Determinantal measures related to big q-Jacobi polynomials

Abstract

Abstract. We define a novel combinatorial object-the extended Gelfand-Tsetlin graph with cotransition probabilities depending on a parameter q. The boundary of this graph admits an explicit description. We introduce a family of probability measures on the boundary and describe their correlation functions. These measures are a q-analogue of the spectral measures studied earlier in the context of the problem of harmonic analysis on the infinite-dimensional unitary group.Key words: Gelfand-Tsetlin graph, determinantal measures, big q-Jacobi polynomials, basic hypergeometric series.Determinantal measures form a special class of probability measures on spaces of locally finite point configurations (see Borodin's survey [1]). The key property of a determinantal measure is that its correlation functions of any order are expressed in a simple way through a function in two variables, called the correlation kernel.A family of determinantal measures, called the zw-measures, was studied by Borodin and Olshanski [2] in connection with the problem of harmonic analysis on the infinite-dimensional unitary group U (∞), posed by Olshanski [12]. Analogues of the zw-measures also exist for other infinitedimensional classical groups and for infinite-dimensional symmetric spaces (see Olshanski and Osinenko's paper [14]). The zw-measures play a fundamental role in infinite-dimensional harmonic analysis, because their scaling limits govern the spectral decomposition of certain distinguished unitary representations.On the other hand, the zw-measures are nice combinatorial objects and have much in common with the so-called z-measures, which are a particular case of Okounkov's Schur measures on partitions. Like Schur measures, the zw-measures admit a generalization involving an additional deformation parameter similar to Dyson's β-parameter in random matrix theory or the continuous parameter of the Jack symmetric functions (see Olshanski's paper [13]).Our aim is to show that the notion of zw-measures can be extended in another direction; namely, there exists a (nonevident) q-analogue of the zw-measures. Our first result says that the "N -particle q-zw-measures" have a large-N limit, and the limiting probability measure is a determinantal measure on a space of infinite point configurations. We find the corresponding correlation kernelit is expressed through the basic hypergeometric function 2 φ 1 . The large-N limit transition is related to another result, which is of independent interest-a description of the q-boundary for an extended version of the Gelfand-Tsetlin graph.1. The q-analogue of zw-measures. We fix a triple (ζ + , ζ − , q) of real parameters, where ζ + > 0, ζ − < 0, and 0 < q < 1. The corresponding double q-lattice is a subsetBy a configuration we mean an arbitrary subset X ⊂ L. For N = 1, 2, . . . , let G N denote the countable set consisting of all N -point configurations. We enumerate the points of every X ∈ G N in increasing order and write X = (x 1 < · · · < x N ).We are going to introduce, for every N = 1, 2, . . . , a ...

$q$-Pearson pair and moments in $q$-deformed ensembles

“…There are many setting involving (1.1) defined a linear discrete lattice {x ∈ Z} in different physically significant models; see for example [28,29,34,35].…”

Infinite-Dimensional Measure Spaces and Frame Analysis

“…In our case, we shall consider the case where the local determinants are computed from Grammians computed from overcomplete frame systems (details below). Determinantal processes arise as important tools in random matrix theory, in combinatorics, and in physics, see e.g., [GoOl15,OlGr11].…”