arXiv subjects

Federico Sau

Publications and source records attributed to Federico Sau.

At least 19 recordsLinked to original sources

arXiv2026

Aldous' spectral gap phenomena in stochastic exchange models

We consider a broad class of exchange dynamics with arbitrary weighted graph or hypergraph update structures, which includes the Kipnis--Marchioro--Presutti (KMP) model and the energies of Kac's walk on the sphere. These are conservative continuous-spin systems whose reversible measures are Dirichlet distributions. We prove that their spectral gap is always attained by a polynomial of degree at most two in the energy variables. Equivalently, through an intertwining with the associated discrete particle systems, the dominant mode is always represented by either a one-particle or a two-particle observable. We interpret this as a manifestation of an Aldous-type spectral gap phenomenon. In particular, the resulting two-particle spectral gap identity settles a recent conjecture of Alon and Puder. We also characterize sharply when one particle suffices and when a genuinely two-particle mode dominates. Under a common rescaling of the Dirichlet parameters, segment-like geometries are precisely those for which the gap is always of one-particle type, mean-field geometries are precisely those for which it is always of two-particle type, and all other geometries exhibit a nontrivial transition. The proof relies on the analysis of the so-called hidden model, a dual representation of the original process. The mechanism underlying the degree-two reduction is remarkably robust and extends to a much broader class of stochastic exchange models and their associated particle systems, including the harmonic process, the immediate exchange model, averaging-type processes, and nonreversible variants. Finally, we analyze boundary-driven versions of the KMP model, in which the bulk exchange dynamics interacts with reservoirs, generally resulting in nonreversible processes. In contrast to the conservative setting, we prove that the spectral gap of a boundary-driven model is always of one-particle type.

math.PR↗

arXiv2026

The critical KPZ scale for the Averaging Process

KPZ-type extremal fluctuations have recently been proved for several models of random walks in space-time random environments (RWRE) in $1+1$ dimensions. A general moment criterion predicts the spatial scale at which this behavior should occur, but does not by itself guarantee non-trivial fluctuations at that scale. In this paper, we show that the averaging process provides an instance in which this criterion is not sharp: because of a degeneracy in the update mechanism, the actual KPZ limit appears only beyond the predicted scale. The relevant critical contribution arises from the interplay of two distinct fluctuation mechanisms, a phenomenon that appears to be rather special within the RWRE setting. Our proof builds on Dobrushin-type local-time limit theorems for zero-sum additive functionals, refined estimates for tilted $k$-point motions, and the recent moment-based axiomatic characterization of Cole--Hopf solutions to the one-dimensional KPZ equation. We also identify the behavior on the two sides of the critical scale, thereby sharply separating the subcritical, critical, and supercritical regimes of the model.

math.PR↗

arXiv2026

Hydrodynamic limit of the directed exclusion process

We derive the Euler (hyperbolic) hydrodynamic limit for the directed exclusion process (DEP), a one-dimensional conservative interacting particle system that preserves particle-hole symmetry while breaking left-right symmetry. The proof relies on an explicit multi-process coupling, which guarantees a strong form of attractiveness and macroscopic stability for the particle system. Further open questions about DEP are briefly discussed.

math.PR↗

arXiv2026

$L^2$-cutoff for the averaging process on random regular graphs

We study the mixing time of the averaging process on a large random $d$-regular graph, $d\ge 3$, and prove an $L^2$-cutoff with an explicit cutoff time. Somewhat surprisingly, we uncover a phase transition at the finite, fixed degree $d=10$: for small degrees, i.e., $d\le 10$, the averaging process mixes as fast as the corresponding random walk on the same graph, whereas for $d> 10$ its $L^2$-mixing is governed by a different, slower mechanism. Our proof relies on a detailed asymptotic analysis of an auxiliary biased birth-and-death chain with a slow bond. We also briefly discuss an analogous phase transition for the $L^1$-mixing.

