Self-repellent branching random walk

Abstract: We consider a discrete-time binary branching random walk with independent standard normal increments subject to a penalty $\b$ for every pair of particles that get within distance $\e$ of each other at any time. We give a precise description of the most likely configurations of the particles under this law for $N$ large and $\b,\e$ fixed. Particles spread out over a distance $2^{2N/3}$, essentially in finite time, and subsequently arrange themselves so that at time $2N/3$ they cover a grid of width $\e$ with one particle per site. After time $2N/3$, the bulk of the particles and their descendants do not move anymore, while the particles in a boundary layer of width $2^{N/3}$ form a ``staircase" to the particles in the bulk. At time $N$, each site in the boundary layer is occupied by $2^{N/3}$ particles.