Dynamical Loop Equation

Vadim Gorin; Jiaoyang Huang

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arXiv · 2205.15785

Dynamical Loop Equation

Vadim Gorin·Jiaoyang Huang

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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.

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BibTeX RIS

Vadim Gorin, Jiaoyang Huang. 2022-05-31. Dynamical Loop Equation. https://arxiv.org/abs/2205.15785

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