Tracy-Widom asymptotics for a river delta model

TL;DR

This paper analyzes a solvable model of river delta evolution, revealing that the delta's width scales as L^{2/3} with Tracy-Widom GUE fluctuations, connecting it to particle systems with similar statistical properties.

Contribution

It introduces an exactly solvable model for river delta growth and derives Tracy-Widom asymptotics for its width and particle system fluctuations.

Findings

Delta width scales as L^{2/3}

Fluctuations follow Tracy-Widom GUE distribution

Particle system exhibits Tracy-Widom fluctuations after finite time

Abstract

We study an oriented first passage percolation model for the evolution of a river delta. This model is exactly solvable and occurs as the low temperature limit of the beta random walk in random environment. We analyze the asymptotics of an exact formula from [4] to show that, at any fixed positive time, the width of a river delta of length $L$ approaches a constant times $L^{2/3}$ with Tracy-Widom GUE fluctuations of order $L^{4/9}$. This result can be rephrased in terms of particle systems. We introduce an exactly solvable particle system on the integer half line and show that after running the system for only finite time the particle positions have Tracy-Widom fluctuations.