ants [14, 15, 16, 17], as well as models where
the signal diffuses away, similar to bacteria [18, 19]. For the latter,
stochastic modeling has provided insights into the behaviour of living
microorganisms and self-propelled colloids, extending beyond traditional
deterministic Keller–Segel equations [20, 21, 22, 23, 24, 25, 26, 27].

Most existing studies focus on the asymptotic properties of SIDs and
autochemotactic systems, explaining typical behaviour. However, there remains
a significant gap in our understanding of atypical and rare events in these
systems, despite their crucial influence on future dynamics. To gain insights
into the dynamics of SIDs, a comprehensive theory of fluctuations and large
deviations of time-averaged quantities, such as (2), is necessary. It is
known, for instance, that long-range memory effects profoundly impact
fluctuation behaviour, diverging from the established large deviation theory
for Markovian cases [28, 29, 30, 31, 32]. Non-Markovian random walks have
revealed memory-driven effects [33, 34, 35, 36, 37], with studies showing a
different speed in the occurrence of large deviations. Recently, a long-time
large deviation theory for a discrete-time version of an SID has been derived
in [38, 39].

In this work, we provide a comprehensive study of fluctuations of the
observable (2) using large deviation theory. In particular,

  * •

In Sec. II we detail the model and give an overview of its typical, or
asymptotic, behaviour. We derive the phase diagram for the SID, revealing a
delocalisation-localisation phase transition from a regime where the SID is
self-repelled to a regime where it is self-attracted by its own trajectory.

  * •

In Sec. III, we introduce two methodologies to analyse the fluctuating
dynamics of (2), namely, (i) the adiabatic approximation, where the system
reaches its stationary distribution before current fluctuations develop, and
(ii) an extension of the so-called level 2.5 large deviations for Markov
processes.

  * •

By comparison with Monte Carlo simulations, we show in Sec. IV that the
latter, combined with perturbation theory, provides an upper bound on current
fluctuations, which is tighter the less localised the SID is, while the former
effectively approximates current fluctuations across all regimes.
Additionally, we also demonstrate that both methods accurately estimate the
asymptotic variance and suggest a phase transition at the onset of the
localised regime.

  * •

Finally, in Sec. V, we summarise our findings and outline open questions for
future research.

##  II Model, typical behaviour and current observable

We begin by formulating a general model for an SID based on the framework of
[1], and then we restrict our focus to the specific SDE displayed in (1).
Subsequently, we examine the asymptotic properties of (1) and discuss a phase
diagram that characterises the long-term behaviour of the empirical occupation
measure $\rho_{t}$.

###  II.1 Model formulation

We consider the general process $\left\\{X_{t}\right\\}_{t\geq
0}\in\mathcal{M}$ (for the moment a non-specified manifold) solution of the
following SDE:

$dX_{t}=\sigma dW_{t}-\frac{1}{t}\left(\int_{0}^{t}\nabla
V_{X_{s}}(X_{t})ds\right)dt,$ (3)

where $W_{t}=\int_{0}^{t}dW_{s}$ is a Brownian motion on $\mathcal{M}$
characterised by independent and Gausssian increments, i.e.,
$\mathbb{E}[dW_{t}]=0$ and
$\mathbb{E}[dW_{t}dW_{t^{\prime}}]=\delta(t-t^{\prime})dt$,
$\sigma:\mathcal{M}\times\mathcal{M}\rightarrow\mathcal{M}$ a matrix that
defines the coupling of the system to Gaussian white noise, and
$V_{x}:\mathcal{M}\times\mathcal{M}\rightarrow\mathbb{R}$ a ‘potential’
function responsible for the interaction between the current state of the
system $X_{t}$ and all previous states $X_{s}$ for $0<s<t$.

By introducing the empirical occupation measure

$\rho_{t}(x)=\frac{1}{t}\int_{0}^{t}\delta(X_{s}-x)\,ds\,,$ (4)

which is the fraction of time spent by the process on every position of the
manifold $\mathcal{M}$, (3) can be rewritten as

$dX_{t}=\sigma dW_{t}-\left(\int_{\mathcal{M}}\nabla
V_{x}(X_{t})\rho_{t}(x)\,dx\right)dt\,,$ (5)

where the time integral has been replaced by a spatial integral over
$\mathcal{M}$, clearly demonstrating the functional dependence of the process
on its own trajectory via the empirical occupation measure $\rho_{t}$. For
simplicity, we will refer to the entire process $\left\\{X_{t}\right\\}_{t\geq
0}$ as $X_{t}$, with context providing clarity.

$X_{t}$ alone is not a Markov process, as long-range interactions in time
directly enter the SDE and may drive the long-term behaviour of the system. We
also introduce the notation

$F_{\rho_{t}}(X_{t})=-\int_{\mathcal{M}}\nabla V_{x}(X_{t})\rho_{t}(x)\,dx\,,$
(6)

for the effective drift acting on the SID, which will be useful later on.

We now restrict ourselves to the case where
$\mathcal{M}\equiv\mathcal{S}^{1}$, i.e., the unidimensional torus or ring
$[0,2\pi)$ with periodic boundary conditions, and consider the periodic
potential

$V_{\theta}(\theta_{t})=2c\cos(\theta-\theta_{t}+\phi)\,,$ (7)

with $c\in\mathbb{R}^{+}$, $\phi\in[0,2\pi)$ and further restrict to
$\sigma=\sqrt{D}$ with $D\in\mathbb{R}^{+}$ (also notice the change
$X_{t}\rightarrow\theta_{t}$). We therefore rewrite (5) as

$d\theta_{t}=\sigma
dW_{t}-\left(\int_{S^{1}}\frac{d}{d\theta_{t}}2c\cos(\theta-\theta_{t}+\phi)\rho_{t}(\theta)\,d\theta\right)dt\,,$
(8)

with the drift in (6) given by the explicit form

$F_{\rho_{t}}(\theta_{t})=-\int_{S^{1}}\frac{d}{d\theta_{t}}2c\cos(\theta-\theta_{t}+\phi)\rho_{t}(\theta)\,d\theta\,.$
(9)

The constant $c$ determines the strenght of the potential, the greater the $c$
the stronger the interaction between the current state $\theta_{t}$ and all
previously visited states in $\rho_{t}$. The phase $\phi$, on the other hand,
determines the type of potential. Consider the diffusion that originates in
$\theta_{0}=0$. If $\phi=0$, the potential is self-repelling because the
particle starts from its maximum and is therefore attracted to the bottom of
the potential, which lies at a distance $\pi$ from the origin. The opposite
occurs when $\phi=\pi$; in this case, the particle originates already at the
bottom of the potential and is most self-attracted. All intermediate cases can
be explored by properly tuning $\phi$. It is also important to note that the
interaction potential is integrated over time, and therefore, the effective
drift experienced by the diffusing particle changes over time and is
ultimately determined by the random realisation of the noise in (8).

###  II.2 Typical behaviour

General results for the asymptotic behaviour of (5) for various classes of
potentials have been investigated in [1, 5, 6] with $\mathcal{M}$ considered
as a compact manifold using stochastic approximation methods [40, 41]. These
methods have been used to describe the asymptotic behaviour of the occupation
measure $\rho_{t}$ by a limiting nonautonomous differential equation (see (19)
below). These results have later been extended to $\mathbb{R}^{d}$ in [8, 9,
10] for strictly convex interaction potentials. Remarkably, [6] showed that
for symmetric potentials, $\rho_{t}$ almost surely converges to a local
minimum of a nonlinear free-energy functional, which has many critical points,
each chosen randomly with a certain probability. Specifically, when the
potential is symmetric and self-repelling (known as Mercer potentials in the
mathematics community), the free-energy functional is strictly convex,
ensuring $\rho_{t}$ almost surely reaches a single global minimum in the long-
time.

####  II.2.1 Stationary Fokker–Planck equation

Restricting to the case of (8) and assuming stationarity, the long-time
behaviour $\rho_{t}\rightarrow\rho_{\text{inv}}$ can also be studied solving
the following stationary non-linear Fokker–Planck equation:

$0=\frac{d}{d\theta}\left(F_{\rho_{\text{inv}}}(\theta)\rho_{\text{inv}}(\theta)\right)+\frac{D}{2}\frac{d^{2}\rho_{\text{inv}}(\theta)}{d\theta^{2}}\,,$
(10)

with the drift given by the spatial convolution

$F_{\rho_{\text{inv}}}(\theta)=\left(\int_{S^{1}}2c\sin(\theta^{\prime}-\theta+\phi)\rho_{\text{inv}}(\theta^{\prime})\,d\theta^{\prime}\right)\,.$
(11)

We remark that a time-dependent Fokker–Planck equation for the stochastic
process $\left\\{\theta_{t}\right\\}_{t\geq 0}$ alone is not defined because
the transition probability function will necessarily depend on the occupation
measure $\rho_{t}$.111A high-dimensional time-dependent Fokker–Planck equation
could in principle be rigorously written considering both the stochastic
process $\left\\{\theta_{t}\right\\}_{t\geq 0}$ and the empirical occupation
measure over time $\left\\{\rho_{t}\right\\}_{t\geq 0}$. Asymptotically,
however, due to the fact that the effective drift experienced by the particle
will eventually converge as $\rho_{t}$ in (4) carries a $t^{-1}$ factor, we
can assume $\rho_{t}\rightarrow\rho_{\text{inv}}$ and consider the stochastic
process $\left\\{\theta_{t}\right\\}_{t\rightarrow\infty}$ as a simple Markov
process whose drift is given by (11). In this limit, (10) is a well defined
object to study, but it will not necessarily explain all asymptotic dynamics
of (1), such as the non-stationary asymptotic evolution of $\rho_{t}$.

A solution of (10) is clearly given by the delocalised function
$\rho_{\text{inv}}(\theta)=(2\pi)^{-1}$, representing the uniform distribution
over the ring. However, this is not the only solution; another localised
solution also exists. By integrating (10) once we obtain

$J_{\rho_{\text{inv}}}=F_{\rho_{\text{inv}}}(\theta)\rho_{\text{inv}}(\theta)+\frac{D}{2}\frac{d\rho_{\text{inv}}(\theta)}{d\theta}\,,$
(12)

where $J_{\rho_{\text{inv}}}$ is a constant expressing the stationary
probability current of the process. A precise form of $J_{\rho_{\text{inv}}}$
can be obtained by imposing periodicity of the solution, viz.
$\rho_{\text{inv}}(0)=\rho_{\text{inv}}(2\pi)$, and reads

$J_{\rho_{\text{inv}}}=\frac{c\left(\alpha_{1}^{2}+\alpha_{2}^{2}\right)\sin\phi}{\pi}\,,$
(13)

where

$\displaystyle\alpha_{1}$
$\displaystyle=\int_{\mathcal{S}^{1}}\rho_{\text{inv}}(\theta)\cos\theta\,d\theta$
(14) $\displaystyle\alpha_{2}$
$\displaystyle=\int_{\mathcal{S}^{1}}\rho_{\text{inv}}(\theta)\sin\theta\,d\theta\,.$
(15)

Noticeably, we expect that if a stationary solution $\rho_{\text{inv}}$ exists
then $J_{\rho_{\text{inv}}}=0$ because the long-time drift
$F_{\rho_{\text{inv}}}$ in (11) is a conservative potential over
$\mathcal{S}^{1}$. This condition over (13) implies either $\phi=0$ or
$\phi=\pi$ for all $\alpha_{1}$ and $\alpha_{2}$ satisfying (14) and (15), or
$\alpha_{1}=\alpha_{2}=0$ for all $\phi\in[0,2\pi)$, which results in the
trivial uniform solution $\rho_{\text{inv}}(\theta)=(2\pi)^{-1}$.

