Some algebraic structures in KPZ universality

IF 1.3 Q2 STATISTICS & PROBABILITY
Nikos Zygouras
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引用次数: 21

Abstract

We review some algebraic and combinatorial structures that underlie models in the KPZ universality class. Emphasis is given on the Robinson-Schensted-Knuth correspondence and its geometric lifting due to A.N.Kirillov and we present how these 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. We also present how fundamental representation theoretic concepts, such as the Cauchy identity, the Pieri rule and the branching rule can be used, alongside RSK correspondences, and can be combined with probabilistic ideas, in order to construct stochastic dynamics on two dimensional arrays called Gelfand-Tsetlin patterns, in ways that couple different one dimensional stochastic processes. The goal of the notes is to expose some of the overarching principles, that have driven a significant number of developments in the field, as a unifying theme.
KPZ泛性中的一些代数结构
我们回顾了KPZ普适性类中模型的一些代数和组合结构。重点讨论了Robinson-Shensted-Knuth对应关系及其由A.N.Kirillov引起的几何提升,我们介绍了如何使用这些对应关系来分析KPZ类中可解模型的结构,并通过连接Schur、Macdonald和Whittaker函数等表示论对象来计算它们的统计量。我们还介绍了如何使用基本的表示论概念,如柯西恒等式、皮耶里规则和分支规则,以及RSK对应关系,并将其与概率思想相结合,以便在称为Gelfand Tsetlin模式的二维阵列上构建随机动力学,以耦合不同的一维随机过程。这些说明的目的是将推动该领域大量发展的一些总体原则作为一个统一的主题加以揭示。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Probability Surveys
Probability Surveys STATISTICS & PROBABILITY-
CiteScore
4.70
自引率
0.00%
发文量
9
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