Computer Science/Discrete Mathematics Seminar I

We introduce a new method of obtaining sharp estimates on mixing for Glauber dynamics for the Ising model, which, in particular, establishes cutoff in three dimensions up to criticality. The new framework, which considers ``information percolation'' clusters in the space-time slab, shows that total-variation mixing exhibits cutoff with an \(O(1)\)-window around the time at which the magnetization is the square-root of the volume. Furthermore, this method opens the door to understanding the effect of the initial state on the mixing time, showing that starting from the uniform (``disordered'') initial distribution asymptotically halves the mixing time, whereas almost every deterministic starting state is asymptotically as bad as starting from the (``ordered'') all-plus state. Joint work with Allan Sly.