Equation (12) for $J_{\rho_{\text{inv}}}=0$ can be rewritten as

$\frac{D}{2}\frac{d\ln\rho_{\text{inv}}(\theta)}{d\theta}=-F_{\rho_{\text{inv}}}(\theta)\,,$
(16)

and formally solved considering (14) and (15) via direct integration. The
solution is

$\begin{split}\rho_{\text{inv}}(\theta)&=\frac{e^{-\frac{2}{D}\int_{0}^{\theta}F_{\rho_{\text{inv}}}(\theta^{\prime})\,d\theta^{\prime}}}{\mathcal{Z}}\\\
&=\frac{e^{-\frac{4c}{D}\left(\cos(\theta-\phi)\alpha_{1}+\sin(\theta-\phi)\alpha_{2}\right)}}{\mathcal{Z}}\,,\end{split}$
(17)

with $\mathcal{Z}$ the normalisation constant. Equation (17), along with (14)
and (15), form a closed system of equations that can be solved as a function
of $c$, $D$, and $\phi$. Summarising, we have found the following solutions

$\rho_{\text{inv}}(\theta)=\begin{cases}(2\pi)^{-1}&\hskip
28.45274pt\forall\;\phi\in[0,2\pi)\\\
(\mathcal{Z})^{-1}e^{-\frac{4c}{D}\left(\cos(\theta-\phi)\alpha_{1}+\sin(\theta-\phi)\alpha_{2}\right)}&\hskip
28.45274pt\text{for}\;\phi\in\left\\{0,\pi\right\\}\end{cases}\,.$ (18)

These stationary solutions are not new. They already appeared in [1] and were
used to define a deterministic flow of a non-autonomous differential equation
of the form

$\frac{d\rho_{t}}{dt}=-\rho_{t}+\frac{e^{-\frac{2}{D}\int_{0}^{\theta}F_{\rho_{t}}(\theta^{\prime})\,d\theta^{\prime}}}{\mathcal{Z}}\,,$
(19)

to study the asymptotic behaviour of $\rho_{t}$. In the following, we discuss
the stability of the solution $\rho_{\text{inv}}(\theta)=(2\pi)^{-1}$, which
will help draw a phase diagram for $\rho_{\text{inv}}$ and discuss an
ergodicity-breaking phase transition controlled by the parameter $c/D$.

####  II.2.2 Phase diagram for $\rho_{\text{inv}}$

It is possible to prove via a linear stability approach (see Appendix A and
[42, 43, 44]) that the solution $\rho_{\text{inv}}(\theta)=(2\pi)^{-1}$ is
stable for

$\frac{c}{D}\cos\phi\geq-\frac{1}{2}\,.$ (20)

We show in the left panel of Fig. 1 the phase diagram of $\rho_{\text{inv}}$.
The region where the uniform solution, first line of (18), is stable is marked
in yellow and named the self-repelling region as the SID is repelled by its
own trajectory. The blue dashed line marks the region where the localised
solution, second line of (18), appears and is stable; this region is named
self-attracting because the SID is attracted by its own past and corresponds
to a non-ergodic phase for reasons that will become clear below. The third
white region cannot be understood by studying the stationary Fokker–Planck
equation in (10), and therefore, it is where we expect non-stationary
behaviour. We remark that for $\phi=0$ the uniform distribution is always
stable because of the positivity of both $c$ and $D$ whereas for $\phi=\pi$,
at the onset $c/D=1/2$, $\rho_{\text{inv}}(\theta)=(2\pi)^{-1}$ loses
stability in favour of
$\rho_{\text{inv}}(\theta)=(\mathcal{Z})^{-1}e^{-\frac{4c}{D}\left(\cos(\theta-\phi)\alpha_{1}+\sin(\theta-\phi)\alpha_{2}\right)}$.
In particular, in the following we show that the latter is not unique but come
with an $O(2)$ symmetry, i.e., it is valid for all $\alpha_{1}$ and
$\alpha_{2}$ such that $\alpha_{1}^{2}+\alpha_{2}^{2}=r^{2}$ where $r$ is a
parameter that only depends on the ratio $c/D$ (see right panel of Fig. 1).
Because of this property, the critical value $c/D=1/2$ marks an ergodicity-
breaking phase transition, viz. the invariant measure of the process is no
longer unique for $c/D\geq 1/2$.

Figure 1: On the left, the phase diagram of $\rho_{\text{inv}}$ shows three
distinguished phases: the self-repelling phase in yellow where the uniform
stationary distribution is stable, the self-attracting phase as a blue dashed
line where a new localised stationary distribution is stable and the uniform
one is no longer and a white region where nonstationary behaviour is expected.
On the right, the order parameter $r=\sqrt{\alpha_{1}^{2}+\alpha_{2}^{2}}$ is
plotted as a function of the control parameter $c/D$. As discussed in the
text, $c/D=1/2$ marks a second-order phase transition characterised by a
square-root singularity (see (28)), transitioning from the self-repelling
phase to a self-attracting phase where the SID is typically localised.

We consider
$\rho_{\text{inv}}(\theta)=(\mathcal{Z})^{-1}e^{-\frac{4c}{D}\left(\cos(\theta-\phi)\alpha_{1}+\sin(\theta-\phi)\alpha_{2}\right)}$.
We remark that $\alpha_{1}$ and $\alpha_{2}$ fully determine where the typical
value $\theta_{\text{max}}$ (the maximum) of the localised solution falls on
the ring via

$\alpha_{1}\tan(\theta_{\text{max}}-\phi)=-\alpha_{2}\,.$ (21)

By moving to polar coordinates $\alpha_{1}=r\cos\psi$ and
$\alpha_{2}=r\sin\psi$, (14) and (15) take the form

$\displaystyle r\cos\psi$
$\displaystyle=\int_{S^{1}}\cos\theta\frac{e^{-\frac{4cr}{D}\left(\cos(\psi+\phi-\theta)\right)}}{\mathcal{Z}}\,d\theta$
(22) $\displaystyle r\sin\psi$
$\displaystyle=\int_{S^{1}}\sin\theta\frac{e^{-\frac{4cr}{D}\left(\cos(\psi+\phi-\theta)\right)}}{\mathcal{Z}}\,d\theta\,.$
(23)

By replacing $\psi+\phi-\theta=\theta^{\prime}$ and recalling that
$\phi\in\left\\{0,\pi\right\\}$ we get

$r=\int_{S^{1}}\cos(\phi-\theta^{\prime})\frac{e^{-\frac{4cr}{D}\cos\theta^{\prime}}}{\mathcal{Z}}\,d\theta^{\prime}\,,$
(24)

for all $\psi\in[0,2\pi)$ or, in the equivalent cartesian formulation,
$\alpha_{1}^{2}+\alpha_{2}^{2}=r^{2}$.

In the following, we calculate the value of the parameter $r$. For $\phi=0$,
(24) can be written in terms of $I_{0}$ and $I_{1}$, i.e., the modified Bessel
functions of the first kind [45], as

$r=\int_{S^{1}}\cos\theta^{\prime}\frac{e^{-\frac{4cr}{D}\cos\theta^{\prime}}}{\mathcal{Z}}\,d\theta^{\prime}=-\frac{I_{1}\left(\frac{4cr}{D}\right)}{I_{0}\left(\frac{4cr}{D}\right)}\,.$
(25)

In particular, as $I_{0}\geq 0$ and $I_{1}\geq 0$ for $r\geq 0$, the only
solution to (25) is $r=0$ and therefore $\alpha_{1}=\alpha_{2}=0$: we re-
obtain the stable uniform distribution. On the other hand, for $\phi=\pi$,
(24) takes the form

$r=-\int_{S^{1}}\cos\theta^{\prime}\frac{e^{-\frac{4cr}{D}\cos\theta^{\prime}}}{\mathcal{Z}}\,d\theta^{\prime}=\frac{I_{1}\left(\frac{4cr}{D}\right)}{I_{0}\left(\frac{4cr}{D}\right)}\,,$
(26)

and in this case it is evident that a solution $r>0$ is admitted. The value of
$r$ can be calculated numerically by solving (26) and depends solely of the
ratio $c/D$. We plot the behaviour of $r$ as a function of $c/D$ in the right
panel of Fig. 1. Evidently, the greater the ratio (e.g., the smaller the
noise), the higher the value of $r$ which asymptotically tends to $1$.
Conversely, for very small ratios (e.g., high noise), $r$ approaches $0$.

The critical value $c/D=1/2$ that is marked in Fig. 1 can be derived from (26)
by taking a derivative with respect to $r$ at $r=0$ on both sides of the
equation and imposing that left and right hand side of the equation must be
the same. If we do so, we get

$1=\frac{d}{dr}\Big{|}_{r=0}\left(\frac{I_{1}(kr)}{I_{0}(kr)}\right)=\frac{k}{2}+\frac{k}{2}\frac{I_{2}(kr)}{I_{0}(kr)}\Big{|}_{r=0}-k\frac{I_{1}^{2}(kr)}{I_{0}^{2}(kr)}\Big{|}_{r=0}=\frac{k}{2}\,,$
(27)

where we use the shorthand notation $k=4c/D$ and the last equality follows
from properties of the modified Bessel functions of the first kind [45].
Evidently, we must have $k=k_{c}\coloneqq 2$ and therefore $c/D=1/2$ at
criticality. By expanding (26) for $r\rightarrow 0$ and keeping the lowest
meaningful order in $r$ we also get

$r=\sqrt{\frac{8(k-2)}{k^{3}}}\stackrel{{\scriptstyle k\approx
k_{c}}}{{\approx}}\left(2\kappa\right)^{\beta}\,,$ (28)

with $\kappa=(k-k_{c})/k_{c}$ and $\beta=1/2$.

In conclusion, $k_{c}=2$ or, equivalently, $c/D=1/2$ marks a second-order
phase transition characterised by a square-root singularity from a self-
repelling phase for $c/D<1/2$ where the only stationary solution of the SID is
uniform over the ring to a self-attracting phase where the SID typically
localises at $\theta_{\text{max}}$ solution of (21) with an $O(2)$ symmetry.
Given the fact that where the process localises depends on the realisation of
the noise and therefore the SID admits more than one stationary distribution,
the self-attracing phase is non-ergodic.

This phase transition is reminiscent of a disorder-order phase transition of a
mean-field XY model (also known as mean-field Kuramoto or Sakaguchi model). In
fact, the asymptotic behaviour is the same and, in the Appendix B, we show
that the stationary Dean’s equation of a mean-field XY model, in the limit of
infinitely many interacting spins, reduces to the stationary non-linear
Fokker-Planck equation in (10). Notably, works in this field, such as [46, 43,
44, 42] use a parameter very similar to $r$ in (24), representing the phase-
overlap responsible for the syncronisation among all spins in the lattice.
Similarly, here $r$ can be interpreted to represent the ‘synchronisation’ of
the SID with its own past trajectory.

The parallel between the SID of (5) and interacting many-particle systems is
more profound, extending as well to the much-studied McKean-Vlasov process in
probability theory. The McKean-Vlasov process is a stochastic process
described by an SDE whose drift depends on the law of the process itself,
i.e., the probability distribution of paths in the configuration space. Under
certain assumptions, such as a self-interacting potential, the law of a
McKean–Vlasov process converges to the invariant measure $\rho_{\text{inv}}$,
the solution of (10) (see [47, 10, 48]). The asymptotic properties of these
two processes can be studied similarly. For instance, it is customary in
probability to study the typical behaviour of an SID or a McKean–Vlasov
process by examining the behaviour of a many-particle system with mean-field
interactions [49].

###  II.3 Current observable

In the following, we focus on the fluctuations of the current observable in
(2) for $\phi=\pi$. We will move along the blue dashed line in Fig. 1 by
varying the ratio $c/D$ and study how the fluctuation behaviour of $j_{t}$
changes as we transition from the self-repelling region, characterised by a
delocalised solution for $\rho_{\text{inv}}$, to the self-attracting region,
where the solution $\rho_{\text{inv}}$ is localised. Although we only study
the behaviour for $\phi=\pi$, the results found to hold in the self-repelling
case extend to the whole self-repelling region of Fig. 1. This work does not
focus on the case where $\rho_{t}$ does not admit a stationary solution (white
region in the left panel of Fig. 1).

