Self-repellent branching random walk

We consider a system of particles performing a discrete-time binary branching random walk with independent standard normal increments subject to a penalty $\text{b}$ for every pair of particles that get within distance $\text{e}$ 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 $\asymp (\text{b}\text{e}{)}^{1\mathrm{/}3}{2}^{2N\mathrm{/}3}$. We also show that the total cost of the optimal configurations up to time $N$ is $\asymp (\text{b}\text{e}{)}^{2\mathrm{/}3}{2}^{4N\mathrm{/}3}$.