Quantum Mechanics of Plancherel Growth

TL;DR

This paper connects the growth of Young diagrams under Plancherel measure to phase transitions in matrix models, providing new insights into their asymptotic shapes, quantum fluctuations, and a novel proof of classical limit shape theorems.

Contribution

It establishes a precise correspondence between Young diagram growth, matrix model phase transitions, and quantum coherent states, offering a new perspective and proof of the limit shape theorem.

Findings

Limit shape of Young diagrams corresponds to GWW transition point.

Growth process is equivalent to phase transition in GWW model.

Quantum fluctuations of diagrams relate to coherent state fluctuations.

Abstract

Growth of Young diagrams, equipped with Plancherel measure, follows the automodel equation of Kerov. Using the technology of unitary matrix model we show that such growth process is exactly same as the growth of gap-less phase in Gross-Witten and Wadia (GWW) model. The limit shape of asymptotic Young diagrams corresponds to GWW transition point. Our analysis also offers an alternate proof of limit shape theorem of Vershik-Kerov and Logan-Shepp. Using the connection between unitary matrix model and free Fermi droplet description, we map the Young diagrams in automodel class to different shapes of two dimensional phase space droplets. Quantising these droplets we further set up a correspondence between automodel diagrams and coherent states in the Hilbert space. Thus growth of Young diagrams are mapped to evolution of coherent states in the Hilbert space. Gaussian fluctuations of large $N$ Young diagrams are also mapped to quantum (large $N$) fluctuations of the coherent states.