Heat Diffusion with Frozen Boundary

Abstract

Consider “frozen random walk” on $\mathbb {Z}$: n particles start at the origin. At any discrete time, the leftmost and rightmost $\lfloor {\frac{n}{4}}\rfloor$ particles are “frozen” and do not move. The rest of the particles in the “bulk” independently jump to the left and right uniformly. The goal of this note is to understand the limit of this process under scaling of mass and time. To this end we study the following deterministic mass splitting process: start with mass 1 at the origin. At each step the extreme quarter mass on each side is “frozen”. The remaining “free” mass in the center evolves according to the discrete heat equation. We establish diffusive behavior of this mass evolution and identify the scaling limit under the assumption of its existence. It is natural to expect the limit to be a truncated Gaussian. A naive guess for the truncation point might be the 1 / 4 quantile points on either side of the origin. We show that this is not the case and it is in fact determined by the evolution of the second moment of the mass distribution.