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Classification of Young diagram local fluctuations for symplectic groups
Classification of Young diagram local fluctuations for symplectic groups
Organizers
Speaker
Anton Selemenchuk
Time
Friday, November 14, 2025 1:00 PM - 2:30 PM
Venue
A3-4-301
Online
Zoom 242 742 6089
(BIMSA)
Abstract
We study local fluctuations or random Young diagrams with respect to the probability measure arising from skew $(\mathrm{Sp}_{2n},\mathrm{Sp}_{2k})$ Howe duality. In the limit $n,k\to\infty$ with $n\sim k$ random Young diagrams converge to the limit shape. Diagrams are represented by random particle configurations described by the orthogonal polynomial ensemble with semiclassical orthogonal polynomials obtained by Christoffel transformation from Krawtchouk polynomials. These semiclassical
orthogonal polynomials satisfy difference equation. By studying spectral theory of the operator that corresponds to this equation we single out four asymptotic regimes of local fluctuations and describe the limits of Christoffel-Darboux correlation kernel.
Besides the universal bulk fluctuations described by the discrete sine kernel and universal Airy fluctuations on the right edge of the limit shape, we observe discrete Hermite kernel if $\lim k/n=1$ and new type of local nonlinear behavior on the left edge of the limit
shape.
orthogonal polynomials satisfy difference equation. By studying spectral theory of the operator that corresponds to this equation we single out four asymptotic regimes of local fluctuations and describe the limits of Christoffel-Darboux correlation kernel.
Besides the universal bulk fluctuations described by the discrete sine kernel and universal Airy fluctuations on the right edge of the limit shape, we observe discrete Hermite kernel if $\lim k/n=1$ and new type of local nonlinear behavior on the left edge of the limit
shape.