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A. Faggionato

Publications and source records attributed to A. Faggionato.

At least 19 recordsLinked to original sources
arXiv2026

Moment bounds and exclusion processes on random Delaunay triangulations with conductances

We consider the Voronoi tessellation associated to a stationary simple point process on $\mathbb{R}^d$ with finite and positive intensity. We introduce the Delaunay triangulation as its dual graph, i.e.~the graph with vertex set given by the point process and with edges between vertices whose Voronoi cells share a $(d-1)$-dimensional face. We also attach to each edge a random weight, called conductance. We provide sufficient conditions ensuring the integrability w.r.t. the Palm distribution of several quantities as weighted degrees and associated moments. These integrability properties are crucial in applications, as they allow to apply existing results on random walks, resistor networks and the symmetric simple exclusion processes with random conductances (cf. [1,11,12,13,14,15]). For the latter, while the moment bounds ensure its well definiteness and several properties, the same does not hold when the jump rates are not symmetric, i.e. for a generic simple exclusion process. In this case, by using a criterion from [13], we recover construction and properties of the simple exclusion process whenever the simple point process has finite range of dependence and the conductances are uniformly upper bounded. This last result relies on a suitable analysis of Bernoulli bond percolation on the Delaunay triangulation inspired by [2,3]. All the above results remain valid if the simple point process is stationary with respect to integer translations.

math.PR↗
arXiv2024

An ergodic theorem with weights and applications to random measures, RW homogenization and IPS hydrodynamics on point processes

We prove a multidimensional ergodic theorem with weighted averages for the action of the group $\mathbb{Z}^d$ on a probability space. At level $n$ weights are of the form $n^{-d} ψ(j/n)$, $ j\in \mathbb{Z}^d$, for real functions $ψ$ decaying suitably fast. We discuss applications to random measures and to quenched stochastic homogenization of random walks on simple point processes with long-range random jump rates, allowing to remove the technical Assumption (A9) from \cite[Theorem~4.4]{Fhom1}. This last result concerns also some semigroup and resolvent convergence particularly relevant for the derivation of the quenched hydrodynamic limit of interacting particle systems via homogenization and duality. As a consequence we show that also the quenched hydrodynamic limit of the symmetric simple exclusion process on point processes stated in \cite[Theorem~4.1]{F_SEP} remains valid when removing the above mentioned Assumption (A9).

math.PR↗
arXiv2023

Scaling limit of the directional conductivity of random resistor networks on simple point processes

We consider random resistor networks with nodes given by a point process on $\mathbb{R}^d$ and with random conductances. The length range of the electrical filaments can be unbounded. We assume that the randomness is stationary and ergodic w.r.t. the action of the group $\mathbb{G}$, given by $\mathbb{R}^d$ or $\mathbb{Z}^d$. This action is covariant w.r.t. translations on the Euclidean space. Under minimal assumptions we prove that a.s. the suitably rescaled directional conductivity of the resistor network along the principal directions of the effective homogenized matrix $D$ converges to the corresponding eigenvalue of $D$ times the intensity of the point process. More generally, we prove a quenched scaling limit of the directional conductivity along any vector $e\in {\rm Ker}(D) \cup {\rm Ker}(D)^\perp$. Our results cover plenty of models including e.g. the standard conductance model on $\mathbb{Z}^d$ (also with long filaments), the Miller-Abrahams resistor network for conduction in amorphous solids (to which we can now extend the bounds in agreement with Mott's law previously obtained in \cite{CP1,FM,FSS} for Mott's random walk), resistor networks on the supercritical cluster in lattice and continuum percolations, resistor networks on crystal lattices and on Delaunay triangulations.

math-ph↗
arXiv2020

Stochastic homogenization in amorphous media and applications to exclusion processes

We consider random walks on marked simple point processes with symmetric jump rates and unbounded jump range. We prove homogenization properties of the associated Markov generators. As an application, we derive the hydrodynamic limit of the simple exclusion process given by multiple random walks as above, with hard-core interaction, on a marked Poisson point process. The above results cover Mott variable range hopping, which is a fundamental mechanism of phonon-induced electron conduction in amorphous solids as doped semiconductors. Our techniques, based on an extension of two-scale convergence, can be adapted to other models, as e.g. the random conductance model.

