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

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.

arXiv comment19+11 pages; 6 figures

References in corpus (11)

• Open string amplitudes and large order behavior in topological string theory
• Developments in 2D String Theory
• Phase transitions, double-scaling limit, and topological strings
• Free Fermions and Thermal AdS/CFT
• Hamiltonian Structures for the Generalized Dispersionless KdV Hierarchy
• Emergent Phase Space Description of Unitary Matrix Model
• Dispersionless Limit of Integrable Models
• Phase Space Distribution for Two-Gap Solution in Unitary Matrix Model
• Phase Space Distribution of Riemann Zeros
• A Differential Model for the Deformation of the Plancherel Growth Process
• Quantum Mechanics of Plancherel Growth