Plenary Lectures

Recently, there has been increasing interest in interacting particle systems on evolving random graphs, respectively in time evolving random environments. In this talk we present results on the contact process in an evolving edge random environment on infinite (random) graphs. We first give an overview over recent results. Then, we in particular consider (infinite) Bienaymé-Galton-Watson (BGW) trees as the underlying random graph. Here, we focus on an edge random environment that is given by a dynamical percolation whose opening and closing rates and probabilities are degree dependent. Our results concern the impact of these parameters on the critical infection rate for weak (global) and strong (local) survival of the infection.

Specifically, we establish conditions under which the contact process undergoes a phase transition: For a general connected locally finite graph we provide sufficient conditions for the critical infection rate to be strictly positive. Furthermore, in the setting of BGW trees, we provide conditions on the offspring distribution as well as on the speed and connection probabilities that the process survives strongly with positive probability for all positive values of the infection rate. In particular, if the offspring distribution follows a power law (or has a stretched exponential tail) and the connection probability is given by a product kernel (or a maximum kernel) and the update speed exhibits polynomial behaviour, we provide quite a complete characterisation of the contact process' behavior.

This talk is based on joint work with Natalia Cardona-Tobon (Universidad Nacional de Colombia, Bogotá), Marcel Ortgiese (University of Bath) and Marco Seiler (University of Frankfurt).

This work was motivated by natural questions related to oriented percolation on a layered environment that introduces long range dependence. As a convenient tool, we are led to deal with questions on the strict decrease of the percolation parameter in the oriented setup when an extra dimension is added.

Joint work with Daniel Ungaretti and Maria Eulália Vares.

We investigate the invariant measure for non-reversible particle systems driven out-of-equilibrium via the action of external reservoirs. This is in general a difficult task, but it has been explicitly given in some special “microscopically integrable” cases. In this talk we are interested in a class of boundary driven zero-range models whose non-equilibrium steady state can be explicitly characterized via a probabilistic mixture of a "hidden variable" of inhomogeneous product measures with means equal to the value of the hidden variable. We further investigate the behaviour of the Gibbs-Shannon entropy of this measure and show, in line with previous results in the literature, that the leading order is that of the entropy of a product measure calculated at the means of the hidden variables, while the first order correction is related to the two-point correlations. We investigate the invariant measure for non-reversible particle systems driven out-of-equilibrium via the action of external reservoirs. This is in general a difficult task, but it has been explicitly given in some special “microscopically integrable” cases. In this talk we are interested in a class of boundary driven zero-range models whose non-equilibrium steady state can be explicitly characterized via a probabilistic mixture of a "hidden variable" of inhomogeneous product measures with means equal to the value of the hidden variable. We further investigate the behaviour of the Gibbs-Shannon entropy of this measure and show, in line with previous results in the literature, that the leading order is that of the entropy of a product measure calculated at the means of the hidden variables, while the first order correction is related to the two-point correlations. 

We discuss a class of nonlinear exchange dynamics for interacting spin systems, inspired by Boltzmann-type equations from kinetic theory. These dynamics provide a nonlinear Monte Carlo approach for sampling Gibbs measures under prescribed magnetization or density constraints. We focus on the Ising and hard-core models and present convergence results showing exponential relaxation to equilibrium in the weak-interaction and low-density regimes, respectively, with optimal decay rates independent of the initial condition. The proofs are based on novel coupling techniques as well as ideas from the Kac program in kinetic theory, including uniform functional inequalities for the associated particle systems. We also discuss some possible algorithmic applications.

We report on work in progress with Guilherme Ost (UFRJ) and Guilherme Reis (UFF) on a class of toy models for neuronal dynamics. These models consist of sets of units taking 0/1 values, which interact pairwise via a matrix of weights. The main question we address is how much of the weight structure can be recovered from the dynamics. We will focus on "mean field" cases where individual pairwise interactions may be quite weak. Interestingly, this property turns out to be helpful. 

The counting of partitions of an integer n under different restrictions appears in many areas, and has been considered from several viewpoints. A family of random variables can be naturally associated to the generating function of the corresponding partition function, and it turns out that those are sums of independent random variables, which are asymptotically normal when conveniently normalized. The asymptotic behaviour of the number of partitions is then related to local limit theorems. We investigate in particular the asymptotic behaviour of the number of partitions of n with k colors, both in the case of unrestricted partitions, and in that into distinct parts. In the case that k grows linearly with n, we show that an asymptotic expansion follows from the classical expansions of the local limit theorems by Gnedenko and Kolmogorov. In the case that k is fixed, precise asymptotic results are deduced by techniques involving expansions in cumulants.

Based in joint work with A. Meyroneinc and Y. Vargas.

In Dynamical Percolation each edge is open with probability p, refreshing its status at rate \mu>0. This process was introduced in the 1990s by Haggstrom, Steif and the speaker, motivated by a question of Malliavin. Remarkable results on exceptional times in two dimensions were obtained by Schramm, Steif, Garban and Pete.

   We study random walk on dynamical percolation in the lattice Z^d, where the  walk moves along open edges at rate 1. Let p_c=p_c(d) denote the critical value for static percolation. For p<p_c and \mu<1,  joint work with Stauffer and Steiff (PTRF, 2015)   showed the mean squared displacement is of order  t \mu.  For  p>p_c, we prove that the mean squared displacement is of order t, uniformly in  0<\mu<1, refining results obtained with Sousi and Steif (PTRF, 2020). In the critical regime  p=p_c, we prove that if d=2 or d>10, then the mean squared displacement is at most O(t \mu^a)  where a=a(d)>0 .  We will show simulations to illustrate the process.

(Joint work with Chenlin Gu, Jianping Jiang, Zhan Shi, Hao Wu and Fan Yang.)