Probability and Stochastic Analysis Seminar 

For a given finite Markov chain, and a given permutation on the state-space, we consider the Markov chain which alternates between random jumps according to the initial chain, and deterministic jumps according to the permutation. In this framework, Chatterjee and Diaconis (2020) showed that when the permutation satisfies some expansion condition with respect to the chain, then the mixing time is logarithmic, and that this expansion condition is satisfied by almost all permutations. We will see that the mixing time can even be characterized much more precisely: for almost all permutations, the permuted chain has cutoff, at a time which only depends on the entropic rate of the initial chain.

We consider the following stochastic partial differential equation in $\mathbb R^d$ $ \partial_t u = \Delta u + \xi \cdot u $ where the unknown $u$ is a function of space and time. The operator $\Delta$ denotes the usual Laplacian in $\mathbb R^d$ and $\xi$ is a space-time Lévy white noise. This equation has been extensively studied in the case where $\xi$ is a Gaussian White noise. In that case, it is known that the equation is well posed only when the space dimension $d$ is equal to one.
In our talk, we consider the case where $\xi$ is a Lévy white noise with no diffusive part and only positive jumps. We identify necessary and sufficient conditions on the Lévy jump measure for the existence of solutions to the equation. We further discuss the connection between the SHE and continuum directed polymer models.
Joint work with Quentin Berger (Sorbonne Université, Paris) and Carsten Chong (Columbia University, New York)

Macroscopic Fluctuations Theory (MFT) developed by the « Italian [Brazil as part of Italy included :-)] group » (Bertini et al.) has been one the biggest progress of the last 20 years in our understanding of 1D non-equilibrium stationary states (NESS) of extended interacting particle systems. While the theory has been developed for diffusive and hyperbolic systems (macroscopically), the theory is still not existing for super-diffusive (fractional diffusive) systems. In this talk I will present recent results about the derivation of one-dimensional fractional reaction-diffusion equations with boundary conditions (Dirichlet, Robin, Neuman) for lattice gas with long jumps (mainly the exclusion process) by a hydrodynamic limit procedure. I will also present MFT results for the NESS of the boundary driven zero range process. These results are the first steps in the long-term project to develop MFT for super-diffusive systems. The talk will not be technical (I hope) and its aim is to explain why we are interested in these problems, more than to present the results obtained. (Collaborators: P. Cardoso, P. Gonçalves, B. Oviedo-Jimenez, S. Scotta).

We consider the large deviations from the hydrodynamic limit of the Totally Asymmetric Simple Exclusion Process (TASEP), which is related to the entropy production in the inviscid Burgers equation. Here we prove the full large deviation principle. Our method relies on the explicit formula of Matetski, Quastel, and Remenik (2016) for the transition probabilities of the TASEP. Joint work with Jeremy Quastel.

We analyse the metastable behaviour of the dilute Curie–Weiss model subject to a Glauber dynamics. The model is a random version of a mean-field Ising model, where the coupling coefficients are replaced by i.i.d. random coefficients, e.g. Bernoulli random variables with fixed parameter p. This model can be also viewed as an Ising model on the Erdos–Renyi random graph with edge probability p. The system is a Markov chain where spins flip according to a Metropolis dynamics at inverse temperature $\beta$. We compute the average time the system takes to reach the stable phase when it starts from a certain probability distribution on the metastable state (called the last-exit biased distribution), in the regime where the system size goes to infinity, the inverse temperature is larger than 1 and the magnetic field is positive and small enough. We obtain asymptotic bounds on the probability of the event that the mean metastable hitting time is approximated by that of the Curie–Weiss model. The proof uses the potential theoretic approach to metastability and concentration of measure inequalities. This is a joint collaboration with Anton Bovier (Bonn) and Saeda Marello (Bonn)

The KPZ fixed point is a scaling invariant, integrable Markov process at the centre of the KPZ universality class. We review how the transition probabilities are derived from those of TASEP, an integrable model in the class, and how they solve classical completely integrable partial differential equations.