In the infinite-time limit we expect $j_{t}$ to converge to the stationary
probability current, or average speed, $J_{\rho_{\text{inv}}}$. This object,
in the relevant case of having a stationary behaviour
$\rho_{t}\rightarrow\rho_{\text{inv}}$, is shown above to be
$J_{\rho_{\text{inv}}}=0$ (see [50, 51] for general derivations in the case of
a drifted particle in a periodic potential). Hence, we have

$j_{t}\rightarrow 0\,,$ (29)

for almost all (in the probability sense) paths of the SID in (1). Although
the current concentrates at $0$ in the long time limit, $j_{t}$ fluctuates
around this typical value. It is this fluctuation behaviour that is of
interest here.

According to large deviation theory, for a (possibly driven) diffusing
particle on the ring subject to a periodic potential (not the SID in (1)), the
distribution of $j_{t}$ is shown in [52] to take the general form

$P(j_{t}=\jmath)=e^{-tI(\jmath)+o(t)}$ (30)

in the limit $t\rightarrow\infty$. The dominant scaling term in (30) is
exponential and its rate function $I$ is given by the limit

$I(j)=-\lim_{t\rightarrow\infty}\frac{1}{t}\ln P(j_{t}=\jmath)\,.$ (31)

This object controls the rate at which the probability $P(j_{t}=\jmath)$
decays to $0$ with time for any $\jmath\neq J_{\text{typ}}$. As a consequence,
$I\geq 0$ with equality reached only at $\jmath=J_{\text{typ}}$. Evidently,
the fact that $I>0$ for any other values of $\jmath$ shows that fluctuations
away from the typical value are exponentially unlikely at large times.

Although the long-time limit of the current $j_{t}$ is understood for the SID
in (1) as given in (29), its fluctuation behaviour has not yet been proved to
satisfy the large deviation relation in (30). In the following, we provide
evidence that such a large deviation behaviour may hold for the SID, at least
under certain assumptions.

##  III Methods

In this Section, we discuss two methods used in Sec. IV to study the
fluctuations of the current observable (2). The first method, adiabatic
approximation, is based on a separation of time scales. The second method is
based on an extension of level 2.5 of large deviation theory to non-Markov
processes of the general form (3).

###  III.1 Adiabatic approximation

By simulating the SID process in (1) in the relevant region of the phase
diagram in Fig. 1, i.e., $\phi=\pi$, we observe that the effective potential
(and consequently the drift) converges rapidly in both the self-repelling and
self-attracting phases. As we are interested in the long time fluctuations of
the current, it seems reasonable to assume a separation of time scales whereby
$\rho_{t}\rightarrow\rho_{\text{inv}}$, solution of (10), more rapidly than
$j_{t}\rightarrow J_{\rho_{\text{inv}}}(\equiv 0)$. Hence, the current in (2)
is now defined on the Markov process

$d\theta_{t}=\sqrt{D}dW_{t}+2c\left(\cos(\theta_{t})\alpha_{2}-\sin(\theta_{t})\alpha_{1}\right)dt\,,$
(32)

which has been obtained from (1) replacing $\rho_{t}$ with $\rho_{\text{inv}}$
and where $\alpha_{1}$ and $\alpha_{2}$ are given by (14) and (15)
respectively.

In the self-repelling case we have $\alpha_{1}=\alpha_{2}=0$ and the process
in (32) reduces to a trivial Brownian motion. In the self-attracting case
instead, $\alpha_{1}$ and $\alpha_{2}$ are not known a priori and their value
depend on the realisation of the noise driving the SID. However, they simply
are constant values and therefore we can study the fluctuation behaviour of
$j_{t}$ considering them as parameters.

As mentioned earlier, due to the Markovian character of (32) it is already
known that the fluctuations of $j_{t}$ take the general large deviation form
in (30). Therefore, we only need to calculate the rate function, namely
$I_{a}$ where the subscript $a$ stands for adiabatic.

To do so we use the Gärtner–Ellis theorem, which states that the rate function
is given by the Legendre–Fenchel transform

$I_{a}(\jmath)=\sup_{k\in\mathbb{R}}\left(k\jmath-\lambda_{a}(k)\right)\,,$
(33)

of the scaled cumulant generating function (SCGF) of $j_{t}$ given by

$\lambda_{a}(k)=\lim_{t\rightarrow\infty}\frac{1}{t}\ln\mathbb{E}\left[e^{tkj_{t}}\right]\,,$
(34)

when the latter exists and is differentiable with respect to $k$. For time-
additive observables of Markov processes, such as $j_{t}$, $\lambda_{a}$ is
known to be given by the dominant eigenvalue of the so-called tilted generator

$\mathcal{L}_{k}=2c\left(\cos(\theta_{t})\alpha_{2}-\sin(\theta_{t})\alpha_{1}\right)\left(\frac{d}{d\theta}+k\right)+\frac{D}{2}\left(\frac{d}{d\theta}+k\right)^{2}\,,$
(35)

which acts on bounded periodic functions of $[0,2\pi)$. $\mathcal{L}_{k}$ is a
linear operator obtained by modifying/tilting the infinitesimal generator of
(32). For further details on the theory we refer the reader to [30, 53] and
for details related to a driven diffusive particle in a periodic potential to
[52].

The eigenvalue problem that we need to solve reads

$\mathcal{L}_{k}r_{k}=\lambda_{a}(k)r_{k}\,,$ (36)

where $r_{k}$ is the dominant right eigenfunction of the tilted operator in
(35). We follow [52] and expand $r_{k}$ in a Bloch–Fourier series as

$r_{k}=\sum_{n=-\infty}^{\infty}c_{n}e^{in\theta}.$ (37)

Replacing (37) in (36) along with (35), using Euler formulas for cosine and
sine, and performing some algebraic manipulations and sum shifts, we obtain
the following tridiagonal system of recurrence relations:

$\begin{split}&c_{n}\left(-\frac{D}{2}n^{2}+kDin+\frac{D}{2}k^{2}-\lambda_{a}(k)\right)+c_{n-1}\left[c\alpha_{2}\left(i(n-1)+k\right)+c\alpha_{1}\left(1-n+ik\right)\right]+\\\
&\hskip
113.81102pt+c_{n+1}\left[c\alpha_{2}\left(i(n+1)+k\right)+c\alpha_{1}\left(n+1-ik\right)\right]=0\,.\end{split}$
(38)

This can be solved by truncating to some finite value of $n$ and studying the
kernel of the truncated tridiagonal matrix. The SCGF $\lambda_{a}$ is left as
a parameter in the system and is determined as the real value for which the
determinant of the matrix becomes zero. This condition is necessary and
sufficient for having a non-trivial kernel).

We notice that in the simple self-repelling case, since (32) reduces to a
Brownian motion, we already know that fluctuations have a Gaussian shape and
therefore

$I_{a}(\jmath)=\frac{\jmath^{2}}{2D}\,.$ (39)

In the self-attracting case instead, $I_{a}$ is a complicated function and it
will be calculated with the method explained above. In Sec. IV we will plot
the rate functions $I_{a}$ obtained in the self-repelling and self-attracting
case and compare them with simulations of the SID.

###  III.2 Extended level 2.5 large deviations

####  III.2.1 Markov processes

Given a Markov diffusion process $\left\\{Y_{t}\right\\}_{t\geq 0}$ in
$\mathbb{R}^{d}$, the level 2.5 of large deviations describes the likelihood
of joint fluctuations of the empirical occupation measure $\rho_{t}$ in (4)
and empirical current given by the formal relation

$\mathfrak{j}_{t}(y)=\frac{1}{t}\int_{0}^{t}\delta(Y_{t}-y)\circ dY_{t}\,.$
(40)

where $\circ$ symbolises the use of Stratonovich integration. The joint
distribution of empirical occupation and current
$P(\rho_{t}=\rho,\mathfrak{j}_{t}=j)$ is known to take a large deviation form
with an explicit rate function, namely $I_{\text{2.5}}$. 222Intuitively, the
form is explicit because for Markov processes drift and diffusion are uniquely
identified by the stationary distribution and the stationary current of the
process. This function can be calculated in different ways, either by
combining tilting with the Girsanov relation or using spectral methods
combined with the Gärtner–Ellis theorem. Regardless of the method applied, the
joint rate function is

$I_{\text{2.5}}[\rho,j]=\begin{cases}\frac{1}{2}\int_{\mathbb{R}^{d}}(j-J_{F,\rho})(\rho
D)^{-1}(j-J_{F,\rho})\,dy&\hskip 28.45274pt\text{if }\nabla\cdot j=0\\\
\infty&\hskip 28.45274pt\text{otherwise}\,,\end{cases}$ (41)

where

$J_{F,\rho}=F\rho-D\frac{\rho}{2}\,,$ (42)

is the probability current associated with the drift $F$ of the Markov process
$Y_{t}$ and density $\rho$ and $D$ is, in general, any invertible noise matrix
(not necessarily a scalar) given by $D=\sigma\sigma^{\top}$ entering (3). We
refer the to [54, 55, 56, 57] for details on the different derivations.

####  III.2.2 Self-interacting diffusions

In the following, we show a heuristic derivation of an extension of level 2.5
large deviations for SIDs with the general form (3). The validity of the
formula obtained below in (52) will be tested in Sec. IV. In the calculation
we apply a tilting approach. However, given that the noise appearing in (3) is
additive, we will not need to resort to the Girsanov relation. Furthermore, we
work on the validity of two assumptions:

  * •

There exists an ergodic process $\left\\{Y_{t}\right\\}_{t\geq 0}$ with
probability path measure $d\mathbb{P}_{Y_{t},\left[0,t\right]}$, such that its
empirical occupation measure $\rho_{t}$ converges to $\rho$ and its empirical
current $\mathfrak{j}_{t}$ in (40) converges to $j\coloneqq
J_{F^{\prime}_{\rho},\rho}$ where

$F^{\prime}_{\rho}(x)=-\int_{\mathbb{R}^{d}}\nabla
V^{\prime}_{y}(x)\rho(x)\,dx\,,$ (43)

is the spatial convolution of $V^{\prime}_{y}$, an auxiliary self-interacting
potential, and $\rho$ the invariant measure of $Y_{t}$. This auxiliary process
is a (Markov) diffusion with noise matrix $D$, but with a modified drift given
by (43) which can be cast as a function of $j$ as

$F^{\prime}_{\rho}(x)=\frac{j(x)}{\rho(x)}+\frac{D}{2}\nabla\ln\rho(x)\,.$
(44)

As a consequence, $\rho$ is unique and $\nabla\cdot j=0$.

  * •

The original path measure of $\left\\{X_{t}\right\\}_{t\geq 0}$ and the
auxiliary one of $\left\\{Y_{t}\right\\}_{t\geq 0}$ are absolutely continuous
with respect to each other. This implies that the Radon–Nikodym derivative is
well defined and its asymptotic behaviour is identified by a function
$I^{\text{(SID)}}_{\text{2.5}}$ such that the relation

$\frac{d\mathbb{P}_{X_{t},\left[0,t\right]}}{d\mathbb{P}_{Y_{t},\left[0,t\right]}}[x]=e^{-tI^{\text{(SID)}}_{\text{2.5}}[x]+o(t)}\,,$
(45)

with $x$ a general path, is well defined.

These two assumptions already seem to impose strict limitations on the types
of fluctuations we can analyse for SIDs. Notably, an SID is not a Markov
process, yet we are assuming that a fluctuation in a non-Markov process can be
represented by a Markov process, where the drift is uniquely generated by the
spatial convolution of an auxiliary interaction potential and the stationary
distribution $\rho$. At best, any information that the function
$I^{\text{(SID)}}_{\text{2.5}}$ provides regarding the likelihood of
fluctuations will only serve as an upper bound in terms of rate functions or a
lower bound in terms of actual probabilities. We will investigate in Sec. IV
how effective this approach is in providing a bound.

