Self-repellent branching random walk

TL;DR

This paper analyzes a branching random walk with a penalty for close particles, deriving the optimal spread and total cost over time, revealing how particles distribute to minimize combined spread and repulsion costs.

Contribution

It introduces a model of self-repellent branching random walk and characterizes the optimal configurations minimizing combined costs over time.

Findings

Particles spread over a distance proportional to ( )^{1/3} 2^{2N/3} at time N.

Total cost of optimal configurations scales as ()^{2/3} 2^{4N/3}.

Optimal configurations balance spread and repulsion costs efficiently.

Abstract

We consider a system of particles performing 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 every time. We study the optimal configurations that minimise the sum of the spread out cost and the repulsion cost up to a given time horizon $N$. We show that at time $N$ particles are spread out over a distance $\asymp (\b\e)^{1/3} 2^{2N/3}$. We also show that the total cost of the optimal configurations up to time $N$ is $\asymp (\b\e)^{2/3} 2^{4N/3}$.