Random Matrices, the Ulam Problem, Directed Polymers & Growth Models, and Sequence Matching

TL;DR

This paper introduces the interconnectedness of four complex problems in mathematics and physics, highlighting the Tracy-Widom distribution as a unifying theme and illustrating how to derive this distribution in specific growth and sequence alignment models.

Contribution

It reveals the common Tracy-Widom distribution underlying diverse problems and demonstrates how to map different models to derive this distribution explicitly.

Findings

Tracy-Widom distribution appears in multiple models

Mapping models helps derive the Tracy-Widom law

Connections between random matrices, growth, and sequence alignment

Abstract

In these lecture notes I will give a pedagogical introduction to some common aspects of 4 different problems: (i) random matrices (ii) the longest increasing subsequence problem (also known as the Ulam problem) (iii) directed polymers in random medium and growth models in (1+1) dimensions and (iv) a problem on the alignment of a pair of random sequences. Each of these problems is almost entirely a sub-field by itself and here I will discuss only some specific aspects of each of them. These 4 problems have been studied almost independently for the past few decades, but only over the last few years a common thread was found to link all of them. In particular all of them share one common limiting probability distribution known as the Tracy-Widom distribution that describes the asymptotic probability distribution of the largest eigenvalue of a random matrix. I will mention here, without mathematical derivation, some of the beautiful results discovered in the past few years. Then, I will consider two specific models (a) a ballistic deposition growth model and (b) a model of sequence alignment known as the Bernoulli matching model and discuss, in some detail, how one derives exactly the Tracy-Widom law in these models. The emphasis of these lectures would be on how to map one model to another. Some open problems are discussed at the end.