math.PR↗
arXiv2018

A unifying picture of generalized thermodynamic uncertainty relations

The thermodynamic uncertainty relation is a universal trade-off relation connecting the precision of a current with the average dissipation at large times. For continuous time Markov chains (also called Markov jump processes) this relation is valid in the time-homogeneous case, while it fails in the time-periodic case. The latter is relevant for the study of several small thermodynamic systems. We consider here a time-periodic Markov chain with continuous time and a broad class of functionals of stochastic trajectories, which are general linear combinations of the empirical flow and the empirical density. Inspired by the analysis done in our previous work [1], we provide general methods to get local quadratic bounds for large deviations, which lead to universal lower bounds on the ratio of the diffusion coefficient to the squared average value in terms of suitable universal rates, independent of the empirical functional. These bounds are called "generalized thermodynamic uncertainty relations" (GTUR's), being generalized versions of the thermodynamic uncertainty relation to the time-periodic case and to functionals which are more general than currents. Previously, GTUR's in the time-periodic case have been obtained in [1, 27, 42]. Here we recover the GTUR's in [1, 27] and produce new ones, leading to even stronger bounds and also to new trade-off relations for time-homogeneous systems. Moreover, we generalize to arbitrary protocols the GTUR obtained in [42] for time-symmetric protocols. We also generalize to the time-periodic case the GTUR obtained in [19] for the so called dynamical activity, and provide a new GTUR which, in the time-homogeneous case, is stronger than the one in [19]. The unifying picture is completed with a comprehensive comparison between the different GTUR's.

cond-mat.stat-mech↗
arXiv2018

Level 2.5 large deviations for continuous time Markov chains with time periodic rates

We consider an irreducible continuous time Markov chain on a finite state space and with time periodic jump rates and prove the joint large deviation principle for the empirical measure and flow and the joint large deviation principle for the empirical measure and current. By contraction we get the large deviation principle of three types of entropy production flow. We derive some Gallavotti-Cohen duality relations and discuss some applications.

math.PR↗
arXiv2017

Large deviation principles and fluctuation theorems for currents in semi-Markov processes

In this short note we consider semi-Markov processes satisfying the condition of direction-time independence (Markov renewal processes). We derive large deviation principles and fluctuation theorems for the empirical current and the empirical currents along cycles. Our derivation is based on the joint LDP for the empirical measure and flow recently proved in \cite{MZ}.

cond-mat.stat-mech↗
arXiv2017

Einstein relation and linear response in one-dimensional Mott variable-range hopping

We consider one-dimensional Mott variable-range hopping with a bias, and prove the linear response as well as the Einstein relation, under an assumption on the exponential moments of the distances between neighboring points. In a previous paper \cite{FGS} we gave conditions on ballisticity, and proved that in the ballistic case the environment viewed from the particle approaches, for almost any initial environment, a given steady state which is absolutely continuous with respect to the original law of the environment. Here, we show that this bias--dependent steady state has a derivative at zero in terms of the bias (linear response), and use this result to get the Einstein relation. Our approach is new: instead of using e.g. perturbation theory or regeneration times, we show that the Radon-Nikodym derivative of the bias--dependent steady state with respect to the equilibrium state in the unbiased case satisfies an $L^p$-bound, $p>2$, uniformly for small bias. This $L^p$-bound yields, by a general argument not involving our specific model, the statement about the linear response.

math.PR↗
arXiv2017

Mixing length scales of low temperature spin plaquettes models

Plaquette models are short range ferromagnetic spin models that play a key role in the dynamic facilitation approach to the liquid glass transition. In this paper we perform a rigorous study of the thermodynamic properties of two dimensional plaquette models, the square and triangular plaquette models. We prove that for any positive temperature both models have a unique infinite volume Gibbs measure with exponentially decaying correlations. We analyse the scaling of three a priori different static correlation lengths in the small temperature regime, the mixing, cavity and multispin correlation lengths. Finally, using the symmetries of the model we determine an exact self similarity property for the infinite volume Gibbs measure.