One of the main difficulties in proving convergence of discrete models of surface growth to the Kardar-Parisi-Zhang (KPZ) equation in dimensions higher than one is that the correct way to take a scaling limit, so that the limit is nontrivial, is not known in a rigorous sense. The same problem has so far prevented the construction of nontrivial solutions of the KPZ equation in dimensions higher than one. To understand KPZ growth without being hindered by this issue, I will introduce a new concept in this talk, called "local KPZ behavior", which roughly means that the instantaneous growth of the surface at a point decomposes into the sum of a Laplacian term, a gradient squared term, a noise term, and a remainder term that is negligible compared to the other three terms and their sum. The main result is that for a general class of surfaces, which contains the model of directed polymers in a random environment as a special case, local KPZ behavior occurs under arbitrary scaling limits, in any dimension.

It has been known for a long time that the Brownian motion is an invariant measure for the Kardar-Parisi-Zhang equation on the real line. For KPZ growth on bounded domains, however, the situation is more complicated. Stationary measures are typically not invariant by translation, non Gaussian, and they depend non trivially on boundary conditions. I will review recent progress on the characterization of invariant measures for the KPZ equation on a segment $[0,L]$ and on the half-line $\mathbb{R}_+$. Based on joint works with Pierre Le Doussal and Ivan Corwin.

Groeneboom in 1989 discovered an explicit formula for the law of the entropy solution to Burgers' equation when the initial condition is a white noise. The method of his proof relied extensively on probabilistic methods and in particular on the sophisticated excursion theory for diffusions. Recently, by verifying a conjecture of Menon and Srinivasan, Kaspar and Rezakhanlou managed to prove a closure theorem for Markovian solutions to scalar conservation laws which bridged the probabilistic problem to kinetic theory. In this talk, I present a new and significantly shorter proof of Groeneboom's results. This approach builds on these recent developments, and a central limit theorem for certain Markovian jump processes. I also discuss how a kinetic theory can be developed when we add an external force to the Burgers' equation.

Let X be a simple random walk in $\mathbb{Z}_n^d$ with $d\geq 3$ and let $t_{\rm{cov}}$ be the expected time it takes for $X$ to visit all vertices of the torus. In joint work with Prévost and Rodriguez we study the set $\mathcal{L}_\alpha$ of points that have not been visited by time $\alpha t_{\rm{cov}}$ and prove that it exhibits a phase transition: there exists $\alpha_*$ so that for all $\alpha>\alpha_*$ and all $\epsilon>0$ there exists a coupling between $\mathcal{L}_\alpha$ and two i.i.d. Bernoulli sets $\mathcal{B}^{\pm}$ on the torus with parameters $n^{-(a\pm\epsilon)d}$ with the property that $\mathcal{B}^-\subseteq \mathcal{L}_\alpha\subseteq \mathcal{B}^+$ with probability tending to $1$ as $n\to\infty$. When $\alpha\leq \alpha_*$, we prove that there is no such coupling.

I will start by presenting a number of known published rigorous upper and lower bounds for the number of collisions of a finite number of billiard balls in free space (i.e., a billiard table with no walls). The known bounds are not sharp. Those questions inspired a model with "pinned billiard balls" that have "velocities" and collide with each other but do not move. The evolution of pseudo-velocities seems to be well represented by modulated white noise. The parameters of white noise, the mean and standard deviation, are deterministic functions of space and time. They satisfy a simple system of PDEs with complicated boundary conditions.
Joint work with J. Athreya, M. Duarte, J. Hoskins, S. Steinerberger and J. Sylvester.

Percolation on the Euclidean d-dimensional lattice has been studied for over sixty years and is still a fascinating source of interesting mathematical problems. The fact that this model undergoes a non-trivial phase transition is well-understood since the early studies in the Bernoulli setting, where the lattice is regular and there are no inhomogeneities on the parameters. One way to introduce random disorder is, for example, either passing to a dilute lattice where lower dimensional affine hyperplanes are removed or, alternately, introducing inhomogeneities on the parameter along such hyperplanes. In these situations, even to establish that non-trivial phase transition takes place may be a hard task. In this talk we review some recent results on this topic and discuss some open problems.

We consider the hydrodynamic mass scaling limit of a zero-range particle system on a $1D$ discrete torus with a defect at one site. In such a model, a particle at $x$ jumps equally likely to a neighbor with rate depending only on a function of $k$, the number of particles at $x$, say $g(k)=k^{-\alpha}$. A defect, however, may be present at specific sites in that the jump rate is slowed down there to $N^{-\beta}g(k)$. Here, in diffusion scale, the grid spacing is seen as $1/N$ and time is speeded up by $N^2$. In three regimes, when $\beta <\alpha$, $\beta=\alpha$, and $\beta>\alpha$, the scaling pde limit is different, with boundary conditions reflecting interaction with the slow site and condensation on it. This is work with Jianfei Xue.