We start off by formally re-expressing the joint probability of density and
current in terms of the path measure of $X_{t}$, i.e.,

$\begin{split}P(\rho_{t}=\rho,\mathfrak{j}_{t}=j)&=\mathbb{E}_{X_{t}}\left[\delta_{\rho_{t}[X_{t}],\rho}\delta_{\mathfrak{j}_{t}[X_{t}],j}\right]\\\
&=\int
d\mathbb{P}_{X_{t},\left[0,t\right]}[x]\delta_{\rho_{t}[x],\rho}\delta_{\mathfrak{j}_{t}[x],j}\,,\end{split}$
(46)

and select paths $\left\\{X_{t}\right\\}_{t\geq 0}=x$ using Kronecker-$\delta$
(or indicator) functions. We now introduce the new tilted path measure
$d\mathbb{P}_{Y_{t},\left[0,t\right]}$ and express the newly-appearing
Radon–Nikodim derivative as in (45) to get

$P(\rho_{t}=\rho,\mathfrak{j}_{t}=j)\sim\mathbb{E}_{Y_{t}}\left[e^{-tI^{\text{(SID)}}_{\text{2.5}}[Y_{t}]}\delta_{\rho_{t}[Y_{t}],\rho}\delta_{\mathfrak{j}_{t}[Y_{t}],j}\right]\,.$
(47)

Eventually, noticing that
$\mathbb{E}_{Y_{t}}\left[\delta_{\rho_{t}[Y_{t}],\rho}\delta_{\mathfrak{j}_{t}[Y_{t}],j}\right]=1$
because of the properties of the auxiliary process $Y_{t}$, we get

$P(\rho_{t}=\rho,\mathfrak{j}_{t}=j)\sim
e^{-tI^{\text{(SID)}}_{\text{2.5}}[\rho,j]}\,,$ (48)

and we are only left with calculating $I^{\text{(SID)}}_{\text{2.5}}$.

In the simple case of additive noise Onsager–Machlup fluctuation theory
applies and we can therefore write

$\ln\left(\frac{d\mathbb{P}_{X_{t},\left[0,t\right]}}{d\mathbb{P}_{Y_{t},\left[0,t\right]}}[X_{t}]\right)=\ln\left(\frac{\exp^{-\int_{0}^{t}ds\frac{1}{2}\left[\left(\dot{X}_{s}-F_{\rho_{s}}\right)^{\top}\cdot
D^{-1}\left(\dot{X}_{s}-F_{\rho_{s}}\right)+\frac{1}{2}\nabla\cdot
F_{\rho_{s}}\right]}}{\exp^{-\int_{0}^{t}ds\frac{1}{2}\left[\left(\dot{X}_{s}-F^{\prime}_{\rho}\right)^{T}\cdot
D^{-1}\left(\dot{X}_{s}-F^{\prime}_{\rho}\right)+\frac{1}{2}\nabla\cdot
F^{\prime}_{\rho}\right]}}\right)\,,$ (49)

where we have used the Itô convention and therefore divergence terms appear in
the integrals and, only in this equation, we mark with $\dot{}$ the temporal
derivative. Furthermore, we notice that at the numerator the drift
$F_{\rho_{t}}$ is defined as in (6) and is non-Markovian in nature, whereas at
the denominator $F^{\prime}_{\rho}$ is given in (44) and is related to the
auxiliary Markov process introduced above. Expanding the products on the right
hand side of (49), merging numerator and denominator in a single expression
and introducing the definitions of $\rho_{t}$ in (4) and $\mathfrak{j}_{t}$ in
(40) we get

$\ln\left(\frac{d\mathbb{P}_{X_{t},\left[0,t\right]}}{d\mathbb{P}_{Y_{t},\left[0,t\right]}}[X_{t}]\right)=-t\int_{\mathbb{R}^{d}}\mathfrak{j}_{t}\cdot
D^{-1}\left(F^{\prime}_{\rho}-F_{\rho_{t}}\right)\,dx-\frac{t}{2}\int_{\mathbb{R}^{d}}\rho_{t}\left[\nabla\cdot\left(F_{\rho_{t}}-F^{\prime}_{\rho}\right)-(F_{\rho_{t}}-F^{\prime}_{\rho})\cdot
D^{-1}(F_{\rho_{t}}+F^{\prime}_{\rho})\right]\,dx\,.$ (50)

By replacing $F^{\prime}_{\rho}$ from (44) and $J_{F_{\rho_{t}},\rho}$ from
(42), with the change $F\rightarrow F_{\rho_{t}}$, we get

$\begin{split}&\ln\left(\frac{d\mathbb{P}_{X_{t},\left[0,t\right]}}{d\mathbb{P}_{Y_{t},\left[0,t\right]}}[X_{t}]\right)=\\\
&\hskip 14.22636pt-t\int_{\mathbb{R}^{d}}\left\\{\mathfrak{j}_{t}\cdot
D^{-1}\left(\frac{j-J_{F_{\rho_{t}},\rho}}{\rho}\right)+\frac{\rho_{t}}{2}\left[\nabla\cdot\left(\frac{-j+J_{F_{\rho_{t}},\rho}}{\rho}\right)-\left(\frac{j-J_{F_{\rho_{t}},\rho}}{\rho}\right)\cdot
D^{-1}\left(\frac{j-J_{F_{\rho_{t}},\rho}}{\rho}-2F^{\prime}_{\rho}\right)\right]\right\\}\,dx\,.\end{split}$
(51)

Finally, integrating by parts the second term in the integral, collecting the
common factor and using again (44) to isolate a current $j$, we get

$I^{\text{(SID)}}_{\text{2.5}}[\rho,j]=\begin{cases}\frac{1}{2}\int_{\mathbb{R}^{d}}(j-J_{F_{\rho},\rho})\cdot(\rho
D)^{-1}(j-J_{F_{\rho},\rho})\,dx&\hskip 28.45274pt\text{if }\nabla\cdot j=0\\\
\infty&\hskip 28.45274pt\text{otherwise}\,.\end{cases}$ (52)

This rate function is similar to the one known for diffusion processes in
(41), with the only difference that the instantaneous current
$J_{F_{\rho},\rho}$ now depends on a drift that is represented by a spatial
convolution with the density $\rho$.

For the case of interest in this paper, that is the SID in (1), (52) reduces
to

$I^{\text{(SID)}}_{\text{2.5}}[\rho,j]=\begin{cases}\frac{1}{2D}\int_{\mathcal{S}^{1}}\rho^{-1}J^{2}_{F_{\rho},\rho}\,d\theta+\frac{j^{2}}{2D}\int_{\mathcal{S}^{1}}\rho^{-1}\,d\theta-\frac{j}{D}\int_{\mathcal{S}^{1}}\rho^{-1}J_{F_{\rho},\rho}\,d\theta&\hskip
28.45274pt\text{if }\frac{dj}{d\theta}=0\\\ \infty&\hskip
28.45274pt\text{otherwise}\end{cases}\,,$ (53)

where it is evident that $j$ must be a constant over the ring. We can further
simplify the expression in (53) expanding $J_{F_{\rho},\rho}$ and manipulating
the expression appropriately to find

$I^{\text{(SID)}}_{\text{2.5}}[\rho,\jmath]=\begin{cases}\frac{D}{2}\int_{\mathcal{S}^{1}}\left[\rho\left(\frac{\rho^{\prime}}{2\rho}-\frac{F_{\rho}}{D}\right)^{2}+\frac{\jmath^{2}}{4\pi^{2}D^{2}}\rho^{-1}\right]\,d\theta&\hskip
28.45274pt\text{if }\frac{d\jmath}{d\theta}=0\\\ \infty&\hskip
28.45274pt\text{otherwise}\end{cases}\,,$ (54)

where, from now on, $\dot{}$ denotes a spatial derivative, and we have used
the fact that integration by parts gives
$\int_{\mathcal{S}^{1}}\rho^{-1}\dot{\rho}\,d\theta=0$. We also consider that
$j_{t}$ in (2) is related to $\mathfrak{j}_{t}$ via
$j_{t}=2\pi\mathfrak{j}_{t}$, and therefore, $\jmath=2\pi j$.

####  III.2.3 Current contraction and the Euler–Lagrange equation

We are interested in the current large deviations. Given the formula for the
extended level 2.5 rate function in (54), we can obtain the current rate
function via contraction as follows:

$I(\jmath)=\inf_{\begin{subarray}{c}\rho\\\
1=\int_{S^{1}}\rho\end{subarray}}I^{\text{(SID)}}_{\text{2.5}}\left[\rho,\jmath\right]\,,$
(55)

where we notice that no minimisation has to be carried out over the current
$\jmath$ because the latter is a constant over the circle as remarked just
below (53).

The minimisation in (55) can be tackled by introducing the Lagrangian
functional

$\mathcal{L}[\rho,\dot{\rho}]=\frac{D}{2}\bigintss_{\mathcal{S}^{1}}\left[\rho\left(\frac{\dot{\rho}}{2\rho}-\frac{F_{\rho}}{D}\right)^{2}+\rho^{-1}\frac{\jmath^{2}}{4\pi^{2}D^{2}}+k\left(\rho-1\right)\right]\,d\theta\,,$
(56)

where $k$ is the constant Lagrangian multiplier that fixes the normalisation
constraint, i.e., $0=\frac{\partial\mathcal{L}_{1}}{\partial
k}=\int_{\mathcal{S}^{1}}\rho\,d\theta-1$. The Euler–Lagrange equation is
calculated as follows:

$\begin{split}0&=\left(\frac{d}{d\epsilon}\mathcal{L}\left[\rho+\epsilon\eta,\dot{\rho}+\epsilon\dot{\eta}\right]\right)_{\epsilon=0}\\\
&=\bigintss_{\mathcal{S}^{1}}\left[\eta\left(\left(\frac{\dot{\rho}}{2\rho}-\frac{F_{\rho}}{D}\right)^{2}+k-\frac{\rho^{-2}\jmath^{2}}{4\pi^{2}D^{2}}\right)+2\rho\left(\frac{\dot{\rho}}{2\rho}-\frac{F_{\rho}}{D}\right)\left(\frac{\dot{\eta}}{2\rho}-\frac{2\dot{\rho}\eta}{(2\rho)^{2}}-\frac{2c}{D}\int_{\mathcal{S}^{1}}\sin(\theta_{1}-\theta)\eta(\theta_{1})\,d\theta_{1}\right)\right]d\theta\,,\\\
\end{split}$ (57)

and is valid for $\eta$-variations such that
$\int_{\mathcal{S}^{1}}\eta\,d\theta=0$. After integrating by parts the term
with $\dot{\eta}$, swapping integrals over the last term, and recognising that
the resulting expression must hold for all $\eta$-variations, we obtain the
following simplified Riccati-like expression for the Euler–Lagrange equation:

$0=\left(\frac{\dot{\rho}}{2\rho}-\frac{F_{\rho}}{D}\right)^{2}+k-\frac{\rho^{-2}\jmath^{2}}{4\pi^{2}D^{2}}-\frac{d}{d\theta}\left(\frac{\dot{\rho}}{2\rho}-\frac{F_{\rho}}{D}\right)-\frac{\dot{\rho}}{\rho}\left(\frac{\dot{\rho}}{2\rho}-\frac{F_{\rho}}{D}\right)-\frac{4c}{D}\int_{\mathcal{S}^{1}}\rho\left(\frac{\dot{\rho}}{2\rho}-\frac{F_{\rho}}{D}\right)\sin(\theta-\theta_{1})\,d\theta_{1}\,,$
(58)

whose solution, namely $\rho^{*}$, is the minimiser of
$I^{\text{(SID)}}_{\text{2.5}}$ for any fixed fluctuation $\jmath$ of the
current.