math-ph↗
arXiv2016

Analysis of random walks in dynamic random environments via $L^2$-perturbations

We consider random walks in dynamic random environments given by Markovian dynamics on $\mathbb{Z}^d$. We assume that the environment has a stationary distribution $μ$ and satisfies the Poincaré inequality w.r.t. $μ$. The random walk is a perturbation of another random walk (called "unperturbed"). We assume that also the environment viewed from the unperturbed random walk has stationary distribution $μ$. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of $L^2$-bounded perturbations of Markov processes by means of the so-called Dyson-Phillips expansion.

math.PR↗
arXiv2014

Flows, currents, and cycles for Markov Chains: large deviation asymptotics

We consider a continuous time Markov chain on a countable state space. We prove a joint large deviation principle (LDP) of the empirical measure and current in the limit of large time interval. The proof is based on results on the joint large deviations of the empirical measure and flow obtained in \cite{BFG}. By improving such results we also show, under additional assumptions, that the LDP holds with the strong L^1 topology on the space of currents. We deduce a general version of the Gallavotti-Cohen (GC) symmetry for the current field and show that it implies the so-called fluctuation theorem for the GC functional. We also analyze the large deviation properties of generalized empirical currents associated to a fundamental basis in the cycle space, which, as we show, are given by the first class homological coefficients in the graph underlying the Markov chain. Finally, we discuss in detail some examples.

math.PR↗
arXiv2014

Universality for one-dimensional hierarchical coalescence processes with double and triple merges

We consider one-dimensional hierarchical coalescence processes (in short HCPs) where two or three neighboring domains can merge. An HCP consists of an infinite sequence of stochastic coalescence processes: each process occurs in a different "epoch" and evolves for an infinite time, while the evolutions in subsequent epochs are linked in such a way that the initial distribution of epoch $n+1$ coincides with the final distribution of epoch $n$. Inside each epoch a domain can incorporate one of its neighboring domains or both of them if its length belongs to a certain epoch-dependent finite range. Assuming that the distribution at the beginning of the first epoch is described by a renewal simple point process, we prove limit theorems for the domain length and for the position of the leftmost point (if any). Our analysis extends the results obtained in [Ann. Probab. 40 (2012) 1377-1435] to a larger family of models, including relevant examples from the physics literature [Europhys. Lett. 27 (1994) 175-180, Phys. Rev. E (3) 68 (2003) 031504]. It reveals the presence of a common abstract structure behind models which are apparently very different, thus leading to very similar limit theorems. Finally, we give here a full characterization of the infinitesimal generator for the dynamics inside each epoch, thus allowing us to describe the time evolution of the expected value of regular observables in terms of an ordinary differential equation.

math-ph↗
arXiv2012

Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models

We consider a family X^{(n)}, n \in \bbN_+, of continuous-time nearest-neighbor random walks on the one dimensional lattice Z. We reduce the spectral analysis of the Markov generator of X^{(n)} with Dirichlet conditions outside (0,n) to the analogous problem for a suitable generalized second order differential operator -D_{m_n} D_x, with Dirichlet conditions outside a given interval. If the measures dm_n weakly converge to some measure dm_*, we prove a limit theorem for the eigenvalues and eigenfunctions of -D_{m_n}D_x to the corresponding spectral quantities of -D_{m_*} D_x. As second result, we prove the Dirichlet-Neumann bracketing for the operators -D_m D_x and, as a consequence, we establish lower and upper bounds for the asymptotic annealed eigenvalue counting functions in the case that m is a self--similar stochastic process. Finally, we apply the above results to investigate the spectral structure of some classes of subdiffusive random trap and barrier models coming from one-dimensional physics.

math.PR↗
arXiv2011

Gallavotti-Cohen-Type symmetry related to cycle decompositions for Markov chains and biochemical applications

