Oriane Blondel (Université Paris Cité)
Title: Facilitated exclusion process: large deviations
Abstract:
The facilitated exclusion process is an interacting particle system with an active/inactive phase transition that arises from the effects of a kinetic constraint. We discuss preliminary results on large deviations in the supercritical setting. One key difficulty is to control the dynamics when the initial configuration is not supported by the stationary measures.
Elisabetta Candellero (Università degli Studi Roma Tre)
Title : First passage percolation with recovery
Abstract:
We study a natural modification of first-passage percolation. Consider a graph $G$ with a reference vertex $o$. Place a red particle at $o$ and colorless particles (which we call seeds) everywhere else. The process spreads at rate 1 and starts at $o$, while seeds are dormant.
Seeds reached by the process turn red and keep spreading first-passage percolation (at rate 1). All vertices have independent exponential clocks which ring at rate $\gamma>0$; when a clock rings, the corresponding red vertex turns black. We establish asymptotic (in time) upper and lower bounds on the size of the longest red path and largest red cluster when $G$ is a tree, deterministic or random. (Joint work with Tom Garcia-Sanchez.)
Alberto Chiarini (Università di Padova)
Title: Hard wall repulsion for the harmonic crystal with bond disorder
Abstract:
The discrete Gaussian Free Field (also known as the harmonic crystal) can be interpreted as a microscopic model describing fluctuations in a homogeneous crystal at non-zero temperatures. In this talk, we examine the impact of impurities by
investigating the maximum fluctuations of the field with random conductances and its behaviour in the presence of a macroscopic hard wall. We will restrict to the supercritical case, that is dimension bigger or equal to three.
First, we derive exact quenched large deviation asymptotics for the hard wall event. These will involve on the one hand the homogenized capacity of a random walk in an environment with random conductances, and on the other hand the essential supremum of the (random) variances of the field. Then, we demonstrate that, conditioned on the presence of the hard wall, the field experiences everywhere an entropic push away from zero. Finally, we characterize the pathwise behavior of the field in the presence of the hard wall. Time permitted, we will also discuss some preliminary results in the case the underlying crystal structure is the Bernoulli supercritical infinite cluster.
This work is a collaboration with E. Pasqui (Università degli Studi di Padova).
Benoit Dagallier (Imperial Collegue, London)
Title: Log-Sobolev inequalities for continuous models on dense graphs
Abstract
I will present a joint work with Roland Bauerschmidt and Thierry Bodineau on log-Sobolev inequalities for models with continuous spins on a dense graph. The prototypical example of this is the complete graph, corresponding to so-called mean-field models. These models are typically non-convex and have phase transitions, meaning: 1) Bakry and Emery's convexity-based criterion does not apply and 2) the speed of convergence of the dynamics varies abruptly with the system size depending on the temperature of the model.
If the underlying graph is sufficiently dense, however, the speed of convergence of the dynamics can be controlled through the convexity of an effective object, related to the free energy of the model.
The talk is based on the article https://arxiv.org/abs/2503.24372 .
Shirshendu Ganguly (University of California, Berkeley)
Title: A Gibbsian line ensemble perspective to entropically repelled surfaces
Gibbsian line ensembles have been the topic of much recent interest at the interface of probability and statistical physics, most prominently via the Airy line ensemble occurring as a scaling limit of Dyson Brownian motion. Recently, Caputo, Ioffe and Wachtel have proposed an area-tilted variant of this ensemble, satisfying a corresponding Gibbs property. This captures the local behavior of level curves in entropically repelled low temperature 3D Ising interfaces, an object of a long line of study including the work of Caputo, Lubetzky, Martinelli, Sly, Toninelli who proved the existence of macroscopic level curves for the related solid-on-solid model. While this line ensemble is out of reach of integrable techniques, in this talk I will review some recent developments in this story.
Reza Gheissari (Northwestern University)
Title : Phase ordering in low-temperature Ising dynamics
Abstract :
We consider the out-of-equilibrium behavior of Ising Glauber dynamics at low temperatures. It is well-known that in its low-temperature regime, the Ising Glauber dynamics takes an exponential time to equilibrate, due to a bottleneck between the mostly plus and mostly minus phases of the model. The question of phase ordering, studied in the physics literature since the 1960s, asks whether bias in the magnetization at initialization is enough to cause the system to quickly (quasi-)equilibrate to the plus phase. Caputo and Martinelli (2005) showed such a result on trees, namely that low-temperature Glauber dynamics initialized from product measures with sufficient bias towards plus, rapidly converge to the plus measure on the tree. We will present progress on this question in $\mathbb Z^2$, joint with Allan Sly.
Ivailo Hartarsky (Université Claude Bernard Lyon 1)
Title: Triangular plaquette model
Abstract:
Consider the following plaquette model from statistical physics: a lamp lies at every vertex of the triangular lattice and a switch lies at every even vertex of the (bipartite) dual hexagonal lattice. Each switch toggles the three lamps on its face. The energy of a configuration is the number of ON lamps. We study the relaxation time of the associated Glauber dynamics, proving its conjectured super-Arrhenius scaling at low temperature. We also reveal highly unusual behaviour in finite volume around the critical length scale.
The talk is based on joint work with Laurent Bartholdi and Ivan Mitrofanov.
Ron Peled (University of Maryland, College Park)
Title : The critical fugacity of the hard-core model in high dimensions
Abstract:
In the hard-core model on a graph G, one samples a random independent subset A of G with probability proportional to λ^|A|, where λ>0 is a parameter, termed the fugacity. The seminal work of Dobrushin (1968) established a phase transition for the hard-core model on Z^d (in the sense of Gibbs measures): At fugacity λ < 1/(2d-1), the model is disordered in the sense that the random set A has half of its vertices on each partite class, while at sufficiently high fugacity, the model exhibits long-range order in the sense that the set A has density strictly larger than one half on one of the two partite classes.
