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Tracer diffusion beyond the ergodicity transition in the North-East model.
Abstract
Can a particle diffuse through an environment containing permanently frozen regions? This talk explores this question in the North-East model, where simple local rules give rise to permanent blocking.
I will begin with a pedagogical introduction to kinetically constrained models. In the North-East model, each site of a lattice is either occupied or empty, and can update only when its neighbours in the positive coordinate directions are empty. At sufficiently high densities of occupied sites, infinite oriented paths of occupied sites appear. These paths remain frozen forever, causing the dynamics to lose ergodicity.
To investigate transport in this environment, we introduce a passive tracer that jumps between neighbouring empty sites without influencing the underlying dynamics. The central question is whether the onset of permanent blocking also marks the loss of tracer diffusion. In two dimensions, frozen structures form barriers that confine the tracer and force its diffusion coefficient to vanish. In higher dimensions, the geometry leaves open the possibility that frozen structures and positive diffusivity coexist.
The main part of the talk will be devoted to ongoing work on positive diffusivity in the non-ergodic regime. I will explain the main ideas of our approach, which compares the tracer with a random walk on an auxiliary percolation cluster, and discuss the geometric properties needed to establish a positive diffusion coefficient. No prior knowledge of kinetically constrained models or percolation will be assumed.
Based on joint work with Oriane Blondel and Cristina Toninelli.
