# Probability and Stochastic Analysis Seminar

## Past sessions

### 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 ρ 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 ρ− and ρ+ 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
https://educast.fccn.pt/vod/clips/2k4jbq7tn5/streaming.html?locale=en

### 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
https://educast.fccn.pt/vod/clips/12qaveluym/streaming.html?locale=en

### 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.