math.PR↗

arXiv2025

A universal cutoff phenomenon for mean-field exchange models

We study a broad class of high-dimensional mean-field exchange models, encompassing both noisy and singular dynamics, along with their dual processes. This includes a generalized version of the averaging process as well as some non-reversible extensions of classical exchange dynamics, such as the flat Kac model. Within a unified framework, we analyze convergence to stationarity from worst-case initial data in Wasserstein distance. Our main result establishes a universal cutoff phenomenon at an explicit mixing time, with a precise window and limiting Gaussian profile. The mixing time and profile are characterized in terms of the logarithm of the size-biased redistribution random variable, thus admitting a natural entropic interpretation.

math.PR↗

arXiv2025

Spectral gap of the KMP and other stochastic exchange models on arbitrary graphs

We present a simple strategy to derive universal bounds on the spectral gap of reversible stochastic exchange models on arbitrary graphs. The Kipnis-Marchioro-Presutti (KMP) model, the harmonic process (HP), and the immediate exchange model (IEM) are all examples that fall into this class. Our upper and lower bounds depend only on two features: worst-case linear statistics and a kinetic factor, which is, in essence, graph-independent. For the three aforementioned examples, these bounds are sharp, and even saturate to an identity for HP and IEM in some log-concave regimes. The proof -- which yields bounds for eigenvalues even in the non-reversible context -- crucially exploits the rigidity of the eigenstructure of these models and quantitative contraction rates of the corresponding hidden parameter models recently introduced in [DMFG24, GRT25].

math.PR↗

arXiv2024

One- and two-particle spectral gap identities for the symmetric inclusion process and related models

The symmetric inclusion process (SIP) models particles diffusing on a graph with mutual attraction. We recently showed that, in the log-concave regime (where diffusivity dominates interaction), the spectral gap of the conservative SIP matches that of a single particle. In this paper, our main result demonstrates that this identity generally fails outside this regime, but always holds for the non-conservative SIP, regardless of the interaction strength. When this one-particle spectral gap identity breaks down, we derive sharp bounds for the gap in terms of diffusivity, and reveal a two-particle spectral gap identity in the vanishing diffusivity limit. Our approach leverages the rigid eigenstructure of SIP, refined comparisons of Dirichlet forms for arbitrary diffusivity and particle numbers, and techniques from slow-fast system analysis. These findings extend to the dual interacting diffusion known as Brownian energy process, and shed some light on the spectral gap behavior for related Dirichlet-reversible systems on general, non-mean-field, geometries.

math.PR↗

arXiv2024

Repeated Block Averages: entropic time and mixing profiles

We consider randomized dynamics over the $n$-simplex, where at each step a random set, or block, of coordinates is evenly averaged. When all blocks have size 2, this reduces to the repeated averages studied in [CDSZ22], a version of the averaging process on a graph [AL12]. We study the convergence to equilibrium of this process as a function of the distribution of the block size, and provide sharp conditions for the emergence of the cutoff phenomenon. Moreover, we characterize the size of the cutoff window and provide an explicit Gaussian cutoff profile. To complete the analysis, we study in detail the simplified case where the block size is not random. We show that the absence of a cutoff is equivalent to having blocks of size $n^{Ω(1)}$, in which case we provide a convergence in distribution for the total variation distance at any given time, showing that, on the proper time scale, it remains constantly 1 up to an exponentially distributed random time, after which it decays following a Poissonian profile.

math.PR↗

arXiv2024

Concentration and local smoothness of the averaging process

We consider the averaging process on the discrete $d$-dimensional torus. On this graph, the process is known to converge to equilibrium on diffusive timescales, not exhibiting cutoff. In this work, we refine this picture in two ways. Firstly, we prove a concentration phenomenon of the averaging process around its mean, occurring on a shorter timescale than the one of its relaxation to equilibrium. Secondly, we establish sharp gradient estimates, which capture its fast local smoothness property. This is the first setting in which these two features of the averaging process -- concentration and local smoothness -- can be quantified. These features carry useful information on a number of large scale properties of the averaging process. As an illustration of this fact, we determine the limit profile of its distance to equilibrium and derive a quantitative hydrodynamic limit for it. Finally, we discuss their implications on cutoff for the binomial splitting process, the particle analogue of the averaging process.

