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

Abstract

We consider the branching random walk $\{{\mathcal{R}}_z^N:z\in V_N\}$ 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 $e^{\gamma {\mathcal{R}}_z^N}$ 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 $O(N^{1-{\gamma }^2/10})$.

14 pages, 6 figures. The current version was accepted for publication in Electronic Communications in Probability

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Cited by in corpus (2)

• Liouville first-passage percolation: subsequential scaling limits at high temperature
• Branching random walk in the presence of a hard wall