Probability and Stochastic Analysis Seminar

Past sessions

Duality, intertwining and orthogonal polynomials for continuum interacting particle systems

Duality is a powerful tool for studying interacting particle systems, i.e., continuous-time Markov processes describing many particles say on the lattice Z^d. In recent years interesting dualities have been proven that involve falling factorials and orthogonal polynomials; the orthogonality measure is the reversible measure of the Markov process. I'll address generalizations to particles moving in the continuum rather than on the lattice. Examples include independent diffusions and free Kawasaki, which have been investigated before, and a continuum version of the symmetric inclusion process, which is new. The falling factorials turn out to be related to Lenard's K-transform. The relevant notion of orthogonal polynomials belongs to infinite-dimensional analysis, Wiener chaos decompositions and multiple stochastic integrals.

Based on joint work with Simone Floreani and Frank Redig (TU Delft) and Stefan Wagner (LMU Munich).

Discrete-time evolution in recombination

We study the discrete-time evolution of a recombination transformation in population genetics acting on the set of measures on genetic sequences. The evolution can be described by a Markov chain on the set of partitions that converges to the finest partition. We describe the geometric decay rate to the limit and the quasi-stationary behavior when conditioned that the chain has not hit the limit.

Failure of the Ornstein-Zernike asymptotics for the pair correlation function at high temperature and small density

After briefly reviewing what is known about the long-distance asymptotic behavior of the 2-point function in lattice spin systems with finite-range interactions, I'll turn to the corresponding result for systems with interactions of infinite range. I'll show that, contrarily to standard expectations in Physics, the classical Ornstein-Zernike asymptotic formula for the 2-point function does not always hold, even in regimes where it was expected to, namely systems with interactions decaying exponentially fast at very high temperature and/or very low density. I'll explain how this is intimately related to the possible non analytic dependence of the correlation length in the relevant parameters (for instance, temperature), a phenomenon that can occur even in one-dimensional systems. This can be also related to a condensation transition in the graphical representations of these correlations. For simplicity, the focus will be on the Ising model, but most of the results hold much more generally.

Lozenge tilings and the Gaussian free field on a cylinder

We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured by Gorin for tiling models on planar domains with holes. This talk is based on joint work with Andrew Ahn and Roger Van Peski.

Kahane's Gaussian Multiplicative Chaos and Circular Random Matrices match exactly

In this talk, I would like to advertise the strict equality between two objects from very different areas of mathematical physics:

• Kahane's Gaussian Multiplicative Chaos (GMC), which uses a log-correlated field as input and plays an important role in certain conformal field theories
• A reference model in random matrices called the Circular Beta Ensemble (CBE).

The goal is to give a precise theorem whose loose form is GMC = CBE. Although it was known that random matrices exhibit log-correlated features, such an exact correspondence is quite a surprise.

Facilitated Exclusion Processes

Facilitated exclusion processes are lattice gasses in which a particle with an empty neighboring site can jump to that site only if it has also an occupied neighboring site. We will discuss three such models in one dimension, for both discrete-time and continuous-time dynamics and with varying degrees of asymmetry. We address two questions: What are the possible translation invariant stationary states? If the initial state is Bernoulli, what is the final state? This is joint work with Arvind Ayyer, Shelly Goldstein, and Joel Lebowitz.

Rough walks in random environment

Random walks in random environment (RWRE) have been extensively studied in the last half-century. Functional central limit theorems (FCLT) hold in some prototypical classes such the reversible and the ballistic ones. The latter are treated using rather different techniques; Kipnis-Varadhan's theory for additive functionals of Markov processes is applicable in the reversible case whereas the main feature exploited in the ballistic class is a regeneration structure. Rough path theory is a deterministic theory which extends classical notions of integration to singular integrators in a continuous manner. It typically provides a framework for pathwise solutions of ordinary and partial stochastic differential equations driven by a singular noise. In the talk we shall discuss FCLT for additive functionals of Markov processes and regenerative processes lifted to the rough path space. The limiting rough path has two levels. The first one is the Brownian motion, whereas in the second we see a new feature: it is the iterated integral of the Brownian motion perturbed by a deterministic linear function called the area anomaly. The aforementioned classes of RWRE are covered as special cases. The results provide sharper information on the limiting path. In addition, the construction of new examples for SDE approximations is an immediate application.