Evidently, the uniform distribution
$\rho^{*}(\theta)=\rho_{\text{inv}}(\theta)=(2\pi)^{-1}$ is always solution of
(58). This means that whenever $\rho_{\text{inv}}(\theta)=(2\pi)^{-1}$ is
stable, i.e., in the self-repelling region, fluctuations of the SID current
far from the typical value ($\jmath\neq J_{\rho_{\text{inv}}}=0$) are
generated by a process that preserves the uniform stationary distribution.
Hence, such a process will simply be a drifted Brownian motion over the ring.

On the opposite, the localised distribution
$\rho^{*}(\theta)=\rho_{\text{inv}}(\theta)=(\mathcal{Z})^{-1}e^{\frac{4c}{D}\left(\cos(\theta)\alpha_{1}+\sin(\theta)\alpha_{2}\right)}$
is only solution for $\jmath=0$. Therefore, in the self-attracting region,
fluctuations $\jmath\neq 0$ are generated by a process whose stationary
distribution $\rho^{*}$ is not given by $\rho_{\text{inv}}$. In the following
we show how to derive the minimiser $\rho^{*}$ solution of (58).

####  III.2.4 Perturbation theory around $\rho_{\text{inv}}$

An exact analytic solution of (58) is hard to find. For this reason, we
consider the perturbative analytic approximation for small currents $\jmath$
that follows:

$\rho=\rho_{\text{inv}}+\jmath\tilde{\rho}+o(\jmath)\ ,$ (59)

with the normalisation condition given by
$\int_{\mathcal{S}^{1}}\rho_{\text{inv}}\,d\theta=1$ and
$\int_{\mathcal{S}^{1}}\tilde{\rho}\,d\theta=0$. We do not know a priory
whether a perturbative approach will work in general, nor whether the current
should appear linearly in (59). In the worst case scenario, this approach will
provide us with a weak(er) upper bound for the current rate function.

Replacing (59) in (58) and keeping only the first order contribution in
$\jmath$ (the zeroth one is $0$) we get

$\displaystyle\begin{split}0&=\frac{\dot{\rho}_{\text{inv}}\dot{\tilde{\rho}}}{2}+\frac{2\rho_{\text{inv}}\tilde{\rho}F_{\rho_{\text{inv}}}^{2}+\rho_{\text{inv}}^{2}F_{\rho_{\text{inv}}}F_{\tilde{\rho}}}{D^{2}}+k\rho_{\text{inv}}\tilde{\rho}-\frac{\tilde{\rho}\ddot{\rho}_{\text{inv}}+\ddot{\tilde{\rho}}\rho_{\text{inv}}}{2}+\frac{2\rho_{\text{inv}}\tilde{\rho}\dot{F}_{\rho_{\text{inv}}}+\rho_{\text{inv}}^{2}\dot{F}_{\tilde{\rho}}}{D}+\\\
&\hskip
142.26378pt-\frac{4c}{D}\rho_{\text{inv}}^{2}\int_{S^{1}}\left[\sin(\theta-\theta_{1})\rho_{\text{inv}}(\theta_{1})\left(\frac{\dot{\tilde{\rho}}}{2\rho_{\text{inv}}}-\frac{\dot{\rho}_{\text{inv}}\tilde{\rho}}{2\rho_{\text{inv}}^{2}}-\frac{F_{\tilde{\rho}}}{D}\right)(\theta_{1})\right]\,d\theta_{1}\,,\end{split}$
(60)

which is still a complicated expression to deal with. We further consider the
following relations:

$\displaystyle\dot{\rho}_{\text{inv}}(\theta)$
$\displaystyle=\rho_{\text{inv}}(\theta)\frac{4c}{D}(\alpha_{2}\cos\theta-\alpha_{1}\sin\theta)$
(61) $\displaystyle\ddot{\rho}_{\text{inv}}(\theta)$
$\displaystyle=\rho_{\text{inv}}(\theta)\frac{16c^{2}}{D^{2}}(\alpha_{2}\cos\theta-\alpha_{1}\sin\theta)^{2}+\rho_{\text{inv}}(\theta)\frac{4c}{D}(-\alpha_{2}\sin\theta-\alpha_{1}\cos\theta)$
(62) $\displaystyle F_{\rho_{\text{inv}}}(\theta)$
$\displaystyle=-\alpha_{1}\sin\theta+\alpha_{2}\cos\theta$ (63)
$\displaystyle\dot{F}_{\rho_{\text{inv}}}(\theta)$
$\displaystyle=2c(-\alpha_{1}\cos\theta-\alpha_{2}\sin\theta)$ (64)
$\displaystyle F_{\tilde{\rho}}(\theta)$
$\displaystyle=2c(\bar{\alpha}_{2}\cos\theta-\bar{\alpha}_{1}\sin\theta)$ (65)
$\displaystyle\dot{F}_{\tilde{\rho}}(\theta)$
$\displaystyle=2c(-\bar{\alpha}_{2}\sin\theta-\bar{\alpha}_{1}\cos\theta)\,,$
(66)

along with the definitions

$\displaystyle\bar{\alpha}_{1}$
$\displaystyle=\int_{S^{1}}\tilde{\rho}(\theta)\cos\theta\,d\theta$ (67)
$\displaystyle\bar{\alpha}_{2}$
$\displaystyle=\int_{S^{1}}\tilde{\rho}(\theta)\sin\theta\,d\theta$ (68)
$\displaystyle\bar{\alpha}_{3}$
$\displaystyle=\int_{S^{1}}\tilde{\rho}(\theta)\cos^{2}\theta\,d\theta$ (69)
$\displaystyle\bar{\alpha}_{4}$
$\displaystyle=\int_{S^{1}}\tilde{\rho}(\theta)\cos\theta\sin\theta\,d\theta$
(70) $\displaystyle\bar{\alpha}_{5}$
$\displaystyle=\int_{S^{1}}\tilde{\rho}(\theta)\sin^{2}\theta\,d\theta$ (71)
$\displaystyle\alpha_{3}$
$\displaystyle=\int_{S^{1}}\rho_{\text{inv}}(\theta)\cos^{2}\theta\,d\theta$
(72) $\displaystyle\alpha_{4}$
$\displaystyle=\int_{S^{1}}\rho_{\text{inv}}(\theta)\cos\theta\sin\theta\,d\theta$
(73) $\displaystyle\alpha_{5}$
$\displaystyle=\int_{S^{1}}\rho_{\text{inv}}(\theta)\sin^{2}\theta\,d\theta\,,$
(74)

and obtain, after some simplifications, the following equation for the
perturbed density $\tilde{\rho}$:

$\begin{split}0&=-\ddot{\tilde{\rho}}(\theta)+\frac{4c}{D}\dot{\tilde{\rho}}(\theta)(\alpha_{2}\cos\theta-\alpha_{1}\sin\theta)-\frac{4c}{D}\tilde{\rho}(\theta)(\alpha_{1}\cos\theta+\alpha_{2}\sin\theta)+g(\theta)\,,\end{split}$
(75)

with

$\begin{split}g(\theta)&=\frac{4c}{D}\rho_{\text{inv}}(\theta)\Bigg{[}\frac{2c}{D}(\alpha_{2}\cos\theta-\alpha_{1}\sin\theta)(\bar{\alpha}_{2}\cos\theta-\bar{\alpha}_{1}\sin\theta)-2(\bar{\alpha}_{2}\sin\theta+\bar{\alpha}_{1}\cos\theta)+\\\
&\hskip
5.69046pt+\frac{4c}{D}\left(\alpha_{2}(\bar{\alpha}_{3}\sin\theta-\bar{\alpha}_{4}\cos\theta)+\alpha_{1}(\bar{\alpha}_{5}\cos\theta-\bar{\alpha}_{4}\sin\theta)+\bar{\alpha}_{2}(\alpha_{3}\sin\theta-\alpha_{4}\cos\theta)+\bar{\alpha}_{1}(\alpha_{5}\cos\theta-\alpha_{4}\sin\theta)\right)\Bigg{]}\
.\end{split}$ (76)

The homogeneous solution of (75) is

$\tilde{\rho}^{\text{G}}(\theta)=c_{1}y_{1}(\theta)+c_{0}y_{0}(\theta)\ ,$
(77)

with

$\displaystyle y_{0}(\theta)$
$\displaystyle=\mathcal{Z}\rho_{\text{inv}}(\theta)e^{-\frac{4c}{D}\alpha_{1}}$
(78)
$\displaystyle\begin{split}y_{1}(\theta)&=\mathcal{Z}\rho_{\text{inv}}(\theta)\left(\int_{0}^{\theta}e^{-\frac{4c}{D}(\alpha_{1}\cos\theta_{1}+\alpha_{2}\sin\theta_{1})}\,d\theta_{1}\right)\,,\end{split}$
(79)

where we remind the reader that
$\mathcal{Z}=\int_{\mathcal{S}^{1}}e^{\frac{4c}{D}\left(\cos\theta\alpha_{1}+\sin\theta\alpha_{2}\right)}\,d\theta$.
Using the Wronskian $W(y_{1},y_{0})=y_{1}\dot{y}_{0}-y_{0}\dot{y}_{1}=-y_{0}$,
we can find a particular solution to the complete heterogeneous problem in
(75), which reads

$\begin{split}\tilde{\rho}^{\text{P}}(\theta)=y_{1}(\theta)\int_{\mathcal{S}^{1}}g(\theta)\,d\theta-
y_{0}(\theta)\int_{\mathcal{S}^{1}}\frac{y_{1}(\theta)g(\theta)}{y_{0}(\theta)}\,d\theta\,.\end{split}$
(80)

The final (formal) solution of (75) is given by

$\begin{split}\tilde{\rho}(\theta)&=\tilde{\rho}^{\text{G}}(\theta)+\tilde{\rho}^{\text{P}}(\theta)\\\
&=\rho_{\text{inv}}(\theta)\left(\tilde{c}_{0}+c_{1}\int_{0}^{\theta}\rho_{\text{inv}}^{-1}(\theta_{1})\,d\theta_{1}+\int_{0}^{\theta}\rho_{\text{inv}}^{-1}(\theta_{1})\,d\theta_{1}\int
g(\theta)\,d\theta-\int\left[g(\theta)\int_{0}^{\theta}\rho_{\text{inv}}^{-1}(\theta_{1})\,d\theta_{1}\right]\,d\theta\right)\
,\end{split}$ (81)

with $\tilde{\rho}^{\text{G}}$ given in (77) and $\tilde{\rho}^{\text{P}}$ in
(80) (and $\tilde{c}_{0}=\mathcal{Z}e^{-\frac{4c}{D}\alpha_{1}}c_{0}$ is just
another constant). Eq. (81) along with (14), (15), and (67)–(74) form a closed
system of equations that can be solved to explicitely determine (81).
Additionally, by imposing the condition $\tilde{\rho}(0)=\tilde{\rho}(2\pi)$
and considering the notation $G(\theta)=\int g(\theta)\,d\theta$ and
$\bar{G}(\theta)=\mathcal{Z}^{-1}\int\left[g(\theta)\int_{0}^{\theta}\rho^{-1}_{\text{inv}}(\theta_{1})d\theta_{1}\right]\,d\theta$
we obtain the constant

$c_{1}=\mathcal{Z}e^{-\frac{4c}{D}\alpha_{1}}\left(\frac{\bar{G}(2\pi)-\bar{G}(0)}{\int_{\mathcal{S}^{1}}\rho^{-1}_{\text{inv}}(\theta)\,d\theta}\right)-G(2\pi)\,.$
(82)

Eventually, the constant $c_{0}$ (or $\tilde{c}_{0}$) is fixed by imposing the
normalisation constraint $\int_{0}^{2\pi}\tilde{\rho}(\theta)\,d\theta=0$.

Figure 2: Exemplary graphs of $\tilde{\rho}(\theta)$ for
$\rho_{\text{inv}}(\theta)=(\mathcal{Z})^{-1}e^{\frac{4cr}{D}\cos(\theta)}$—obtained
setting $\alpha_{1}=r$ solution of (26) and $\alpha_{2}=0$—localised where the
SID starts, at increasing values of $c/D$.

