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 satisfy 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.
Highlights
In this paper we construct a unitary matrix model that captures the asymptotic growth of Young diagrams under q-deformed Plancherel measure
We study the asymptotic growth of Young diagrams, equipped with q analog of Plancherel measure and show that one can write a unitary matrix model (UMM) to capture such growth processes
Using the connection between UMM and 2D droplets we show that there exists an equivalent U (N ) UMM for the partition function (3.3) whose no-gap phase captures the growth of Young diagrams under q-deformed Plancherel measure
Summary
Suvankar Duttaa, Debangshu Mukherjeeb, Neetua, Sanhita Parihara aDepartment of Physics Indian Institute of Science Education and Research Bhopal, Bhopal bypass, Bhopal 462066, India bIndian Institute of Science Education and Research Thiruvananthapuram, Maruthamala PO, Vithura, Thiruvananthapuram - 695551, Kerala, India.
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