Bristol–Warwick–QMUL informal online probability seminar 2021–2022

Bristol and Warwick webpages

Talks in 2020–21

Please contact one of the organisers for the Zoom link.
The seminar will be followed by an online social gathering with the speaker.

Term 1
Wed 13.10 16:00
Alexander Povolotsky (Dubna)
Generalized TASEP between KPZ and jamming regimes Totally Asymmetric Simple Exclusion Process (TASEP) with generalized update is an integrable stochastic model of interacting particles, which differs from the standard TASEP by the presence of an additional interaction controlling the degree of particle clustering. As the strength of the interaction varies, the system suffers the transition from the regime in which the fluctuations of the particle flow are described by standard random processes associated with the Kardar-Parisi-Zhang (KPZ) universality class, to the jamming regime, in which all particles stick to one cluster and move synchronously as the simple random walk. I will focus on the limiting laws of fluctuations of distances travelled by tagged particles both in the KPZ and in the transitional regime for two types of initial conditions. In particular new transitional processes interpolating between the known limiting cases will be discussed.
Wed 20.10 16:00
Eveliina Peltola (Bonn and Espoo)
Large deviations of SLEs, real rational functions, and zeta-regularized determinants of Laplacians When studying large deviations (LDP) of Schramm-Loewner evolution (SLE) curves, we recently introduced a “Loewner potential” that describes the rate function for the LDP. This object turned out to have several intrinsic, and perhaps surprising, connections to various fields. For instance, it has a simple expression in terms of zeta-regularized determinants of Laplace-Beltrami operators. On the other hand, minima of the Loewner potential solve a nonlinear first order PDE that arises in a semiclassical limit of certain correlation functions in conformal field theory, arguably also related to isomonodromic systems. Finally, and perhaps most interestingly, the Loewner potential minimizers classify rational functions with real critical points, thereby providing a novel proof for a version of the now well-known Shapiro-Shapiro conjecture in real enumerative geometry.
This talk is based on joint work with Yilin Wang (MIT).
Wed 27.10 16:00
Matan Harel (Boston)
Quantitative estimates on the effect of disorder on low-dimensional lattice models In their seminal work, Imry and Ma predicted that the addition of an arbitrarily small random external field to a low-dimensional statistical physics model causes the usual first-order phase transition to be `rounded-off.' This phenomenon was proven rigorously by Aizenman and Wehr in 1989 for a vastly general class of spin systems and random perturbations. Recently, the effect was quantified for the random-field Ising model, proving that it exhibits exponential decay of correlations at all temperatures. Unfortunately, the analysis relies on the monotonicity (FKG) properties which are not present in many other classical models of interest. This talk will present quantitative versions of the Aizenman-Wehr theorems for general spin systems with random disorder, including Potts, spin O(n), spin glasses. This is joint work with Paul Dario and Ron Peled.
Wed 03.11 16:00
Ariel Yadin (Beer Sheva)
Realizations of random walk entropy Random walk entropy is a numerical measure of the behaviour of the random walk at infinity. Out of all random walks on groups generated by d elements, the free group has the maximal entropy. One may ask naturally which intermediate values between 0 and the full entropy of the free group can be realized as entropies of random walks on groups. This question is still open. We analyze a related question, which is a "stochastic" version of the above open question. Here we are able to provide a full answer for quotients of the free group, and even a bit further than that. Generalizing results of Bowen (Inventiones 2014), we show that all possible "IRS entropy" values can be realized on the free group. These notions will be precisely explained during the talk.
Based on joint works with Yair Hartman and Liran Ron-George.
Wed 10.11 16:00
Slim Kammoun (Toulouse)
Longest Common Subsequence of Random Permutations Bukh and Zhou conjectured that the expectation of the length of the longest common subsequence (LCS) of two i.i.d random permutations of size n is greater than √ n.
This problem is related to the Ulam-Hammersley problem; Ulam conjectured that the expectation of the length of the longest increasing subsection (LIS) for a uniform permutation behaves like c √ n . The conjecture was solved in 1977, but few results are known for non-uniform permutations. The LIS and LCS are closely related, and solving the conjecture of Bukh and Zhou is equivalent to minimize the expected value of LIS for random permutations that can be written as ρn○ σn, where σn and ρn are i.i.d. random permutations.
We recall the classical results for the uniform case as well as partial answers for the conjugation invariant case.
Wed 17.11 16:00
Fabio Toninelli (Vienna)
Diffusion in the curl of the 2-dimensional Gaussian Free Field I will discuss the large time behaviour of a Brownian diffusion in two dimensions, whose drift is divergence-free, ergodic and given by the curl of the 2-dimensional Gaussian Free Field. Together with G. Cannizzaro and L. Haundschmid, we prove the conjecture by B. Toth and B. Valko that the mean square displacement is of order √ log t;. The same type of superdiffusive behaviour has been predicted to occur for a wide variety of (self)-interacting diffusions in dimension d = 2: the diffusion of a tracer particle in a fluid, self-repelling polymers and random walks, Brownian particles in divergence-free random environments, and, more recently, the 2-dimensional critical Anisotropic KPZ equation. To the best of our authors’ knowledge, ours is the first instance in which √ log t; superdiffusion is rigorously established in this universality class.
Wed 24.11 16:00
Sarah Penington (Bath)
Genealogy of the N-particle branching random walk with polynomial tails The N-particle branching random walk is a discrete time branching particle system with selection consisting of N particles located on the real line. At every time step, each particle is replaced by two offspring, and each offspring particle makes a jump from its parent's location, independently from the other jumps, according to a given jump distribution. Then only the N rightmost particles survive; the other particles are removed from the system to keep the population size constant. I will discuss recent results and open conjectures about the long-term behaviour of this particle system when N, the number of particles, is large. In the case where the jump distribution has regularly varying tails, building on earlier work of J. Bérard and P. Maillard, we prove that at a typical large time the genealogy is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale. Based on joint work with Matt Roberts and Zsófia Talyigás.
Wed 01.12 16:00
Ofer Zeitouni (Rehovot and New York)
High moments of partition function for 2D polymers in the weak disorder regime (joint with Clement Cosco) abstract
Wed 8.12 16:00
Cyril Labbé (Paris)
Mixing points on an interval Consider N points on the unit interval. Resample each point at rate one uniformly in between its two nearest neighbours. The question is: how much time is needed to reach equilibrium (the uniform measure on the simplex) starting from the worst initial condition ? I will present results in collaboration with Pietro Caputo and Hubert Lacoin where we identified the asymptotic of the mixing times and showed a cutoff phenomenon for the distance to equilibrium.
Term 2
Wed 12.01 16:00
Zied Ammari (Rennes)
Wed 19.01 16:00
Shahar Mendelson (Warwick)

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