UBC Probability Seminar: Leo Gayral

A Subshift of Finite Type (SFT) is a translational-invariant set of colourings of a lattice by a finite alphabet, induced by a finite set of local forbidden patterns. Ergodic theory allows for a bridge between SFTs and a more probabilistic approach, by focusing on “generic” configurations in some sense.

This presentation will focus on what happens when a small amount of random noise, of mismatching patterns, is allowed, in particular as the amount of noise goes to 0. I will first introduce some general ideas on symbolic dynamics and ergodic theory, and then move onto the noisy framework to highlight the main results and ideas of the article On the Besicovitch-Stability of Noisy Random Tilings (https://arxiv.org/abs/2104.09885).

More precisely, after introducing a topology adapted to problem at hand, we will see how we can obtain stability for both periodic and aperiodic tilings. In doing so, we will use mostly elementary probabilistic methods, and in particular study percolations on Zdwith finite-range dependences. Slides will be available beforehand on my webpage (https://lgayral.pages.math.cnrs.fr/2022/Slides-SeminaireFourier2022.pdf).