2013/03/12 by Alexei Borodin, Borodin, Alexei, Grigori Olshanski +1 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Representation Theory (math.RT) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1303.2794
openalex publication_date 2013/03/12 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
The Thoma cone is an infinite-dimensional locally compact space, which is\nclosely related to the space of extremal characters of the infinite symmetric\ngroup. In another context, the Thoma cone appears as the set of parameters for\ntotally positive, upper triangular Toeplitz matrices of infinite size.\n The purpose of the paper is to construct a family of continuous time Markov\nprocesses on the Thoma cone, depending on two continuous parameters. Our\nconstruction largely exploits specific properties of the Thoma cone related to\nits representation-theoretic origin, although we do not use representations\ndirectly. On the other hand, we were inspired by analogies with random matrix\ntheory coming from models of Markov dynamics related to orthogonal polynomial\nensembles.\n We show that our processes possess a number of nice properties, namely: (1)\nevery process X is a Feller process; (2) the infinitesimal generator of X, its\nspectrum, and the eigenfunctions admit an explicit description; (3) in the\nequilibrium regime, the finite-dimensional distributions of X can be\ninterpreted as (the laws of) infinite-particle systems with determinantal\ncorrelations; (4) the corresponding time-dependent correlation kernel admits an\nexplicit expression, and its structure is similar to that of time-dependent\ncorrelation kernels appearing in random matrix theory.\n