Probability and Stochastic Analysis Seminar 

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.

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.