Extremal scaling limits for random walks in space-time random environments

In this talk, we will consider the model of random walks in a space-time random environment, which can be thought of as a discrete model for diffusing particles in a time-dependent random medium. We will study the scaling limits of these models in certain moderate deviation scaling regimes for the quenched probability density and show that they are described by stochastic PDEs. The solutions to these SPDEs are Gaussian processes up to a dimension-dependent critical scale. In d=1 we prove that the critical fluctuations are given by the KPZ equation. In d=2, we conjecture that the scaling limit at criticality is given by the 2d critical stochastic heat flow recently constructed by Caravenna, Sun, and Zygouras.

This is based on joint works with Sayan Das and Shalin Parekh.