We slightly extend the fluctuation theorem obtained in \cite{LS} for sums of generators, considering continuous-time Markov chains on a finite state space whose underlying graph has multiple edges and no loop. This extended frame is suited when analyzing chemical systems. As simple corollary we derive in a different method the fluctuation theorem of D. Andrieux and P. Gaspard for the fluxes along the chords associated to a fundamental set of oriented cycles \cite{AG2}. We associate to each random trajectory an oriented cycle on the graph and we decompose it in terms of a basis of oriented cycles. We prove a fluctuation theorem for the coefficients in this decomposition. The resulting fluctuation theorem involves the cycle affinities, which in many real systems correspond to the macroscopic forces. In addition, the above decomposition is useful when analyzing the large deviations of additive functionals of the Markov chain. As example of application, in a very general context we derive a fluctuation relation for the mechanical and chemical currents of a molecular motor moving along a periodic filament.

math-ph↗
arXiv2010

Aging through hierarchical coalescence in the East model

We rigorously analyze the low temperature non-equilibrium dynamics of the East model, a special example of a one dimensional oriented kinetically constrained particle model, when the initial distribution is different from the reversible one and for times much smaller than the global relaxation time. This setting has been intensively studied in the physics literature to analyze the slow dynamics which follows a sudden quench from the liquid to the glass phase. In the limit of zero temperature (i.e. a vanishing density of vacancies) and for initial distributions such that the vacancies form a renewal process we prove that the density of vacancies, the persistence function and the two-time autocorrelation function behave as staircase functions with several plateaux. Furthermore the two-time autocorrelation function displays an aging behavior. We also provide a sharp description of the statistics of the domain length as a function of time, a domain being the interval between two consecutive vacancies. When the initial renewal process has finite mean our results confirm (and generalize) previous findings of the physicists for the restricted case of a product Bernoulli measure. However we show that a different behavior appears when the initial domain distribution is in the attraction domain of a $α$-stable law. All the above results actually follow from a more general result which says that the low temperature dynamics of the East model is very well described by that of a certain hierarchical coalescence process, a probabilistic object which can be viewed as a hierarchical sequence of suitably linked coalescence processes and whose asymptotic behavior has been recently studied in [14].

math-ph↗
arXiv2010

A representation formula for large deviations rate functionals of invariant measures on the one dimensional torus

We consider a generic diffusion on the 1D torus and give a simple representation formula for the large deviation rate functional of its invariant probability measure, in the limit of vanishing noise. Previously, this rate functional had been characterized by M.I. Freidlin and A.D.\ Wentzell as solution of a rather complex optimization problem. We discuss this last problem in full generality and show that it leads to our formula. We express the rate functional by means of a geometric transformation that, with a Maxwell-like construction, creates flat regions. We then consider piecewise deterministic Markov processes on the 1D torus and show that the corresponding large deviation rate functional for the stationary distribution is obtained by applying the same transformation. Inspired by this, we prove a universality result showing that the transformation generates viscosity solution of stationary Hamilton--Jacobi equation associated to any Hamiltonian $H$ satisfying suitable weak conditions.

math.PR↗
arXiv2010

Hydrodynamic limit of zero range processes among random conductances on the supercritical percolation cluster

We consider i.i.d. random variables {ω(b):b \in E_d} parameterized by the family of bonds in Z^d, d>1. The random variable ω(b) is thought of as the conductance of bond b and it ranges in a finite interval [0,c_0]. Assuming the probability m of the event {ω(b)>0} to be supercritical and denoting by C(ω) the unique infinite cluster associated to the bonds with positive conductance, we study the zero range process on C(ω) with ω(b)-proportional probability rate of jumps along bond b. For almost all realizations of the environment we prove that the hydrodynamic behavior of the zero range process is governed by the nonlinear heat equation $\partial_t ρ= m \nabla \cdot (D \nablaϕ(ρ/m))$, where the matrix D and the function ϕare ω--independent. We do not require any ellipticity condition.

math.PR↗