Random words, quantum statistics, central limits, random matrices

TL;DR

This paper proves that the shape of a semi-standard tableau from a random word asymptotically matches the spectrum of a specific random matrix, using quantum and probabilistic methods, generalizing to Lie algebra representations.

Contribution

It provides two novel proofs of the asymptotic shape distribution, connecting quantum random variables and local limit theorems, extending results to arbitrary Lie algebra representations.

Findings

The shape distribution converges to the spectrum of a traceless GUE matrix.

Quantum random variable realization offers a new perspective on the problem.

Asymptotic Gaussian behavior is shown with additional polynomial weights and reflections.

Abstract

Recently Tracy and Widom conjectured [math.CO/9904042] and Johansson proved [math.CO/9906120] that the expected shape \lambda of the semi-standard tableau produced by a random word in k letters is asymptotically the spectrum of a random traceless k by k GUE matrix. In this article we give two arguments for this fact. In the first argument, we realize the random matrix itself as a quantum random variable on the space of random words, if this space is viewed as a quantum state space. In the second argument, we show that the distribution of \lambda is asymptotically given by the usual local limit theorem, but the resulting Gaussian is disguised by an extra polynomial weight and by reflecting walls. Both arguments more generally apply to an arbitrary finite-dimensional representation V of an arbitrary simple Lie algebra g. In the original question, V is the defining representation of g = su(k).