An Anisotropic Ballistic Deposition Model with Links to the Ulam Problem and the Tracy-Widom Distribution

TL;DR

This paper establishes a connection between an anisotropic ballistic deposition model and the Ulam problem, demonstrating that the scaled height distribution follows the Tracy-Widom distribution, supporting universality in growth models within the KPZ class.

Contribution

The paper provides an exact asymptotic distribution for the model's height by linking it to the Ulam problem, revealing Tracy-Widom distribution as a universal feature.

Findings

Scaled height follows Tracy-Widom distribution

Supports universality in KPZ class growth models

Links growth models to random matrix theory

Abstract

We compute exactly the asymptotic distribution of scaled height in a (1+1)--dimensional anisotropic ballistic deposition model by mapping it to the Ulam problem of finding the longest nondecreasing subsequence in a random sequence of integers. Using the known results for the Ulam problem, we show that the scaled height in our model has the Tracy-Widom distribution appearing in the theory of random matrices near the edges of the spectrum. Our result supports the hypothesis that various growth models in $(1+1)$ dimensions that belong to the Kardar-Parisi-Zhang universality class perhaps all share the same universal Tracy-Widom distribution for the suitably scaled height variables.