To summarise, the function $\tilde{\rho}$ in (81) is the small-current change
to the stationary distribution $\rho_{\text{inv}}$ characterising the optimal
solution to the minimisation problem (55). Therefore, according to the driven
process theory [30] for a small current $\jmath$, the density
$\rho^{*}=\rho_{\text{inv}}+\jmath\tilde{\rho}$ characterises the stationary
distribution of the auxiliary process that typically generates the current
$\jmath$. Notice that, because of the first assumption made in III.2.2, i.e.,
the existence of an auxiliary dynamics which typically converges to a
fluctuation of $\rho_{t}$ and $\mathfrak{j}_{t}$, and the general linear-
in-$\jmath$ form of the expansion (59), the inherently-Markovian dynamics of
the auxiliary process just determined may not be the most likely dynamics that
generate the current $\jmath$ in an SID. Therefore, in this sense, the
auxiliary process is only sub-optimal, as much as the rate function
$I(\jmath)$ in (55) is expected to only be an upper bound of the real current
rate function for the SID.

####  III.2.5 Current rate function for small $\jmath$

As mentioned in the last paragraph of III.2.2, in the self-repelling region
fluctuations of the SID current are generated by an auxiliary drifted Brownian
dynamics over the ring whose stationary distribution is uniform. Therefore, we
expect $\tilde{\rho}=0$. To check this we plug
$\rho_{\text{inv}}=(2\pi)^{-1}$, along with
$\alpha_{1}=\alpha_{2}=\alpha_{4}=0$ and $\alpha_{3}=\alpha_{5}=1/2$ from
(14), (15) and (72)–(74), in (81) and obtain

$\tilde{\rho}(\theta)=\left(\frac{4c}{\pi D}-\frac{4c^{2}}{\pi
D^{2}}\right)\left(\bar{\alpha}_{1}\cos\theta+\bar{\alpha}_{2}\sin\theta\right)\,,$
(83)

where $\bar{\alpha}_{1}$ and $\bar{\alpha}_{2}$ are the only two parameters
left to be determined. Solving for them from (67) and (68) we get
$\bar{\alpha}_{1}=\bar{\alpha}_{2}=0$ and, as a consequence, $\tilde{\rho}=0$
as expected.

In the self-attracting region where a localised distribution
$\rho_{\text{inv}}$ is reached, the form of $\tilde{\rho}$ will have to be
determined by solving the system of equations given by (81) along with (14),
(15) and (67)–(74). To build intuition, we plot exemplary graphs of
$\tilde{\rho}(\theta)$ in Fig. 2 for
$\rho_{\text{inv}}(\theta)=(\mathcal{Z})^{-1}e^{\frac{4cr}{D}\cos(\theta)}$
localised at the origin of the ring, where the SID starts, at increasing
values of $c/D$. From the plots, we notice that the form of $\tilde{\rho}$ is
always antisymmetric with respect to the origin with a positive part just to
the left of the origin and a negative one to the right. This means that the
final form of $\rho^{*}=\rho_{\text{inv}}+\jmath\tilde{\rho}$, which generates
the small current $\jmath$ will still be highly localised at $0$, but with a
small bulge on the left and a dip on the right. This form breaks the symmetry
of $\rho_{\text{inv}}$ and favours the generation of a counterclockwise
current $\jmath$. The opposite behaviour, i.e., clockwise current, is
generated similarly by considering the change
$\tilde{\rho}\rightarrow-\tilde{\rho}$. Additionally, from Fig. 2 we also
notice that the higher the ratio $c/D$ (greater localisation) the thinner the
peaks of $\tilde{\rho}$ and therefore the more localised the bulge and the dip
in $\rho^{*}$.

Now that we have a perturbative form for the optimal $\rho^{*}$ solution of
(55), we can use this to determine the shape of the rate function $I(\jmath)$
for small currents $\jmath$. To do so, we take the ansatz (59) (plus a
‘bookkeeping’ term of $O(\jmath^{2})$ that will anyway disappear), plug it
into $I^{\text{(SID)}}_{\text{2.5}}[\rho,\jmath]$ in (54) and keep terms up to
$O(\jmath^{2})$ to eventually get

$\displaystyle I_{\text{c}}(\jmath)$
$\displaystyle=I^{\text{(SID)}}_{\text{2.5}}[\rho_{\text{inv}}+\jmath\tilde{\rho}+\frac{\jmath^{2}}{2}\tilde{\tilde{\rho}}+o(\jmath^{2})]$
(84)
$\displaystyle=\frac{\jmath^{2}}{2}D\int_{\mathcal{S}^{1}}\left[\rho_{\text{inv}}(\theta)\left(\frac{1}{2}\frac{d}{d\theta}\left(\frac{\tilde{\rho}(\theta)}{\rho_{\text{inv}}(\theta)}\right)-\frac{F_{\tilde{\rho}}(\theta)}{D}\right)^{2}+\frac{\rho_{\text{inv}}^{-1}(\theta)}{4\pi^{2}D^{2}}\right]\,d\theta+o(\jmath^{2})\,.$
(85)

In doing so, we realise that the form of the rate function
$I_{\text{c}}(\jmath)$ should explicitly depend on the parameters $\alpha_{1}$
and $\alpha_{2}$, which fully determine $\rho_{\text{inv}}$ (along with other
parameters) around which we are expanding. However, as we explain later, this
dependence does not change the form of the rate function for small currents,
and therefore, we do not keep track of it. We remark that the factor

$\left(D\int_{\mathcal{S}^{1}}\left[\rho_{\text{inv}}\left(\frac{1}{2}\frac{d}{d\theta}\left(\frac{\tilde{\rho}}{\rho_{\text{inv}}}\right)-\frac{F_{\tilde{\rho}}}{D}\right)^{2}+\frac{\rho_{\text{inv}}^{-1}}{4\pi^{2}D^{2}}\right]\right)^{-1}\eqqcolon\sigma^{2}_{\text{c}}[\tilde{\rho}]\,,$
(86)

which multiplies $\jmath^{2}/2$ in (85), is a lower bound on the so-called
asymptotic variance, i.e,

$\sigma^{2}\coloneqq\lim_{t\rightarrow\infty}t\mathbb{E}\left[(j_{t}-J_{\rho_{\text{inv}}})^{2}\right]\,,$
(87)

of the current of the SID process converging to $\rho_{\text{inv}}$.
Evidently, from the quadratic form of $I_{c}(\jmath)$, fluctuations $\jmath$
are Gaussian, with a width determined by (86), around
$J_{\rho_{\text{inv}}}=0$.

##  IV Results

###  IV.1 Asymptotic variance

We discuss the behaviour of the asymptotic variance in (86) in both the self-
repelling ($c/D\leq 1/2$) and self-attracting ($c/D>1/2$) region. In the self-
repelling region, $\rho_{\text{inv}}(\theta)=(2\pi)^{-1}$ and
$\tilde{\rho}=0$, and therefore

$\sigma^{2}_{\text{c}}[0]=D\,,$ (88)

proving once again that the auxiliary process determining small-current
fluctuations is a simple drifted Brownian motion over the ring. In the self-
attracting region, for generic values of $c$ and $D$, $\sigma^{2}_{\text{c}}$
can only be evalueated numerically. Remarkably, although
$\sigma^{2}_{\text{c}}[\tilde{\rho}]$ has a ‘hidden’ dependence on
$\rho_{\text{inv}}$ through the parameters $\alpha_{1}$ and $\alpha_{2}$ that
fully determine $\rho_{\text{inv}}$, we can show numerically that
$\sigma^{2}_{\text{c}}$ does not change with varying $\alpha_{1}$ and
$\alpha_{2}$ given the relation $\alpha_{1}^{2}+\alpha_{2}^{2}=r^{2}$ with $r$
solution of (26). In other words, the $O(2)$ symmetry of $\rho_{\text{inv}}$
is inherited by the rate function and, in particular, by the asymptotic
variance.

In Fig. 3 we compare the diffusion coefficient of a pure Brownian motion
$dB_{t}=\sqrt{D}dW_{t}$ with the lower bound $\sigma^{2}_{\text{c}}$ we have
on the asymptotic variance of the SID in (86) and with the asymptotic variance
$\sigma^{2}_{\text{a}}$ of the adiabatic process in (32). The latter is
calculated as $\sigma^{2}_{\text{a}}=\lambda_{a}^{\prime\prime}(k)|_{k=0}$,
where $\lambda_{a}$ is given in (34). It shows numerically the same $O(2)$
symmetry of $\sigma^{2}_{\text{c}}$ and is therefore invariant as well with
respect to the choice of $\alpha_{1}$ and $\alpha_{2}$. Evidently, for
$c/D<1/2$ we have (88) and the three curves fully overlap. On the other hand,
for $c/D>1/2$, $\sigma^{2}_{c}$ and $\sigma^{2}_{\text{a}}$ start decreasing
exponentially fast, deviating from the pure diffusive behaviour. A sudden
transition in the behaviour of $\sigma^{2}_{c}$ and $\sigma^{2}_{\text{a}}$
arises at $c/D=1/2$, which we believe is consequence of the delocalisation-
localisation phase transition discussed in II.2.2.

The current state of our research does not include an analytical proof to
support the claim of the transition at the level of the asymptotic variance.
However, the numerical results are clear. Furthermore, the overall description
aligns with the transition observed in [58] (see Sect. 1.10 and Eq. 1.34)
regarding the transport coefficient of a noisy mean-field Kuramoto model. This
model’s associated Hamiltonian is the same as that of the $XY$ model discussed
in the last paragraph of II.2.2 and in Appendix B).

Figure 3: Lower bound $\sigma^{2}_{\text{c}}$ on the asymptotic variance of
the SID in (86) and asymptotic variance $\sigma^{2}_{\text{a}}$ of the
adiabatic process in (32) compared with the diffusion coefficient $D$ of a
pure Brownian motion. Given $c/D$, $\sigma^{2}_{\text{c}}$ and
$\sigma^{2}_{\text{a}}$ are both constant with varying $\alpha_{1}$ and
$\alpha_{2}$ such that $\alpha_{1}^{2}+\alpha_{2}^{2}=r^{2}$ with $r$ solution
of (26). [For the plot we chose $\alpha_{1}=r$ and $\alpha_{2}=0$.]

Additionally, the fact that the lower bound on the asymptotic variance
$\sigma^{2}_{\text{c}}$ of the SID always overlap with the asymptotic variance
of the adiabatic process $\sigma^{2}_{\text{a}}$ means that for small current
fluctuations the adiabatic process and the SID behave similarly at long times.
A more detailed comparison of these fluctuations follows.

###  IV.2 Comparison with Monte-Carlo simulations

In the self-repelling scenario, the long-time behaviour of the current
observable $j_{t}$ in (2) associated to an SID process is equivalent to
considering the ensemble average. In general, in a self-attracting scenario,
this is no longer true. By assuming an ensemble point of view, i.e.,
considering infinitely many realisations of the SID process in (1), the
distribution of the current $P_{e}(j_{t}=\jmath)$ can be written as

$P_{e}(j_{t}=\jmath)=\int_{\alpha_{1}^{2}+\alpha_{2}^{2}=r^{2}}P_{e}(j_{t}=\jmath|a_{t,1}=\alpha_{1},a_{t,2}=\alpha_{2})P(a_{t,1}=\alpha_{1},a_{t,2}=\alpha_{2})\,d\alpha_{1}d\alpha_{2},$
(89)

where we have introduced new random variables $a_{t,1}$ and $a_{t,2}$, whose
values depend on the entire realisation of the noise up to time $t$ in (1),
along with their joint distribution, and made use of the law of conditioning.
In general, a different convergence of the empirical measure of the SID, fully
determined by the values $\alpha_{1}$ and $\alpha_{2}$, could lead to
different fluctuations in the current.

