First passage percolation on the exponential of two-dimensional branching random walk

TL;DR

This paper investigates first passage percolation on a two-dimensional branching random walk with exponential weights, showing that for small gamma, the expected crossing distance scales sublinearly with the size of the domain.

Contribution

It provides bounds on the expected FPP distance in a branching random walk model with exponential weights, revealing how the distance scales with domain size for small gamma.

Findings

Expected FPP distance is at most O(N^{1 - b^2/10}) for small b>0.

The model exhibits sublinear growth in crossing distances as domain size increases.

Results contribute to understanding of percolation in correlated random environments.

Abstract

We consider the branching random walk $\{\mathcal R^N_z: z\in V_N\}$ with Gaussian increments indexed over a two-dimensional box $V_N$ of side length $N$, and we study the first passage percolation where each vertex is assigned weight $e^{\gamma \mathcal R^N_z}$ for $\gamma>0$. We show that for $\gamma>0$ sufficiently small but fixed, the expected FPP distance between the left and right boundaries is at most $O(N^{1 - \gamma^2/10})$.