Quantum mechanics of Plancherel growth

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