Probability and Stochastic Analysis Seminar 

12/01/2021, 14:00 — 15:00 — Room P3.10, Mathematics Building Online
Serguei Popov, Universidade de Porto

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.

See also

talk_hatS_Lis_slides.pdf

15/12/2020, 14:00 — 15:00 — Room P3.10, Mathematics Building Online
Claudio Landim, Instituto Nacional de Matemática Pura e Aplicada

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.

See also

Stat-LD-RD.pdf

10/11/2020, 14:00 — 15:00 — Room P3.10, Mathematics Building Online
Nina Gantert, Technische Universität München

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).

13/10/2020, 14:00 — 15:00 — Room P3.10, Mathematics Building Online
Persi Diaconis, Stanford University

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.

See also

Diaconis_notes.pdf