Fluctuations of Young diagrams for symplectic groups and semiclassical orthogonal polynomials

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 $G{L}_n\times G{L}_k$-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 $G{L}_n\times G{L}_k$-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 $S{p}_{2n}\times S{p}_{2k}$. 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 $S{p}_{2n}\times S{p}_{2k}$ from Krawtchouk polynomials that describe $G{L}_{2n}\times G{L}_{2k}$ 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.

29 pages, 1 figure, 1 table. v2 fixes incorrect statement about tensor power and normalization of polynomials, adds table of polynomials

References in corpus (2)

• Asymptotics of Plancherel-type random partitions
• Skew Howe duality and limit shapes of Young diagrams