Statistical Self-Similarity of One-Dimensional Growth Processes

TL;DR

This paper investigates the distribution of heights in one-dimensional growth processes, demonstrating that self-similar growth from a single seed follows the Tracy-Widom distribution, linking growth models to random matrix theory.

Contribution

It establishes the universal Tracy-Widom distribution as the height distribution for certain growth processes, connecting growth phenomena with random matrix theory and identifying specific cases.

Findings

Height distribution in self-similar growth is Tracy-Widom.

Growth from a flat substrate has a different, numerically determined distribution.

Polynuclear growth model height maps to the longest increasing subsequence, following Tracy-Widom distribution.

Abstract

For one-dimensional growth processes we consider the distribution of the height above a given point of the substrate and study its scale invariance in the limit of large times. We argue that for self-similar growth from a single seed the universal distribution is the Tracy-Widom distribution from the theory of random matrices and that for growth from a flat substrate it is some other, only numerically determined distribution. In particular, for the polynuclear growth model in the droplet geometry the height maps onto the longest increasing subsequence of a random permutation, from which the height distribution is identified as the Tracy-Widom distribution.