math.PR↗

arXiv2024

Tiny fluctuations of the averaging process around its degenerate steady state

We analyze nonequilibrium fluctuations of the averaging process on $\mathbb T_\varepsilon^d$, a continuous degenerate Gibbs sampler running over the edges of the discrete $d$-dimensional torus. We show that, if we start from a smooth deterministic non-flat interface, recenter, blow-up by a non-standard CLT-scaling factor $\theta_\varepsilon=\varepsilon^{-(d/2+1)}$, and rescale diffusively, Gaussian fluctuations emerge in the limit $\varepsilon\to 0$. These fluctuations are purely dynamical, zero at times $t=0$ and $t=\infty$, and non-trivial for $t\in (0,\infty)$. We fully determine the correlation matrix of the limiting noise, non-diagonal as soon as $d\ge 2$. The main technical challenge in this stochastic homogenization procedure lies in a LLN for a weighted space-time average of squared discrete gradients. We accomplish this through a Poincar\'e inequality with respect to the underlying randomness of the edge updates, a tool from Malliavin calculus in Poisson space. This inequality, combined with sharp gradients' second moment estimates, yields quantitative variance bounds without prior knowledge of the limiting mean. Our method avoids higher (e.g., fourth) moment bounds, which seem inaccessible with the present techniques.

math.PR↗

arXiv2023

Scaling Limits of Random Walks, Harmonic Profiles, and Stationary Non-Equilibrium States in Lipschitz Domains

We consider the open symmetric exclusion (SEP) and inclusion (SIP) processes on a bounded Lipschitz domain $Ω$, with both fast and slow boundary. For the random walks on $Ω$ dual to SEP/SIP we establish: a functional-CLT-type convergence to the Brownian motion on $Ω$ with either Neumann (slow boundary), Dirichlet (fast boundary), or Robin (at criticality) boundary conditions; the discrete-to-continuum convergence of the corresponding harmonic profiles. As a consequence, we rigorously derive the hydrodynamic and hydrostatic limits for SEP/SIP on $Ω$, and analyze their stationary non-equilibrium fluctuations. All scaling limit results for SEP/SIP concern finite-dimensional distribution convergence only, as our duality techniques do not require to establish tightness for the fields associated to the particle systems.

math.PR↗

arXiv2023

On the meeting of random walks on random DFA

We consider two random walks evolving synchronously on a random out-regular graph of $n$ vertices with bounded out-degree $r\ge 2$, also known as a random Deterministic Finite Automaton (DFA). We show that, with high probability with respect to the generation of the graph, the meeting time of the two walks is stochastically dominated by a geometric random variable of rate $(1+o(1))n^{-1}$, uniformly over their starting locations. Further, we prove that this upper bound is typically tight, i.e., it is also a lower bound when the locations of the two walks are selected uniformly at random. Our work takes inspiration from a recent conjecture by Fish and Reyzin in the context of computational learning, the connection with which is discussed.

math.PR↗

arXiv2023

Spectral gap of the symmetric inclusion process

We consider the symmetric inclusion process on a general finite graph. Our main result establishes universal upper and lower bounds for the spectral gap of this interacting particle system in terms of the spectral gap of the random walk on the same graph. In the regime in which the gamma-like reversible measures of the particle systems are log-concave, our bounds match, yielding a version for the symmetric inclusion process of the celebrated Aldous' spectral gap conjecture originally formulated for the interchange process. Finally, by means of duality techniques, we draw analogous conclusions for an interacting diffusion-like unbounded conservative spin system known as Brownian energy process.

math.PR↗

arXiv2023

From quenched invariance principle to semigroup convergence with applications to exclusion processes

Consider a random walk on $\mathbb{Z}^d$ in a translation-invariant and ergodic random environment and starting from the origin. In this short note, assuming that a quenched invariance principle for the opportunely-rescaled walks holds, we show how to derive an $L^1$-convergence of the corresponding semigroups. We then apply this result to obtain a quenched pathwise hydrodynamic limit for the simple symmetric exclusion process on $\mathbb{Z}^d$, $d\ge 2$, with i.i.d. symmetric nearest-neighbors conductances $\omega_{xy}\in [0,\infty)$ only satisfying $$\mathbb{Q}(\omega_{xy}>0)>p_c\ ,$$ where $p_c$ is the critical value for bond percolation.

