Gaussian fluctuations of Young diagrams and structure constants of Jack characters

Gaussian fluctuations of Jack-deformed random Young diagrams

We introduce a large class of random Young diagrams which can be regarded as a natural one-parameter deformation of some classical Young diagram ensembles; a deformation which is related to Jack polynomials and Jack characters. We show that each such a random Young diagram converges asymptotically to some limit shape and that the fluctuations around the limit are asymptotically Gaussian.

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…

Jack变形随机Young图在不同温度下的全局渐近性的普遍性

Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures

本文研究了Jack变形随机Young图在不同温度下的全局渐近性，揭示了其极限形状和波动的普遍性质，证明了在高温和低温下极限形状为一侧无限阶梯形。

Coherent permutations with descent statistic and the boundary problem for the graph of zigzag diagrams

The graph of zigzag diagrams is a close relative of Young's lattice. The boundary problem for this graph amounts to describing coherent random permutations with descent-set statistic, and is also related to certain positive characters on the algebra of quasi-symmetric functions. We establish connections to some further relatives of Young's lattice and solve the boundary problem by reducing it to the classification of spreadable total orders on integers, as recently obtained by Jacka and Warren.

An Inductive Proof of the Berry-Esseen Theorem for Character Ratios

Bolthausen used a variation of Stein's method to give an inductive proof of the Berry-Esseen theorem for sums of independent, identically distributed random variables. We modify this technique to prove a Berry-Esseen theorem for character ratios of a random representation of the symmetric group on transpositions. An analogous result is proved for Jack measure on partitions.

双自由边界麦克唐纳过程：反射恒等式与杰克渐近性

Doubly Free-Boundary Macdonald Processes: Reflection Identities and Jack Asymptotics

该研究引入双自由边界麦克唐纳过程，通过反射演算得出多点轮廓公式。在特定标度和数据下有相关极限形状，可观测量收敛到高斯向量，协方差有边界 - 变形分离，还对\(\beta = 1\)时的情况进行刻画，恢复半空间公式。