Probability and Stochastic Analysis Seminar 

In this joint work with M. Falconnet and N. Gantert, we define interacting particle systems on configurations of the integer lattice (with values in some finite alphabet) by the superimposition of two dynamics: a substitution process with finite range rates, and a circular permutation mechanism (called “cut-and-paste”) with possibly unbounded range.

The model is motivated by the dynamics of DNA sequences: we consider an ergodic model for substitutions, the RN+YpR model, introduced by Berard et al. in 2008, as well as three particular cases. We investigate whether they remain ergodic with the additional cut-and-paste mechanism, which models insertions and deletions of nucleotides. Using either duality or attractiveness techniques, we provide various sets of sufficient conditions, concerning only the substitution rates, for ergodicity of the superimposed process.

The exclusion process is one of the best-studied examples of an interacting particle system. In this talk, we consider simple exclusion processes on finite graphs. We give an overview over some recent results on the mixing time of the totally asymmetric simple exclusion process (TASEP). In particular, we provide bounds on the mixing time of the TASEP on the circle, using a connection to periodic last passage percolation. This talk is based on joint work with Allan Sly.

Understanding the diffusive or superdiffusive behavior of the energy in classical physical systems is challenging because of the non-linearity of the interactions between the particles. A way to reduce the difficulty is to replace the nonlinearity by a stochastic noise. In this presentation I will consider a noisy harmonic chain subjected to a magnetic field. We will see that according to the intensity of the magnetic field, the superdiffusive nature of the system changes.

The Kardar-Parisi-Zhang (KPZ) equation is a stochastic partial differential equation with universality, and it has been derived from several microscopic models through scaling limits. When the temperature of a system tends to infinity, we can often extract a heat diffusion part with some residual perturbation by a Taylor expansion argument, which decomposition is crucial for the derivation. We will show through some particular models that we can thereby obtain the KPZ equation as a limit in a robust way.

Consider a sequence of continuous-time Markov chains $X^n_t$ evolving on a fixed finite state space $V$. Let $I_n$ be its level two large deviations rate functiona. Under a general hypothesis on the jump rates, we derive an expansion for $I_n$: we show that $I_n$ can be written as$ I^0 + \sum_{1\ le p\le q} (\theta^p_n)^{-1} I^p$ In this formula, $\theta^p_n$ are the time-scales at which a meta-stable behavior is observed and I^p the level two large deviations rate function of the Markov chain which describes the evolution of the chain $X^n_t$ in the time-scale $\theta^p_n$.

We will study a model of noisy units in mean field interaction, which is described in the large population limit by a non linear PDE. Relying on a slow/fast approach we will show the emergence of noise induced periodic behaviors. We will focus in particular on the case when each unit evolves according to the FiztHugh Nagumo model.

Spin glasses are disordered spin systems originally introduced to explain some unusual magnetic behavior of certain alloys. Although their formulations are typically easy to state, many of them enjoy several profound mathematical and physical principles that are extremely challenging to prove. In this talk, I will focus on the Sherrington-Kirkpatrick model and consider its local magnetization in the computational perspective. While it has been well-understood that this quantity satisfies so-called cavity method and Thouless-Anders-Palmer equations in the literature, I will explain how they give rise to novel iterative algorithms and are helpful in simulating the local magnetization in the high temperature regime. Based on a joint work with Si Tang.

The Kardar-Parisi-Zhang (KPZ) equation is the stochastic partial differential equation that models interface growth. In the talk I will present the construction of a stationary measure for the KPZ equation on a bounded interval with general inhomogeneous Neumann boundary conditions. Along the way, we will encounter classical orthogonal polynomials, the asymmetric simple exclusion process, and precise asymptotics of q-Gamma functions. This construction is a joint work with Ivan Corwin.

We will introduce a periodic version of the Polynuclear growth model (PNG) and show that it is a solvable model. We can give stationary measures for the model at a fixed time and for the distribution of the space-time paths, which in this model are up-down paths that form rings. This is joint work with Pablo Ferrari (UBA).

