BC type z-measures and determinantal point processes

Abstract

The (BC type) z-measures are a family of four parameter z, z ′ , a, b probability measures on the path space of the nonnegative Gelfand-Tsetlin graph with Jacobi-edge multiplicities. We can interpret the z-measures as random point processes P z,z ′ ,a,b on the punctured positive real line X = R >0 \ {1}. Our main result is that these random processes are determinantal and moreover we compute their correlation kernels explicitly in terms of hypergeometric functions.For very special values of the parameters z, z ′ , the processes P z,z ′ ,a,b on X are essentially scaling limits of Racah orthogonal polynomial ensembles and their correlation kernels can be computed simply from some limits of the Racah polynomials. Thus, in the language of random matrices, we study certain analytic continuations of processes that are limits of Racah ensembles, and such that they retain the determinantal structure. Another interpretation of our results, and the main motivation of this paper, is the representation theory of big groups. In representation-theoretic terms, this paper solves a natural problem of harmonic analysis for several infinite-dimensional symmetric spaces.

Skew Howe duality and limit shapes of Young diagrams

“…We then sketch a possible unification of these as a manifestation of the super Howe duality for (GL k , gl(m|n)) [How89]. We conclude with showing that other skew Howe dual pairs from our paper are specializations of the type BC z-measure introduced by Cuenca [Cue18a]. We discuss the relationship with orthogonal polynomials and possible related super Howe dualities.…”

“…Note that Z l (z, z , α, β) is the normalization constant. In [Cue18a,Cue18b], Cuenca constructed an explicit kernel for the corresponding point process and showed a relation to the one for the zw-measure. Furthermore, as described in [Cue18a], there are special values of the pairs (α, β) (there denoted (a, b)) that correspond to the limits of symmetric spaces first examined in [OO06], where the BC z-measure describes an approximation of the spectral measure from a generalization of the biregular representation at finite values.…”

Markov processes on the duals to infinite-dimensional classical Lie groups

“…The conditions on the pair (z, z ′ ), that we give in Definition 6.2, are sufficient for all expressions of the form (1.3) to be nonnegative (and in fact, strictly positive). The result of this paper and others, [8,11,30], suggests that the z-measures with four parameters z, z ′ , a, b is the most general object that can be analyzed thoroughly, despite the fact that most quadruples (z, z ′ , a, b) do not have any representation theoretic origin. The recent paper [31] indicates that the BC type z-measures admit further a natural one-parameter θ > 0 degeneration (all papers cited before treat BC z-measures with θ = 1) and it would be interesting to study them.…”

“…Importantly in the theory of harmonic analysis of big groups, the z-measures can be viewed as probability measures on point configurations in R >0 \ {1} with infinitely many particles (see [7] for a deep study of the zw-measures from this point of view). The author has proved that the resulting point processes for z-measures are determinantal and their kernels have closed forms in terms of hypergeometric functions, [11]. Thus the dynamics we construct on point configurations that preserve the determinantal structure is one of many that have been studied in the literature, mostly within the context of random matrix theory, see the introduction in [9] and further references therein.…”

“…Models whose underlying integrability is based on "type BC" representation theory have been less popular in the probability and integrable systems literature, though there are some interesting works showing that models of both types can be studied by similar techniques, but display interesting and different features, see e.g. [5,11,33]. Our hope is that the theory of integrable models with BC type integrability yield a wealth of applications to various branches of probability theory and mathematical physics, just like their counterparts of type A.…”