Markov processes on the path space of the Gelfand-Tsetlin graph and on its boundary
The authors construct a Feller process on the space \(\Omega\) of extreme characters of the infinite-dimensional unitary group \(U(\infty)\), preserving the so-called \(zw\)-measure on \(\Omega\). The four parameter family of \(zw\)-measures arises in a problem of harmonic analysis on \(U(\infty)\) as the decomposing measures for a distinguished family of characters [the second author, J. Funct. Anal. 205, No. 2, 464--524 (2003; Zbl 1036.43002)]. A \(zw\)-measure gives rise to a determinantal point process on the real line with two punctures, the corresponding correlation kernel being given in terms of the Gauss hypergeometric function [the authors, Ann. Math. (2) 161, No. 3, 1319--1422 (2005; Zbl 1082.43003)]. The construction of the infinite-dimensional stochastic dynamics is based in the fact that the probability measures on \(\Omega\) are in one-to-one correspondence with Gibbs measures on the infinite Gelfand-Tsetlin schemes that can also be viewed as stepped surface or lozenge tilings of a half plane.NEWLINENEWLINE The projections of a \(zw\)-measure to suitably defined slices of these infinite schemes yield orthogonal polynomial ensembles with weight functions corresponding to hypergeometric Askey-Lesky orthogonal polynomials. These are eigenfunctions for a birth and death process on \(\mathbb{Z}\) with quadratic jump rates. The \(N\)-dimensional Askey-Lesky orthogonal polynomial ensemble is then preserved by a Doob's \(h\)-transform of \(N\) independent birth and death processes. Markov processes on the slices are consistent with respect to stochastic projections of the \(N\)th slice to the \((N-1)\)st one, and this consistency turns out to be essentially sufficient for defining the corresponding Markov process on \(\Omega\).
scientific article; zbMATH DE number 6055166
scientific article