Dynamical Loop Equation

2022/05/31 by Vadim Gorin, Jiaoyang Huang, Gorin, Vadim +1
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2205.15785

openalex publication_date 2022/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce dynamical versions of loop (or Dyson-Schwinger) equations for large families of two--dimensional interacting particle systems, including Dyson Brownian motion, Nonintersecting Bernoulli/Poisson random walks, β--corners processes, uniform and Jack-deformed measures on Gelfand-Tsetlin patterns, Macdonald processes, and (q,κ)-distributions on lozenge tilings. Under technical assumptions, we show that the dynamical loop equations lead to Gaussian field type fluctuations. As an application, we compute the limit shape for (q,κ)--distributions on lozenge tilings and prove that their height fluctuations converge to the Gaussian Free Field in an appropriate complex structure.

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