On the other hand, our analytical and numerical results for the long-time
limit of $P(j_{t}=\jmath)$ show no dependence on $\alpha_{1}$ and
$\alpha_{2}$, i.e., the large deviation form

$P(j_{t}=\jmath)\approx e^{-tI}\,,$ (90)

holds, and the rate functions $I\equiv I_{\text{a}}$ or $I\equiv
I_{\text{c}}$, actually only the small-current limit of the latter, are
invariant with respect to a change of $\alpha_{1}$ and $\alpha_{2}$ as
mentioned above. We therefore expect that by running long simulations

$P_{e}(j_{t}=\jmath)\approx P(j_{t}=\jmath)\approx
e^{-tI_{\text{a}/\text{c}}(\jmath)}\,,$ (91)

which means that the current observable behaves ergodically.

To check this statement and our analytical and numerical results above, we run
Monte-Carlo simulations for long times $t$ for different values of $c/D$. For
each of these, starting from the histogram of currents $f_{t}(\jmath)$, which
is expected to converge (at long times) to $P_{e}(j_{t}=\jmath)$ in (89), the
function

$I_{t}(\jmath)=-\frac{1}{t}\ln f_{t}(\jmath)\,$ (92)

should converge to the true rate function of the SID process. In Fig. 4, we
compare this function with the current rate functions $I_{a}$ in (33)
associated to the adiabatic process, $I_{c}$ derived from the contraction over
the extended level 2.5 large deviation theory in the small-current limit of
(85), and $I_{D}(\jmath)=\jmath^{2}/(2D)$ of a pure Brownian motion over the
ring.

Figure 4: Monte Carlo simulations, plotted as $I_{t}(\jmath)$ in (92), were
obtained by running the SID process in (1) for $t=1000$ with $dt=0.025$,
c=$1$, and $D=4,1.67,1.25,1$ respectively for the four plots from top left to
bottom right. These simulations are compared with the current rate functions:
$I_{a}$ in (33) for the adiabatic process, $I_{c}$ derived from extended level
2.5 large deviation theory in the small-current limit of (85), and
$I_{D}(\jmath)$ of a pure Brownian motion over the ring.

From the plots, we notice that in the self-repelling phase ($c/D\leq 1/2$),
all curves and simulation data points perfectly overlap: long-time
fluctuations of the current are simply generated by pure Brownian motion over
the ring. In the self-attracting phase ($c/D>1/2$), we investigate three
scenarios with increasing localisation: $c/D=0.6$, $c/D=0.8$, and $c/D=1$.
Recall that the greater the localisation the sharper $\rho_{\text{inv}}$
around its typical value. In all cases, $I_{D}$ greatly underestimates $I_{t}$
in (92) providing a lower-bound that becomes weaker the greater the
localisation. For weak localisation, both $I_{a}$ and $I_{c}$ well estimate
simulations. For strong localisation, $I_{c}$ overestimates $I_{t}$, whereas
$I_{a}$ still captures the global behaviour of fluctuations. However, for
small current fluctuations, the rate functions $I_{a}$ and $I_{c}$ are
equivalent, as suggested by the asymptotic variances in Fig. 3. In conclusion,
$I_{a}$ effectively captures the behaviour of current fluctuations in all
regimes investigated, whereas $I_{c}$ provides an upper bound that is tight in
the self-repelling region and becomes weaker with increased localisation,
though it remains a good estimate.

##  V Conclusion

In this study, we have explored the fluctuations of the average speed or
current of a self-interacting diffusion on a ring. Our analysis revealed a
delocalisation-localisation phase transition characterised by a shift from a
self-repelling to a self-attracting regime. We employed two methods to analyse
the current fluctuations: the adiabatic approximation and an original
extension of level 2.5 large deviation theory along with perturbation theory.

Monte Carlo simulations confirmed our analytical predictions, showing that in
the self-repelling phase ($c/D<1/2$), the fluctuations of the current are
well-described by pure Brownian motion over the ring. In the self-attracting
phase ($c/D>1/2$), increasing localisation sharpened the stationary
distribution $\rho_{\text{inv}}$ around its typical value, influencing the
fluctuation behaviour. Our results indicated that while $I_{D}$, the current
rate function of a pure Brownian motion, can only be accounted as a weak lower
bound of simulations $I_{\text{t}}$, the adiabatic approximation
$I_{\text{a}}$ effectively captures the behaviour of current fluctuations
across all regimes and $I_{\text{c}}$ derived from the extended level 2.5
along with perturbation theory serves as an upper bound, especially tight for
weakly localised regimes. Finally, both the methods employed accurately
estimated the asymptotic variance and suggested a phase transition at the
onset of the localised regime.

Several open questions remain for future research. One key area of interest is
determining whether rare events, beyond small-to-moderate current
fluctuations, can be effectively captured by the adiabatic approximation and
accurately estimated using the extended level 2.5 large deviation theory.
Initial and transient noise paths could significantly influence the memory
term, potentially leading to rare events that are not easily detected by the
current framework. Additionally, gaining a deeper understanding of the non-
stationary behaviour in the white region of the phase diagram, where the
stationary Fokker–Planck equation fails, is crucial. Exploring how different
potential forms and interaction strengths affect the fluctuation behaviour of
SIDs could also provide valuable insights. Extending these methods to other
systems could also help uncover more about the universal properties of self-
interacting diffusions and assess whether the level 2.5 large deviation theory
can generally apply beyond the perturbative approach introduced in this work.
Lastly, alongside theoretical work, there is considerable scope for applied
research, particularly in understanding fluctuations of relevant observables
in reinforcement mechanisms used in autochemotactic agents [16], swarm
intelligence [59], and machine learning algorithms [60].

## Acknowledgments

The author is deeply grateful to R. Chetrite for introducing him to the topic
and for the initial discussions and insights that fueled this work. The author
also expresses gratitude to J. Barré, R. Harris, Sarah A. M. Loos, K. S.
Olsen, G. Pavliotis, H. Touchette and P. Vivo. for their engaging and
enlightening discussions. This work was partially supported by the Swedish
Research Council Grant No. 638-2013-9243.

##  Appendix A Linear stability of $\rho_{\text{inv}}=(2\pi)^{-1}$

We study the time-dependent Fokker–Planck equation

$\frac{\partial\rho_{t}(\theta)}{\partial
t}=-\frac{\partial}{\partial\theta}\left(\rho_{t}(\theta)F_{\rho_{t}}(\theta)\right)+\frac{D}{2}\frac{\partial^{2}\rho_{t}(\theta)}{\partial\theta^{2}}\,,$
(93)

with

$\rho_{t}(\theta)=\frac{1}{2\pi}+\epsilon\eta_{t}(\theta)\,,$ (94)

for $\epsilon\ll 1$ and $F_{\rho_{t}}$ defined in (9). Notice that although
(93) is not valid in general as discussed in the main text, it is valid in the
linear stability study that follows as we look at a perturbation in time only
around the stationary solution $\rho_{\text{inv}}=(2\pi)^{-1}$ as in (94).

Due to the normalisation of $\rho_{t}(\theta)$ it follows that $\eta_{t}$ has
zero mean, i.e.,

$\int_{S^{1}}\eta_{t}(\theta)\,d\theta=0\,.$ (95)

Additionally, $\eta_{t}$ is also $2\pi$-periodic, i.e.,
$\eta_{t}(0)=\eta_{t}(2\pi)$.

Replacing (94) in (93) we get, at order $\epsilon$,

$\begin{split}\frac{\partial\eta_{t}(\theta)}{\partial
t}=-\frac{c}{\pi}\int_{S^{1}}\cos(\theta_{1}-\theta+\phi)\eta_{t}(\theta_{1})\,d\theta_{1}+\frac{D}{2}\frac{\partial^{2}\eta_{t}(\theta)}{\partial\theta^{2}}\,.\end{split}$
(96)

We now expand the perturbation $\eta_{t}(\theta)$ in a Fourier series, i.e.,

$\eta_{t}(\theta)=d(t)e^{i\theta}+d^{*}(t)e^{-i\theta}+\eta^{\perp}_{t}(\theta)\,,$
(97)

where we only highlight the first oscillation mode.

We replace (97) in (96) and obtain

$\begin{split}\left(\frac{\partial d(t)}{\partial t}e^{i\theta}+\frac{\partial
d^{*}(t)}{\partial
t}e^{-i\theta}+\frac{\partial\eta^{\perp}_{t}(\theta)}{\partial
t}\right)&=-\frac{c}{\pi}\int_{S^{1}}\cos(\theta_{1}-\theta+\phi)\left(d(t)e^{i\theta_{1}}+d^{*}(t)e^{-i\theta_{1}}+\eta_{t}^{\perp}(\theta_{1})\right)\,d\theta_{1}+\\\
&\hskip
85.35826pt+\frac{D}{2}\left(-d(t)e^{i\theta}-d^{*}(t)e^{-i\theta}+\frac{\partial^{2}\eta_{t}^{\perp}(\theta)}{\partial\theta^{2}}\right)\,.\end{split}$
(98)

We only focus on the equation for the amplitude $d$ of the fundamental
oscillatory mode, which reads

$\begin{split}\frac{\partial d(t)}{\partial
t}e^{i\theta}&=-\frac{cd(t)}{\pi}\int_{S^{1}}\cos(\theta_{1}-\theta+\phi)e^{i\theta_{1}}\,d\theta_{1}-\frac{Dd(t)}{2}e^{i\theta}\\\
&=-\frac{cd(t)}{2\pi}\int_{S^{1}}\left(e^{i\theta_{1}}e^{i(-\theta+\phi)}+e^{-i\theta_{1}}e^{i(\theta-\phi)}\right)e^{i\theta_{1}}d\theta_{1}-\frac{Dd(t)}{2}e^{i\theta}\\\
&=-cd(t)e^{i\theta}e^{-i\phi}-\frac{Dd(t)}{2}e^{i\theta}\,.\end{split}$ (99)

Hence, the amplitude equation eventually becomes

$\frac{dd(t)}{dt}=-cd(t)e^{-i\phi}-\frac{D}{2}d(t)\,.$ (100)

It is interesting to notice that higher modes do not influence the linear
stability. This can be seen extrapolating the equation for $\eta_{t}^{\perp}$
from (98), which reads

$\begin{split}\frac{\partial\eta_{t}^{\perp}(\theta)}{\partial
t}&=-\frac{c}{\pi}\int_{S^{1}}\cos(\theta_{1}-\theta+\phi)\eta_{t}^{\perp}(\theta_{1})\,d\theta_{1}+\frac{D}{2}\frac{\partial^{2}\eta_{t}^{\perp}(\theta)}{\partial\theta^{2}}\\\
&\stackrel{{\scriptstyle(\ref{eq:PropertiesHigherMoodsLinearStability})}}{{=}}\frac{D}{2}\frac{\partial^{2}\eta_{t}^{\perp}(\theta)}{\partial\theta^{2}}\,.\end{split}$
(101)

If we now replace333$a_{k}=a_{-k}$ as $\eta_{t}^{\perp}\in\mathbb{R}$.

