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 -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 –distributions on lozenge tilings and prove that their height fluctuations converge to the Gaussian free field in an appropriate complex structure.
Funding Statement
The work of V.G. was partially supported by NSF Grants DMS-1664619, DMS-1949820, DMS-2152588, and by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin–Madison with funding from the Wisconsin Alumni Research Foundation.
The research of J.H. is supported by the Simons Foundation as a Junior Fellow at the Simons Society of Fellows and NSF Grants DMS-2054835 and DMS-2331096.
Acknowledgments
The authors thank A. Aggarwal, E. Dimitrov, and N. Nekrasov for helpful discussions. We are grateful to the Galileo Galilei Institute for Theoretical Physics, where some of these discussions took place. We thank anonymous referees for proofreading the manuscript and useful suggestions.
Citation
Vadim Gorin. Jiaoyang Huang. "Dynamical loop equation." Ann. Probab. 52 (5) 1758 - 1863, September 2024. https://doi.org/10.1214/24-AOP1685
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