We discuss some recent developments in the analysis of convergence to stationarity for the Gibbs sampler of general spin systems on arbitrary graphs. These are based on two recently introduced concepts: Spectral Independence and Block Factorization of Entropy. We show that the existence of a contractive coupling for a local Markov chain implies that the system is spectrally independent, and that if a system is spectrally independent then its entropy functional satisfies a general block factorization. As a corollary, we obtain new optimal bounds on the mixing time of a large class of sampling algorithms for the ferromagnetic Ising/Potts models in the so-called tree-uniqueness regime, including non-local chains such as the Swendsen-Wang dynamics. The methods apply to systems with hard constraints such as proper colorings and the hard core gas. We also discuss the entropy factorization for the uniform distribution over permutations and its role in the proof of a conjectured bound on the permanent of arbitrary matrices. Based on some recent joint works with Alexandre Bristiel, Antonio Blanca, Zongchen Chen, Daniel Parisi, Alistair Sinclair, Daniel Stefankovic, and Eric Vigoda.

I will review results on fluctuations of the height function (particle density) for the open Asymmetric Simple Exclusion Process in steady state as the size of the system goes to infinity. I will discuss the cases where the parameters of ASEP are constant or vary with the size of the system. The talk is based on several papers with Alexey Kuznetsov, Yizao Wang and Jacek Wesolowski.

This talk aims to report on the fluctuations of traces of powers of a random $n$ by $n$ matrix U distributed according to the Haar measure on the unitary group. This random matrix problem has been extensively studied using several different methods such as asymptotics of Toeplitz determinants, representation theory, loop equations etc. It turns out that for any $k≥1$, $Tr[U^k]$ converges as $n$ tends to infinity to a Gaussian random variable with a super exponential rate of convergence. In this talk, I will explain some of these results and present some recent work with Klara Courteaut and Kurt Johansson (KTH) in which we revisited this classical problem.

The cylinder’s percolation model arises from a Poissonian soup of infinite lines in $R^d$ and it is a stationary process under the isometries of the underlying space. Each such line is then thickened, becoming the axis of a cylinder of radius one. The associated percolation picture exhibits long range correlations and the rigidity of the underlying objects hampers direct attempts at proving decorrelation inequalities via sprinkling of the intensity parameter. We obtain such inequalities by exploiting the continuity of the process, taking the radii of the cylinders as a parameter and using it in a sprinkling argument. As an application, we prove that for small intensities of the cylinder’s process the simple random walk on the vacant set is transient. The talk will also go over similar decoupling inequalities for other models, their applications and open problems.
This talk is based on a joint work with Caio Alves.

We introduce and study an interacting particle system evolving on the $d$-dimensional torus $\mathbb{Z}^d_N$. Each vertex of the torus can be either empty or occupied by an individual of a given type; the space of all types is the positive real line. An individual of type $\lambda$ dies with rate one and gives birth at each neighbouring empty position with rate $\lambda$. Moreover, when the birth takes place, the new born individual is likely to have the same type as the parent, but has a small chance to be a mutant; the mutation rate and law of the type of the mutant both depend on $\lambda$. We consider the asymptotic behaviour of this process when the size of the torus is taken to infinity and the overall rate of mutation tends to zero fast enough that mutations are sufficiently separated in time, so that the amount of time spent on configurations with more than one type becomes negligible. We show that, after a suitable projection (which extracts just the dominant type from the configuration of individuals in the torus) and time scaling, the process converges to a Markov jump process on the positive real lines, whose rates we characterize. Joint work with Adrián González Casanova and András Tobias.