$\eta_{t}^{\perp}(\theta)=\sum_{k\geq 2}a_{k}(t)e^{ik\theta}\,,$ (102)

into (101), we get

$\frac{da_{k}}{dt}=-\frac{Dk^{2}}{2}a_{k}\,,$ (103)

which has solution

$a_{k}(t)=a_{k}(0)e^{-\frac{Dk^{2}}{2}t}\,.$ (104)

Eventually, we find that

$\eta_{t}^{\perp}(\theta)=\sum_{k\geq
2}a_{k}(0)e^{-\frac{Dk^{2}}{2}t}e^{ik\theta}\,,$ (105)

which means that higher harmonics decay to zero exponentially fast in time for
$D>0$, and so everything is controlled by the first oscillation mode. To study
the spectrum of (100) we write $d(t)=be^{\lambda t}$ and replace it in (100).
Linear stability is guaranteed by $\operatorname{Re}(\lambda)<0$, that is

$\operatorname{Re}(\lambda)=\operatorname{Re}\left(-ce^{-i\phi}-\frac{D}{2}\right)<0\iff\frac{c}{D}\cos\phi\geq-\frac{1}{2}\,.$
(106)

##  Appendix B Connection to the mean-field XY model

We introduce the mean-field XY model (or $\equiv O(2)$ model) on a finite
complete graph $\mathcal{G}$ of $N$ vertices. At each site $i$ of
$\mathcal{G}$ there is a spin taking values in $S^{1}$. The state space of the
whole system is $(S^{1})^{N}$. The corresponding mean-field Hamiltonian
$H_{N}:(S^{1})^{N}\rightarrow\mathbb{R}$ is

$H_{N}=-\frac{B}{N}\sum_{i,j\in\mathcal{G}}\cos(\theta^{j,N}-\theta^{i,N}+\phi)\,,$
(107)

where $\theta^{i,N}$ represents the phase of spin $i$, and $\phi$ is a phase-
shift affecting the two-body interaction between spins. In the following, we
will set $B=-2c$, and $\phi=\pi$.

We now consider the dynamics of each site individually [28], i.e.,

$\begin{split}d\theta_{t}^{i,N}&=-\frac{dH_{N}}{d\theta_{t}^{i,N}}+\sqrt{D}dW_{t}^{i,N}\\\
&=\frac{2c}{N}\sum_{j\in
V}\frac{d}{d\theta_{t}^{i,N}}\cos(\theta_{t}^{j,N}-\theta_{t}^{i,N})+\sqrt{D}dW_{t}^{i,N}\,,\end{split}$
(108)

where $\left(W_{t}^{i,N}\right)_{i=1}^{N}$ are $N$ independent Brownian
motions on $S^{1}$ such that

$\mathbb{E}\left[dW_{t}^{i,N},dW_{t+s}^{j,N}\right]=2\delta_{ij}\delta(s).$
(109)

Following [61], the Dean’s equation, i.e., the evolution equation for the
overall distribution $\rho_{t}^{N}$ of the spins given by

$\rho_{t}^{N}(\theta)=\frac{1}{N}\sum_{i=1}^{N}\rho_{t}^{i,N}=\frac{1}{N}\sum_{i=1}^{N}\delta(\theta_{t}^{i,N}-\theta)\,,$
(110)

where $\rho_{t}^{i,N}$ is the density function of a single diffusing particle,
reads

$\frac{\partial\rho_{t}^{N}(\theta)}{\partial
t}=-\frac{\partial}{\partial\theta}\left(2c\rho_{t}^{N}(\theta)\int_{S^{1}}\sin(\theta_{1}-\theta)\rho_{t}^{N}(\theta_{1})\,d\theta_{1}\right)+\frac{D}{2}\frac{\partial^{2}}{\partial\theta^{2}}\rho_{t}^{N}(\theta)-\frac{\partial}{\partial\theta}\left(\eta_{t}(\theta)\left(\rho_{t}^{N}\right)^{\frac{1}{2}}\right)\,,$
(111)

where $\eta_{t}$ is an uncorrelated white noise such that

$\mathbb{E}\left[\eta_{t}(\theta),\eta_{t+s}(\theta^{\prime})\right]=\frac{2}{N}\delta(\theta-\theta^{\prime})\delta(s)\,.$
(112)

Remarkably, in the limit for $N\rightarrow\infty$ the multiplicative noise
term in (111) disappears and, along with $t\rightarrow\infty$, we recover the
stationary Fokker–Planck equation (10) for the SID. Hence, the infinite-time
and infinite-size limit of a mean-field XY model is equivalent to the
stationary limit of an SID.

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# A Simple Condition for the Existence of Transversals

Arindam Biswas Chennai Mathematical Institute<EMAIL_ADDRESS>
###### Abstract

Hall’s Theorem is a basic result in Combinatorics which states that the
obvious necesssary condition for a finite family of sets to have a transversal
is also sufficient. We present a sufficient (but not necessary) condition on
the sizes of the sets in the family and the sizes of their intersections so
that a transversal exists. Using this, we prove that in a bipartite graph $G$
(bipartition $\\{A,B\\}$), without 4-cycles, if $\deg(v)\geq\sqrt{2e|A|}$ for
all $v\in A$, then $G$ has a matching of size $|A|$.

##  1 Introduction and Preliminaries

In this paper, we look at a sufficient (but not necessary) condition for a
finite family of sets to have a transversal, which only involves the sizes of
the sets and the sizes of their intersections.

We begin by recalling some theorems and definitions necessary for our
presentation.

######  Definition 1 (Transversal).

Let $\mathcal{F}=\\{S_{1},\ldots,S_{n}\\}$ be a finite family of sets. A
transversal for $\mathcal{F}$ is a tuple $(T,\phi)$, where
$T\subseteq\bigcup\mathcal{F}$, and $\phi:T\to\mathcal{F}$ is a bijection such
that $x\in\phi(x)$ for all $x\in T$.

######  Theorem 1 (Hall’s Theorem).

Let $\mathcal{F}=\\{S_{1},\ldots,S_{n}\\}$ be a finite family of sets. If for
each $\mathcal{F^{\prime}}\subseteq\mathcal{F}$,

$\left|\bigcup\mathcal{F^{\prime}}\right|\geq|\mathcal{F^{\prime}}|,$

then $\mathcal{F}$ has a transversal.

For an elegant proof of the above result, the reader may refer to [1].

Consider a finite set of objects $S$ and a property $P$. Suppose we have a
probability distribution on $S$. If an element $x\in S$ picked randomly
according to the distribution has property $P$ with positive probability, then
the set $S$ has atleast one element with property $P$. This is the basic idea
behind the “Probabilistic Method”, which has been used to produce elegant
proofs of many combinatorial results.

In what follows, we formalise this idea.

######  Definition 2 (Dependency Digraph).

Let $E_{1},\ldots,E_{n}$ be events in a probability space. The _dependency
digraph_ for these events is the graph $G=(V,E)$, where $V=\\{1,\ldots,n\\}$
and $E=\\{(i,j)\mid 1\leq i,j\leq n,\ i\neq j,\ E_{i}\textrm{ and
}E_{j}\textrm{ are dependent events}\\}$.

######  Lemma 1 (The Lovász Local Lemma).

Let $E_{1},\ldots,E_{n}$ be events in a probability space and $D$ be their
dependency digraph. Suppose there are numbers $0\leq x_{i}<1,\ 1\leq i\leq n$
such that

$P(E_{i})\leq x_{i}\prod_{(i,j)\in E(D)}(1-x_{j}),\textrm{ for all }1\leq
i\leq n.$

Then

$P\left(\bigwedge_{i=1}^{n}E_{i}^{c}\right)\geq\prod_{i=1}^{n}(1-x_{i}).$

In particular, the probability that none of the events $E_{i}$ occur is
positive.

######  Corollary 1.

Let $E_{1},\ldots,E_{n}$ be events in a probability space such that each event
is dependent on at most $d$ other events and $P(E_{i})\leq p,\textrm{ for all
}1\leq i\leq n$. If

$ep(d+1)\leq 1,$

then

$P\left(\bigwedge_{i=1}^{n}E_{i}^{c}\right)>0.$

Proofs of the above lemma and its corollary can be found in [2].

Now suppose we want to prove that in a set of objects, there is an object with
a certain property. If we can identify “bad events”, which prevent a randomly
picked object from having the property, then by the above corollary, it
suffices to show that each “bad event”, occurs with small enough probability
and it does not depend on too many other “bad events”.

In the next section, using the corollary, we obtain a simple sufficient
condition for the existence of a transversal for a finite family of _finite_
sets.

##  2 The Condition

We are now ready to present our result.

######  Theorem 2.

Let $\mathcal{F}=\\{S_{1},\ldots,S_{n}\\}$ be a finite family of finite sets.
Suppose there are numbers $l$ and $m$ such that $|S_{i}|\geq l$ and
$|S_{i}\cap S_{j}|\leq m$, for all $1\leq i,j\leq n$. If

$\sqrt{em(2n-3)}\leq l,$

then $\mathcal{F}$ has a transversal.

###### Proof.

Consider a random n-tuple $(X_{1},\ldots,X_{n})$, where $X_{i}$ is a uniform
random variable over $S_{i}$, for $1\leq i\leq n$. For $1\leq i,j\leq n$,
denote by $E_{ij}$ the event $\\{X_{i}=X_{j}\\}$. The random tuple is a
transversal when none of the events $E_{ij},\ 1\leq i<j\leq n$ occur. This is
exactly the event $\displaystyle\smashoperator[]{\bigwedge_{1\leq i<j\leq
n}^{}}E_{ij}^{c}$.

We have

$\displaystyle P(E_{ij})$ $\displaystyle=\frac{|S_{i}\cap
S_{j}|}{|S_{i}||S_{j}|}$ $\displaystyle\leq\frac{m}{l^{2}}.$

Each event $E_{ij}$ only depends on another event $E_{i^{\prime}j^{\prime}}$
if either $i=i^{\prime}$ or $j=j^{\prime}$. This can happen in
$(n-2)+(n-2)=2n-4$ ways. Applying the Local lemma with $p=\frac{m}{l^{2}}$ and
$d=2n-4$, we have

$P\left(\smashoperator[r]{\bigwedge_{1\leq i<j\leq n}^{}}E_{ij}^{c}\right)>0,$

if

$\displaystyle ep(d+1)=e\frac{m}{l^{2}}(2n-4+1)$ $\displaystyle\leq 1,\textrm{
i.e.}$ $\displaystyle em(2n-3)$ $\displaystyle\leq l^{2},\textrm{i.e.}$
$\displaystyle\sqrt{em(2n-3)}$ $\displaystyle\leq l.$

Thus the event $\displaystyle\smashoperator[]{\bigwedge_{1\leq i<j\leq
n}^{}}E_{ij}^{c}$ occurs with positive probability, i.e. the family
$\mathcal{F}$ has a transversal when $\sqrt{m(2n-3)}\leq l$. ∎

##  3 Matchings in Graphs

By considering the set of neighbours of each vertex of a graph as an element
of a family of sets, we can use the previous theorem to obtain the following
result about the existence of matchings which saturate one of the vertex sets
in a bipartition of a graph.

######  Theorem 3.

Let $G=(V,E)$ be a bipartite graph with no 4-cycles. Let $\\{A,B\\}$ be a
bipartition of $V$ and $n=|A|$. If $\deg(v)\geq\sqrt{2en}$ for all $v\in A$,
then $G$ has a matching which saturates $A$.

###### Proof.

For each $v\in A$, define

$S_{v}=\\{u\in B\mid\textrm{ u is a neighbour of v}\\}.$

Since $\\{A,B\\}$ is a bipartition, all neighbours of vertices in $A$ are in
$B$. Thus we have

$|S_{v}|=\deg(v)\geq\sqrt{2en}.$

######  Claim 1 (For all $u,v\in A,\ |S_{u}\cap S_{v}|\leq 1$).

Suppose $|S_{u}\cap S_{v}|\geq 2$. Then there are vertices
$u^{\prime},v^{\prime}\in B$ such that
$uu^{\prime},uv^{\prime},vu^{\prime},vv^{\prime}$ are edges in $G$. Thus $G$
contains the 4-cycle $uu^{\prime}vv^{\prime}u$. This contradicts our
assumption that $G$ is 4-cycle-free.

Now we apply our theorem from the previous section with $l=\sqrt{2en}$ and
$m=1$. Since

$\sqrt{em(2n-3)}=\sqrt{e(2n-3)}\leq\sqrt{2en}=l,$

the family $\mathcal{F}=\\{S_{v}\mid v\in A\\}$ has a transversal. Let
$(B^{\prime},\phi)$ be the transversal. Then

$M=\\{\\{v,\phi^{-1}(S_{v})\\}\mid v\in A\\}$

is a matching which saturates $A$. ∎

## References

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