Dynamical Loop Equation

TL;DR

This paper develops dynamical loop equations for various two-dimensional particle systems, demonstrating their connection to Gaussian fluctuations and applying them to derive limit shapes and Gaussian Free Field fluctuations in lozenge tilings.

Contribution

It introduces dynamical loop equations for a broad class of particle systems and shows their role in deriving Gaussian fluctuations and limit shapes.

Findings

Gaussian field type fluctuations derived from dynamical loop equations

Limit shape for $(q, abla ext{Kappa})$--distributions on lozenge tilings computed

Height fluctuations converge to Gaussian Free Field in complex structure

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, $\beta$--corners processes, uniform and Jack-deformed measures on Gelfand-Tsetlin patterns, Macdonald processes, and $(q,\kappa)$-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 $(q,\kappa)$--distributions on lozenge tilings and prove that their height fluctuations converge to the Gaussian Free Field in an appropriate complex structure.