Moments of characteristic polynomials for compact symmetric spaces and Jack polynomials

Stein's Method, Jack Measure, and the Metropolis Algorithm

The one parameter family of Jack(alpha) measures on partitions is an important discrete analog of Dyson's beta ensembles of random matrix theory. Except for special values of alpha=1/2,1,2 which have group theoretic interpretations, the Jack(alpha) measure has been difficult if not intractable to analyze. This paper proves a central limit theorem (with an error term) for Jack(alpha) measure which works for arbitrary values of alpha. For alpha=1 we recover a known central limit theorem on the distribution of character ratios of random representations of the symmetric group on transpositions. Th…

Jack polynomials and free cumulants

We study the coefficients in the expansion of Jack polynomials in terms of power sums. We express them as polynomials in the free cumulants of the transition measure of an anisotropic Young diagram. We conjecture that such polynomials have nonnegative integer coefficients. This extends recent results about normalized characters of the symmetric group.

Anisotropic Young diagrams and infinite-dimensional diffusion processes with the Jack parameter

We construct a family of Markov processes with continuous sample trajectories on an infinite-dimensional space, the Thoma simplex. The family depends on three continuous parameters, one of which, the Jack parameter, is similar to the beta parameter in random matrix theory. The processes arise in a scaling limit transition from certain finite Markov chains, the so called up-down chains on the Young graph with the Jack edge multiplicities. Each of the limit Markov processes is ergodic and its stationary distribution is a symmetrizing measure. The infinitesimal generators of the processes are exp…

Multilevel Dyson Brownian motions via Jack polynomials

We introduce multilevel versions of Dyson Brownian motions of arbitrary parameter $β>0$, generalizing the interlacing reflected Brownian motions of Warren for $β=2$. Such processes unify $β$ corners processes and Dyson Brownian motions in a single object. Our approach is based on the approximation by certain multilevel discrete Markov chains of independent interest, which are defined by means of Jack symmetric polynomials. In particular, this approach allows to show that the levels in a multilevel Dyson Brownian motion are intertwined (at least for $β\ge 1$) and to give the corresponding link …

Jack过程与拓扑展开的艾里极限

Airy limit for the Jack process and topological expansion

该研究建立Jack-Plancherel过程的多时间软边矩收敛性，推导其极限矩的布朗桥展开，确定等时极限与Airy_β过程的关联，得到边际b猜想的渐近β拓扑展开及β=2时Witten-Kontsevich相交数的显式非负公式。

$β$-Laguerre 过程的交织性质与 Jack 多项式的积分算子

The intertwining property for $β$-Laguerre processes and integral operators for Jack polynomials

本文研究$β$-Laguerre过程的交织关系，引入依赖$β$和$α$的Markov核并利用Jack对称多项式作为其特征函数，建立了新的交织关系，同时推导出与Jack多项式相关的多元Laguerre多项式和多元超几何函数的积分公式。