Invariant measures for the one-dimensional asymmetric exclusion process (ASEP) are fairly well (though not entirely) understood. Under broad assumptions they consist of homogeneous Bernoulli measures and blocking measures. In several dimensions, the characterization of invariant measures (outside translation invariant ones) is still open. Bramson and Liggett (AOP 2005) laid important foundations and formulated conjectures. In particular they introduced a family of d-dimensional blocking-type measures as reasonable candidates for characterization. Here we study an intermediate model, the multilane ASEP or ladder process,that is a 2d ASEP where one direction is finite and the dynamics is translation invariant in the infinite direction. We obtain characterization results for invariant measures involving product measures homogeneous in the infinite direction and 2d blocking measures with the same flavour as Bramson and Liggett's.

Joint work with G. Amir, O. Busani and E. Saada.

In the mid 70's Giorgio Parisi proposed a revolutionary solution for the Sherrington-Kirkpatrick (SK) model. His proposed phase diagram of this spin glass model depends on one functional ordered parameter, now known as the Parisi measure. In this talk, I will survey what is rigorously known about this measure and highlight the remaining challenging open questions. I will present results both for the SK model, the mixed p-spin and for their spherical version. Based on joint works with Wei-Kuo Chen (Minnesota), Qiang Zeng (Macau) and Yuxin Zhou (Northwestern).

See here: https://spmes.impa.br

The Fredrickson-Andersen 2-spin facilitated model (FA-2f) on $Z^d$ is a paradigmatic interacting particle system with kinetic constraints (KCM) featuring cooperative and glassy dynamics. For FA-2f vacancies facilitate motion: a particle can be created/killed on a site only if at least $2$ of its nearest neighbors are empty. We will present sharp results for the first time, $\tau$, at which the origin is emptied for the stationary process when the density of empty sites ($q$) is small: in any dimension $d \geq 2$ it holds $\tau \sim   \exp \left( \tfrac{ d \lambda(d,2) +o(1)} {q^{1/(d-1)}} \right)$ w.h.p., with  $\lambda (d,2)$ the threshold constant for the 2-neighbour bootstrap percolation on $Z^d$. This is the  first sharp result for a critical KCM and settles various controversies accumulated in physics literature over the last four decades. Joint work with I. Hartarsky and F. Martinelli

The paradigmatic 2-neighbour bootstrap percolation model is the following cellular automaton. Given a set of infected sites in $\mathbb Z^2$, we iteratively infect each site with at least 2 infected neighbours, while infections never heal. We are then interested whether and when the origin becomes infected under this dynamics starting from an i.i.d. Bernoulli initial infection. There is a naturally associated stochastic non-monotone model: the Fredrickson-Andersen 2-spin facilitated one, in which the state of each site is resampled to a Bernoulli variable at rate 1, provided it has at least 2 infected neighbours. Of course, many related models have been considered, replacing the 2-neighbour constraint by an increasing local translation-invariant constraint (e.g. both the North and East neighbours are infected). In this talk we will overview recent universality results for this class of bootstrap percolation and its non-monotone stochastic counterpart called kinetically constrained models. The outcome is a classification of all rules in terms of the scaling of the infection time of the origin when the density of infections approaches a possibly degenerate critical value.

We consider the asymmetric simple exclusion process (ASEP) on the one-dimensional lattice of size $n$. Particles can enter or exit the system from the boundares with given time-dependent rates. Those rates are regulated by a factor $n^\theta$. We investigate the hydrodynamic limit under the hyperbolic time scale in three cases: (1) $\theta > 0$ (fast boundary), (2) $\theta = 0$, the bulk dynamics performs rightward flux, but particles enter only from the right and exit only from the left (boundary against the flux), (3) $\theta < 0$ (slow boundary). In all cases, a weak symmetry (w.r.t. the hyperbolic time scale) is technically necessary. The macroscopic equation is given by Burgers equation with boundary conditions that allow typical discontinuities at boundaries (boundary layer). The boundary conditions are obtained through a grading scheme in case (1) and a vanishing current argument in case (2) and (3).

Based on the arXiv preprints 2108.09345 and 2203.15091.

We will report on some recent results linking three models of disordered media which permeate in several forms in multiplicative noise stochastic PDEs, directed polymers and Gaussian multiplicative chaos.