Kinetically constrained models are interacting particle systems on $\mathbb{Z}^d$, in which particles can appear/disappear only if a given local constraint is satisfied. This condition complexifies significantly the dynamics. In particular, it deprives the system of monotonicity properties, which leaves us with few tools to study the dynamics when it is initially not at equilibrium. I will review the results and techniques we have in this direction.

We consider Bernoulli percolation on transitive graphs of polynomial growth. In the subcritical regime ($p < p_c$), it is well known that the connection probabilities decay exponentially fast. In this talk, we discuss the supercritical phase $p > p_c$, where we prove the exponential decay of the truncated connection probabilities (probabilities that two points are connected by an open path, but not to infinity). This sharpness result was established by Chayes, Chayes and Newman on $\mathbb{Z}^d$ and uses the difficult slab result of Grimmett and Marstrand. However, the techniques used there are very specific to hypercubic lattices and do not extend to more general geometries. Our approach involves new robust techniques based on the recent progress in the theory of sharp thresholds and the sprinkling method of Benjamini and Tassion. In this talk, we will mainly discuss the methods on $ \mathbb{Z}^d $, and give a completely new proof of the slab result of Grimmett and Marstrand.

Based on joint work with Sébastien Martineau and Vincent Tassion.

We discuss deviations for the number of intersections of two independent infinite-time ranges in dimensions five and more. This settles a conjecture of van den Berg, Bolthausen and den Hollander. Moreover, we obtain the scenario leading to this deviation. (joint work with B. Schapira).

We consider diffusive particle systems evolving on a lattice of mesh $1/N$ and study the large deviations for the current on a time window $[0,T]$. We consider two different asymptotics. In the first case we send before $N\to +\infty$ with a diffusive hydrodynamic rescaling keeping $T$ fixed, and then send $T\to +\infty$. In the second case we send before $T\to +\infty$ keeping $N$ fixed and then send $N\to +\infty$.

We rigorously formulate and prove for a relatively general class of interactions the characterization of shift-invariant closed $L^2$-forms for a large scale interacting system. Such characterization of closed forms has played an essential role in proving the hydrodynamic limit of nongradient systems. The universal expression in terms of conserved quantities was sought from observations for specific models, but a precise formulation or rigorous proof up until now had been elusive. To obtain this, we first prove the universal characterization of shift-invariant closed “local” forms in a completely geometric way, that is, in a way that has nothing to do with probability measures. Then, we apply the result to characterize the $L^2$-forms. Our result is applicable for many important models including generalized exclusion processes, multi-species exclusion processes, exclusion processes on crystal lattices and so on. This talk is based on a joint work with Kenichi Bannai and Yukio Kametani, and a joint work with Kenichi Bannai.

Dynamic large deviations for additive path functionals of stochastic processes have attracted recent research interest, in particular in the context of stochastic particle systems and statistical physics. Efficient numerical 'cloning' algorithms have been developed to estimate the scaled cumulant generating function, based on importance sampling via cloning of rare event trajectories. Adapting previous results from the literature of particle filters and sequential Monte Carlo methods, we use Feynman-Kac models to establish fully rigorous bounds on systematic and random errors of cloning algorithms in continuous time. To this end we develop a method to compare different algorithms for particular classes of observables, based on the martingale characterization and related to the propagation of chaos for mean-field models. Our results apply to a large class of jump processes on locally compact state space, and provide a framework that can also be used to evaluate and improve the efficiency of algorithms. This is joint work with Letizia Angeli, Adam Johansen and Andrea Pizzoferrato.

The Airy_2 process is a universal distribution which describes fluctuations in models in the Kardar--Parisi--Zhang (KPZ) universality class, such as the asymmetric simple exclusion process (ASEP) and the Gaussian Unitary Ensemble (GUE). Despite its ubiquity, there are no proven results for analogous fluctuations of multi--species models. Here, we will discuss one model in the KPZ universality class, the q--Boson. We will show that the joint multi--point fluctuations of the single--species q--Boson match the single--point fluctuations of the multi--species q--Boson. Therefore the single--point fluctuations of multi--species models in the KPZ class ought to be the Airy_2 process. The proof utilizes the underlying algebraic structure of the multi--species q--Boson, namely the quantum group symmetry and Coxeter group actions.