We introduce a stochastic dynamics related to the measures that arise in harmonic analysis on the infinite-dimensional unitary group. Our dynamics is obtained as a limit of a sequence of natural Markov chains on Gelfand-Tsetlin graph. We compute finite-dimensional distributions of the limit Markov process, the generator and eigenfunctions of the semigroup related to this process. The limit process can be identified with Doob h-transform of a family of independent diffusions. Space-time correlation functions of the limit process have a determinantal form.
The boundary of the Gelfand-Tsetlin graph: A new approach
Gelfand-Tsetlin图是一个编码酉群的不可约特征标分支的无限分次图。Gelfand-Tsetlin图的边界至少有三种表现形式--离散位势论边界、无限维酉群的有限不可分解特征标集和双无限全正序列集. Albert Edrei和Dan Voiculescu的一个旧的深度结果提供了边界的明确描述;它可以实现为无限维坐标空间中的一个区域。本文包含了一个新的方法Edrei-Voiculescu定理。它是基于一个新的显式公式的数量的半标准杨tableaux一个给定的斜形状（或Gelfand-Tsetlin计划的梯形）。该公式是通过对称函数理论得到的，新的Schur类对称函数在推导中起着关键作用。
A continuation of the joint work by Vadim Gorin and the author, J. Funct. Anal. 270 (2016), 375-418; arXiv:1504.06832. The extended Gelfand-Tsetlin graph, introduced in that paper, is a novel combinatorial object. Its q-boundary is formed by infinite point configurations on a two-sided q-lattice. The q-boundary carries a continuous family of probability measures that are a q-analogue of the so-called zw-measures, which originated in the problem of harmonic analysis on the infinite-dimensional unitary group. In the present paper, it is proved that certain transition Markov kernels, linked to th…
Using Okounkov's $q$-integral representation of Macdonald polynomials we construct an infinite sequence $Ω_1,Ω_2,Ω_3,\dots$ of countable sets linked by transition probabilities from $Ω_N$ to $Ω_{N-1}$ for each $N=2,3,\dots$. The elements of the sets $Ω_N$ are the vertices of the extended Gelfand-Tsetlin graph, and the transition probabilities depend on the two Macdonald parameters, $q$ and $t$. These data determine a family of Markov chains, and the main result is the description of their entrance boundaries. This work has its origin in asymptotic representation theory. In the subsequent paper…
The thesis focuses on processes on symplectic Gelfand-Tsetlin patterns. In chapter 4, a process with dynamics inspired by the Berele correspondence [Ber86] is presented. It is proved that the shape of the pattern is a Doob $h$-transform of independent random walks with $h$ given by the symplectic Schur function. This is followed by an extension to a $q$-weighted version. This randomised version has itself a branching structure and is related to a $q$-deformation of the $so_{2n+1}$-Whittaker functions. In chapter 5, we present a fully randomised process. This process $q$-deforms a process propo…
We study the combinatorial structure of the irreducible characters of the classical groups ${\rm GL}_n(\mathbb{C})$, ${\rm SO}_{2n+1}(\mathbb{C})$, ${\rm Sp}_{2n}(\mathbb{C})$, ${\rm SO}_{2n}(\mathbb{C})$ and the "non-classical" odd symplectic group ${\rm Sp}_{2n+1}(\mathbb{C})$, finding new connections to the probabilistic model of Last Passage Percolation (LPP). Perturbing the expressions of these characters as generating functions of Gelfand-Tsetlin patterns, we produce two families of symmetric polynomials that interpolate between characters of ${\rm Sp}_{2n}(\mathbb{C})$ and ${\rm SO}_{2n…
We prove that the boundary of the Hall-Littlewood $t$-deformation of the Gelfand-Tsetlin graph is parametrized by infinite integer signatures, extending results of Gorin and Cuenca on boundaries of related deformed Gelfand-Tsetlin graphs. In the special case when $1/t$ is a prime $p$ we use this to recover results of Bufetov-Qiu and Assiotis on infinite $p$-adic random matrices, placing them in the general context of branching graphs derived from symmetric functions. Our methods rely on explicit formulas for certain skew Hall-Littlewood polynomials. As a separate corollary to these, we obtain …
We consider consistent diffusion dynamics, leaving the celebrated Hua-Pickrell measures, depending on a complex parameter $s$, invariant. These, give rise to Feller-Markov processes on the infinite dimensional boundary $Ω$ of the "graph of spectra", the continuum analogue of the Gelfand-Tsetlin graph, via the method of intertwiners of Borodin and Olshanski. In the particular case of $s=0$, this stochastic process is closely related to the $\mathsf{Sine_2}$ point process on $\mathbb{R}$ that describes the spectrum in the bulk of large random matrices. Equivalently, these coherent dynamics are a…
We review some algebraic and combinatorial structures that underlie models in the KPZ universality class.Emphasis is placed on the Robinson-Schensted-Knuth correspondence and its geometric lifting due to A.N.Kirillov. We present how these combinatorial constructions are used to analyse the structure of solvable models in the KPZ class and lead to computation of their statistics via connecting to representation theoretic objects such as Schur, Macdonald and Whittaker functions, Young tableaux and Gelfand-Tsetlin patterns. We also present how fundamental representation theoretic concepts, such a…
We show that for $β\ge 1$ the semigroups of $β$-Laguerre and $β$-Jacobi processes of different dimensions are intertwined in analogy to a similar result for $β$-Dyson Brownian motion recently obtained by Ramanan and Shkolnikov. These intertwining relations generalize to arbitrary $β\ge 1$ the ones obtained for $β=2$ by the author, O'Connell and Warren between $h$-transformed Karlin-McGregor semigroups. Moreover they form the key step towards constructing a multilevel process in a Gelfand-Tsetlin pattern leaving certain Gibbs measures invariant. Finally as a by product we obtain a relation betw…