Based on collaborations (some still in progress) with Johannes Bäumler, Noam Berger, Jean-Dominique Deuschel, Olga Lopusanschi, Nicolas Perkowski and Martin Slowik.

References:

1) Additive functionals as rough paths, with Jean-Dominique Deuschel and Nicolas Perkowski, Ann. Probab. 49(3): 1450-1479 (May 2021). DOI: 10.1214/20-AOP1488.

2) Ballistic random walks in random environment as rough paths: convergence and area anomaly, with Olga Lopusanschi, ALEA, Lat. Am. J. Probab. Math. Stat. 18, 945–962 (April 2021) DOI: 10.30757/ALEA.v18-34.

3) Rough invariance principle for delayed regenerative processes, arXiv:2101.05222.

Sharp thresholds for Glauber dynamics percolation

Sharp threshold is a phenomena that characterizes abrupt change of behavior in a phase transition. In this talk, we prove that the two-dimensional model obtained by running Glauber dynamics up to some finite time yields a percolation model that presents a sharp threshold. The proof is based on special properties of a graphical construction of the model that allows us to pass from annealed to quenched measurements. Based on a joint work with C. Alves, G. Amir, and A. Teixeira.

A tale of two balloons

From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Will balloons reach the origin infinitely often or not? We answer this question for various underlying spaces. En route we find a new(ish) 0-1 law, and generalize bounds on independent sets that are factors of IID on trees. Joint work with Omer Angel and Gourab Ray.

Note that we are now a joint probability seminar with IMPA. See here: https://sbp.impa.br

Condensation of SIP particles and sticky Brownian motions

The symmetric inclusion process (SIP) is a particle system with attractive interaction. We study its behavior in the condensation regime attained for large values of the attraction intensity. Using Mosco convergence of Dirichlet forms, we prove convergence to sticky Brownian motion for the distance of two SIP particles. We use this result to obtain, via duality, an explicit scaling for the variance of the density field in this regime, for the SIP initially started from a homogeneous product measure. This provides relevant new information on the coarsening dynamics of condensing particle systems on the infinite lattice. Joint work with M. Ayala and F. Redig.

Note that we are now a joint probability seminar with IMPA. See here: https://sbp.impa.br

Hydrodynamic equations for the classical Toda lattice

The Toda lattice is one of the most famous integrable system of classical mechanics. For N lattice sites there are N+1 conserved quantities. We are interested in the hydrodynamic scale, which means to start with suitably adjusted random initial data. In my talk I will outline the general structure and the resulting hydrodynamic Euler type equations. An unexpected connection to the repulsive log gas in one dimension is discussed.

Random walk on the simple symmetric exclusion process

In a joint work with Marcelo R. Hilário and Augusto Teixeira, we investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole. The asymptotic behavior is expected to depend on the density $\rho$ in $[0, 1]$ of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities ρ except for at most two values $\rho^-$ and $\rho^+$ in $[0, 1]$, where the speed (as a function fo the density) possibly jumps from, or to, $0$. Second, we prove that, for any density corresponding to a non-zero speed regime, the fluctuations are diffusive and a Central Limit Theorem holds. Our main results extend to environments given by a family of independent simple symmetric random walks in equilibrium.

Exact solution of an integrable particle system

We consider the family of boundary-driven models introduced in [FGK] and show they can be solved exactly, i.e. the correlations functions and the non-equilibrium steady-state have a closed-form expression.

