Coconut

In this project, we propose to study the geometry of random models on lattices such as percolation and the Ising model. In these models, when microscopic interactions reach a certain critical threshold, they propagate to all scales, giving rise to a fractal geometric structure. More precisely, our aim is to study the effect of introducing noise into these models, through two phenomena: noise sensitivity and chaos. The introduction of noise also provides methods to better understand the correlations between different geometric properties of the models. Our starting points are: (1) The work of Benjamini, Kalai and Schramm, who introduced the notion of noise sensitivity in 1999. In particular, we aim to better understand the links between this notion and the universality conjecture in planar percolation; (2) The links between superconcentration and chaos proposed by Chatterjee in 2008. We will then have in mind in particular the study of fluctuations of the random metric in first passage percolation and that of chaos phenomena in the Ising model; (3) More recent work indicating that it would be fruitful to create more bridges between the study of these two phenomena.