Following Dobrushin, the question of determining the minimal fugacity λ_c(d) at which long-range order arises, and its asymptotics as d → ∞ has remained a challenge of enduring interest. In a breakthrough work, Galvin–Kahn (2004) proved that λ_c(d) < d^{-1/4 + o(1)}, thus showing that the critical fugacity decays to zero with the dimension. Their bound was improved by Samotij–Peled (2014) who showed λ_c(d) < d^{-1/3 + o(1)}.
Galvin–Kahn suggested that λ_c(d) = d^{-1 + o(1)}. In this talk I will discuss the proof of this fact, obtained jointly with Daniel Hadas.
Matteo Quattropani (Università degli Studi Roma Tre)
Title : Convergence to equilibrium of Stochastic Exchange Models.
Abstract:
A Stochastic Exchange Model (SEM) is a Markov process in which a number of particles (or agents) repeatedly exchange a conserved quantity (discrete or continuous), such as energy or money, according to a random dynamics. We will imagine the particles as the vertices of an undirected, finite, connected graph and equip each edge with a Poisson clock: when the edge connecting x and y rings, the associated particles perform an exchange of their current amount of energy, which might be deterministic or random, depending on the model.
The simplest continuous model one can imagine is the one in which, whenever an exchange occurs, the two particles split their total energy deterministically in two halves. This model usually goes under the name of Averaging process (AVG), and has been studied extensively in recent years. In this case, regardless of the underlying graph, in the long run the system will converge to a flat configuration in which all particles carry the same amount of energy.
It is also natural to consider variants in which the exchange mechanism is itself random. A well-known instance of this class of models is the so-called Kipnis-Marchioro-Presutti model (KMP), in which, at an exchange time, the total amount of energy at the two particles is divided randomly according to a uniform distribution. In contrast to the AVG, the stationary distribution of such a dynamics is not singular, being in fact the uniform distribution on the energy simplex.
Of course, even though the graph geometry does not affect the equilibrium of these dynamics, it can have a major impact of the time the process needs to approach such a stationary state.
In this talk I will survey recent results on the convergence to equilibrium of general SEMs. I will emphasize, on the one hand, the effect of the underlying graph geometry on the convergence time of AVG and, on the other hand, the analysis of the mean-field case (i.e., the case in which the underlying graph is complete), where AVG, KMP, and several other SEMs can be shown to exhibit a universal out-of-equilibrium behavior when convergence is measured in a suitable Wasserstein metric.
The talk is based on joint work with Pietro Caputo and Federico Sau.
Justin Salez (Univ. Dauphine PSL)
Title : The cutoff phenomenon follows from a non-equilibrium product condition
Abstract :
In the community of mixing times, a famously wrong conjecture asserts that a sequence of ergodic Markov chains exhibits cutoff as soon as the product of their Poincaré constant and mixing time tends to infinity. In this talk, I will prove that this conjecture is actually valid, provided one replaces the Poincaré constant with its natural non-equilibrium version. This is based on a (very recent) joint work with Francesco Pedrotti.
Federico Sau (Università degli Studi di Milano)
Title: Aldous’ spectral gap phenomenon for the KMP model
Abstract :
Aldous’ spectral gap conjecture, proved by Caputo, Liggett and Richthammer, states that, on any weighted graph, the spectral gaps of the interchange process and of the underlying random walk coincide. In this talk, we present an analogue for the Kipnis-Marchioro-Presutti model on arbitrary weighted hypergraphs: its spectral gap is not identified with that of one random walk, but with that of a two-particle dual dynamics.
Based on joint work with Pietro Caputo and Matteo Quattropani.
Assaf Shapira (Université Paris Cité)
Title : Diffusion coefficients via variational principles
Abstract :
The talk will concentrate on reversible interacting particle systems with conserved number of particles. Generically, the macroscopic evolution of such systems happens in diffusive time scales, and characterized by the diffusion coefficient. This coefficient can be found explicitly for models satisfying the gradient condition, but unfortunately such models are rare, and cannot describe some important qualitative properties. We will discuss the formulation of the diffusion coefficient as a variational quantity, which could be used in order to estimate its value in the non-gradient case.
Vittoria Silvestri (Sapienza, Università di Roma)
Title : Anisotropic Hastings-Levitov growth
Abstract:
The formation of fractal structures in nature is, from a mathematical perspective, still far from being understood. Among the many models which were introduced with this aim, random growth models defined via conformal maps are particularly amenable to a rigorous analysis. In this talk I will focus on the class of Hastings-Levitov models in anisotropic regimes, for which I will discuss hydrodynamic limit and fluctuations in the small particle limit. Based on joint work with Luca Cattani (University College London).
Alexandre Stauffer (King's College London)
Title : Uniqueness of the infinite cluster for monotone percolation models without insertion tolerance
Abstract:
We will study the infinite cluster of toppled sites during the stabilization of the Abelian sandpile model. We will show that in the supercritical regime the infinite cluster is unique almost surely. The main challenge in this model is that it does not have the insertion tolerance property.
Based on a joint work with Christoforos Panagiotis (Univ. of Bath).
Vincent Tassion (ETH Zurich)
Title : Supercritical Sharpness of Percolation
Abstract:
Bernoulli percolation undergoes a phase transition between a
subcritical phase, where all clusters are finite, and a supercritical
phase, where an infinite component exists. A classical result states
that in the subcritical phase, the cluster sizes have an exponential
tail. In this talk, we discuss the supercritical analogue, where the
geometry of the underlying graph plays a role. Based on joint work with
Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, and Benny
Sudakov.