The solution relies on probabilistic arguments and techniques inspired by integrable systems. As in the context of bulk-driven systems (scaling to KPZ), it is obtained in two steps: i) the introduction of a dual process; ii) the solution of the dual dynamics by Bethe ansatz.

For boundary-driven systems, a general by-product of duality is the existence of a direct mapping (a conjugation) between the generator of the non-equilibrium process and the generator of the associated reversible equilibrium process. Macroscopically, this mapping was observed years ago by Tailleur, Kurchan and Lecomte in the context of the Macroscopic Fluctuation Theory.

[FGK] R. Frassek, C. Giardinà, J. Kurchan, Non-compact quantum spin chains as integrable stochastic particle processes, Journal of Statistical Physics 180, 366-397 (2020).

giardina-lisbon-17march-2021.pdf

Please note change of day and time. Joint session with Seminario Brasileiro de Probabilidade: https://sbp.impa.br

Reinforced random walks and statistical physics

We explain how the Edge-reinforced random walk, introduced by Coppersmith and Diaconis in 1986, is related to several models in statistical physics, namely the supersymmetric hyperbolic sigma model studied by Disertori, Spencer and Zirnbauer (2010), the random Schrödinger operator and Dynkin's isomorphism. We also discuss recent non-reversible generalizations of the ERRW and the VRJP.
Based on joint works (or work in progress) with C. Sabot, X. Zeng, T. Lupu, M. Disertori and S. Baccalado.

Slide of the talk.pdf

Conditioned SRW in two dimensions and some of its surprising properties

We consider the two-dimensional simple random walk conditioned on never hitting the origin. This process is a Markov chain, namely it is the Doob $h$-transform of the simple random walk with respect to the potential kernel. It is known to be transient and we show that it is "almost recurrent" in the sense that each infinite set is visited infinitely often, almost surely. After discussing some basic properties of this process (in particular, calculating its Green's function), we prove that, for a "large" set, the proportion of its sites visited by the conditioned walk is approximately a Uniform $[0,1]$ random variable. Also, given a set $G\subset R^2$ that does not "surround" the origin, we prove that a.s. there is an infinite number of $k$'s such that $kG \cap Z^2$ is unvisited. These results suggest that the range of the conditioned walk has "fractal" behavior. Also, we obtain estimates on the speed of escape of the walk to infinity, and prove that, in spite of transience, two independent copies of conditioned walks will a.s. meet infinitely many times.

This talk is based on joint papers with Francis Comets, Nina Gantert, Leonardo Rolla, Daniel Ungaretti, and Marina Vachkovskaia.

talk_hatS_Lis_slides.pdf

Static large deviations for a reaction-diffusion model

We examine the stationary state of an interacting particle system whose macroscopic evolution is described by one-dimensional reaction-diffusion equations.

Stat-LD-RD.pdf

Mixing times for the simple exclusion process with open boundaries

We study mixing times of the symmetric and asymmetric simple exclusion process on the segment where particles are allowed to enter and exit at the endpoints. We consider different regimes depending on the entering and exiting rates as well as on the rates in the bulk, and show that the process exhibits pre-cutoff and in some special cases even cutoff.

No prior knowledge is assumed.

Based on joint work with Evita Nestoridi (Princeton) and Dominik Schmid (Munich).

lissabon.pdf

Projecto FCT UIDB/04459/2020.

The Mathematics of making a mess (an introduction to random walk on groups)

How many random transpositions does it take to mix up $n$ cards? This is a typical question of random walk on finite groups. The answer is $\frac{1}{2}n \log{n} + Cn$ and there is a sharp phase transition from order to chaos as $C$ varies. The techniques involve Fourier analysis on non-commutative groups (which I will try to explain for non specialists). As you change the group or change the walk, new analytic and algebraic tools are required. The subject has wide applications (people still shuffle cards, but there are applications in physics, chemistry,biology and computer science — even for random transpositions). Extending to compact or more general groups opens up many problems. This was the first problem where the ‘cutoff phenomenon’ was observed and this has become a healthy research area.