Fluctuations of Young diagrams for symplectic groups and semiclassical orthogonal polynomials
Anton Nazarov, Anton Selemenchuk
TL;DR
This work analyzes the local fluctuations of random Young diagrams arising from skew Howe duality for symplectic groups $(Sp_{2n},Sp_{2k})$. It constructs a determinantal point process via Proctor/Berele-type insertion, expressing the measure through symplectic dimensions and Krawtchouk-weighted determinants, and then uses a Christoffel transformation to build a semiclassical family of orthogonal polynomials that encode the correlation kernel. Through double-scaling asymptotics of these polynomials, the authors derive the bulk limit of the kernel, showing convergence to the discrete sine kernel with an explicit density $\rho(x)$, thereby establishing universality in the symplectic setting and linking to the limit shape results. The analysis combines Christoffel-Darboux kernels, QR dynamics, and saddle-point methods for Krawtchouk polynomials to connect representation-theoretic measures with integrable probability. These results extend the understanding of fluctuations from the general linear to the symplectic symmetry class and align with prior limit-shape findings while highlighting new techniques for semiclassical ensembles.
Abstract
Consider an $n\times k$ matrix of i.i.d. Bernoulli random numbers with $p=1/2$. Dual RSK algorithm gives a bijection of this matrix to a pair of Young tableaux of conjugate shape, which is manifestation of skew Howe $GL_{n}\times GL_{k}$-duality. Thus the probability measure on zero-ones matrix leads to the probability measure on Young diagrams proportional to the ratio of the dimension of $GL_{n}\times GL_{k}$-representation and the dimension of the exterior algebra $\bigwedge\left(\mathbb{C}^{n}\otimes\mathbb{C}^{k}\right)$. Similarly, by applying Proctor's algorithm based on Berele's modification of the Schensted insertion, we get skew Howe duality for the pairs of groups $Sp_{2n}\times Sp_{2k}$. In the limit when $n,k\to\infty$ $GL$-case is relatively easily studied by use of free-fermionic representation for the correlation kernel. But for the symplectic groups there is no convenient free-fermionic representation. We use Christoffel transformation to obtain the semiclassical orthogonal polynomials for $Sp_{2n}\times Sp_{2k}$ from Krawtchouk polynomials that describe $GL_{2n}\times GL_{2k}$ case. We derive an integral representation for semiclassical polynomials. The study of the asymptotic of this integral representation gives us the description of the limit shapes and fluctuations of the random Young diagrams for symplectic groups.