Convergence to the Tracy-Widom distribution for longest paths in a directed random graph

TL;DR

This paper proves that the maximum path length in a directed lattice graph converges to the Tracy-Widom distribution under certain scaling conditions, extending to non-constant probabilities.

Contribution

It establishes the convergence of the longest path length distribution to Tracy-Widom in a directed lattice graph, including a generalization for variable edge probabilities.

Findings

Convergence to Tracy-Widom distribution under specific scaling.

Identification of a positive exponent relating rectangle dimensions.

Extension to graphs with non-constant edge probabilities.

Abstract

We consider a directed graph on the 2-dimensional integer lattice, placing a directed edge from vertex $(i_1,i_2)$ to $(j_1,j_2)$, whenever $i_1 \le j_1$, $i_2 \le j_2$, with probability $p$, independently for each such pair of vertices. Let $L_{n,m}$ denote the maximum length of all paths contained in an $n \times m$ rectangle. We show that there is a positive exponent $a$, such that, if $m/n^a \to 1$, as $n \to \infty$, then a properly centered/rescaled version of $L_{n,m}$ converges weakly to the Tracy-Widom distribution. A generalization to graphs with non-constant probabilities is also discussed.