MARK KAC SEMINAR

The OSSS inequality [O'Donnell, Saks, Schramm and Servedio, 2005] gives an upper bound for the variance of a function f of independent 0-1 valued random variables, in terms of the influences of these random variables and the computational complexity of a (randomised) algorithm for determining the value of f. Since the break-through work around 2017 by Duminil-Copin, Raoufi and Tassion, who obtained a generalization to monotonic measures, the inequality has become one of the main tools to prove sharp phase transition (and other results) in a number of percolation-like and other important models from statistical mechanics. Their generalization of the OSSS inequality raises the question if there are still other measures for which it holds.

We derive a version of the OSSS inequality for a family of measures that are far from monotonic, namely the k-out-of-n measures (these measures correspond with drawing k elements from a set of size n uniformly). During the first part of the talk I will present some introduction and background, state the original OSSS inequality and briefly discuss its proof. After the break I will state the version for k-out-of-n measures and illustrate it by studying the event that there is an occupied horizontal crossing of a large box, in a percolation model where exactly half of the vertices in the box are occupied.

The talk is based on my recent paper https://arxiv.org/abs/2210.16100

1) In this presentation we focus on billiards tables with chaotic behaviour. We will consider several examples of dispersive billiard tables. Some of these tables will be given by finite domains (Sinai billiards in the torus with "round" obstacles, the Bunimovich billiard in a stadium, billiards with cusps), others by infinite domains (Z- or Z^2-periodic Lorentz gas). We will be interested in the asymptotic behaviour of a point particle evolving in these domains according to Descartes' law of reflection (incident angle = reflected angle), with a small randomness on the initial position and velocity (randomness corresponding for example to a small uncertainty on their exact values). We will see that, depending on the billiard table, ergodic sums of natural observables converge, after suitable normalization, to different processes : Brownian motions (with standard or non-standard normalisation), Lévy processes, etc.

2) The second part of the talk deals more specifically with the mixing properties of the Z- and Z^2-periodic Lorentz gas. Because of the Z^d-extension structure, these results are related to some mixing local limit theorem for the Sinai billiard. We will be interested in mixing results for the collision map as well as for the flow, in the finite and in the infinite horizon cases. In particular, we will present results in collaboration with Dmitry Dolgopyat and Péter Nándori, with Dalia Terhesiu, and with Ian Melbourne and Dalia Terhesiu.