math.PR↗

arXiv2023

Full $\Gamma$-expansion of reversible Markov chains level two large deviations rate functionals

Let $\Xi_n \subset \mathbb R^d$, $n\ge 1$, be a sequence of finite sets and consider a $\Xi_n$-valued, irreducible, reversible, continuous-time Markov chain $(X^{(n)}_t:t\ge 0)$. Denote by $\mathscr P(\mathbb R^d) $ the set of probability measures on $\mathbb R^d$ and by $I_n\colon \mathscr P(\mathbb R^d) \to [0,+\infty)$ the level two large deviations rate functional for $X^{(n)}_t$ as $t\to\infty$. We present a general method, based on tools used to prove the metastable behaviour of Markov chains, to derive a full expansion of $I_n$ expressing it as $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/\theta^{(p)}_n)\, I^{(p)}$, where $I^{(p)}\colon \mathscr P(\mathbb R^d) \to [0,+\infty]$ represent rate functionals independent of $n$ and $\theta^{(p)}_n$ sequences such that $\theta^{(1)}_n \to\infty$, $\theta^{(p)}_n / \theta^{(p+1)}_n \to 0$ for $1\le p< q$. The speed $\theta^{(p)}_n$ corresponds to the time-scale at which the Markov chains $X^{(n)}_t$ exhibits a metastable behavior, and the $I^{(p-1)}$ zero-level sets to the metastable states. To illustrate the theory we apply the method to random walks in potential fields.

math.PR↗

arXiv2023

Fractional kinetics equation from a Markovian system of interacting Bouchaud trap models

We consider a partial exclusion process evolving on $\mathbb Z^d$ in a random trapping environment. In dimension $d\ge 2$, we derive the fractional kinetics equation \begin{equation*}\frac{\partial^\beta\rho_t}{\partial t^\beta} = \Delta \rho_t \end{equation*} as a hydrodynamic limit of the particle system. Here, $\frac{\partial^\beta}{\partial t^\beta}$, $\beta\in(0,1)$, denotes the fractional derivative in the Caputo sense. We thus exhibit a Markovian interacting particle system whose empirical density field rescales to a sub-diffusive equation corresponding to a non-Markovian process, the Fractional Kinetics process. In contrast, we show that, when $d=1$, the system rescales to the solution to \begin{equation*} \frac{\partial \rho_t}{\partial t}= \mathcal L_\beta \rho_t\ , \end{equation*} where $\mathcal L_\beta$ is the random generator of the singular quasi-diffusion known as FIN diffusion.

math.PR↗

arXiv2022

Cutoff for the Averaging process on the hypercube and complete bipartite graphs

We consider the averaging process on a graph, that is the evolution of a mass distribution undergoing repeated averages along the edges of the graph at the arrival times of independent Poisson processes. We establish cutoff phenomena for both the $L^1$ and $L^2$ distance from stationarity when the graph is a discrete hypercube and when the graph is complete bipartite. Some general facts about the averaging process on arbitrary graphs are also discussed.

math.PR↗

arXiv2022

Mixing of the Averaging process and its discrete dual on finite-dimensional geometries

We analyze the $L^1$-mixing of a generalization of the Averaging process introduced by Aldous. The process takes place on a growing sequence of graphs which we assume to be finite-dimensional, in the sense that the random walk on those geometries satisfies a family of Nash inequalities. As a byproduct of our analysis, we provide a complete picture of the total variation mixing of a discrete dual of the Averaging process, which we call Binomial Splitting process. A single particle of this process is essentially the random walk on the underlying graph. When several particles evolve together, they interact by synchronizing their jumps when placed on neighboring sites. We show that, given $k$ the number of particles and $n$ the (growing) size of the underlying graph, the system exhibits cutoff in total variation if $k\to\infty$ and $k=O(n^2)$. Finally, we exploit the duality between the two processes to show that the Binomial Splitting satisfies a version of Aldous' spectral gap identity, namely, the relaxation time of the process is independent of the number of particles.

math.PR↗