A Unitary Matrix Model for $q$-deformed Plancherel Growth

TL;DR

This paper introduces a $q$-deformed unitary matrix model that describes the asymptotic growth of Young diagrams, revealing a phase transition and connecting droplet descriptions with quantum states.

Contribution

It constructs a novel $q$-analog matrix model for Young diagram growth and analyzes its phase transition and droplet quantization.

Findings

Identifies a $q$-deformed GWW phase transition.

Derives a $q$-automodel differential equation.

Connects Young diagrams with coherent quantum states.

Abstract

In this paper we construct a unitary matrix model that captures the asymptotic growth of Young diagrams under $q$-deformed Plancherel measure. The matrix model is a $q$ analog of Gross-Witten-Wadia (GWW) matrix model. In the large $N$ limit the model exhibits a third order phase transition between no-gap and gapped phases, which is a $q$-deformed version of the GWW phase transition. We show that the no-gap phase of this matrix model captures the asymptotic growth of Young diagrams equipped with $q$-deformed Plancherel measure. The no-gap solutions also satisfies a differential equation which is the $q$-analogue of the automodel equation. We further provide a droplet description for these growing Young diagrams. Quantising these droplets we identify the Young diagrams with coherent states in the Hilbert space. We also elaborate the connection between moments of Young diagrams and the infinite number of commuting Hamiltonians obtained from the large $N$ droplets and explicitly compute the moments for asymptotic Young diagrams.