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VOL. 87 | 2021 Gaussian free fields coupled with multiple SLEs driven by stochastic log-gases
Makoto Katori, Shinji Koshida

Editor(s) Yuzuru Inahama, Hirofumi Osada, Tomoyuki Shirai

Abstract

Miller and Sheffield introduced the notion of an imaginary surface as an equivalence class of pairs of simply connected proper subdomains of $\mathbb{C}$ and Gaussian free fields (GFFs) on them under the conformal equivalence. They considered the situation in which the conformal maps are given by a chordal Schramm–Loewner evolution (SLE). In the present paper, we construct GFF-valued processes on $\mathbb{H}$ (the upper half-plane) and $\mathbb{O}$ (the first orthant of $\mathbb{C}$) by coupling a GFF with a multiple SLE evolving in time on each domain. We prove that a GFF on $\mathbb{H}$ and $\mathbb{O}$ is locally coupled with a multiple SLE if the multiple SLE is driven by the stochastic log-gas called the Dyson model defined on $\mathbb{R}$ and the Bru–Wishart process defined on $\mathbb{R}_{+}$, respectively. We obtain pairs of time-evolutionary domains and GFF-valued processes.

Information

Published: 1 January 2021
First available in Project Euclid: 20 January 2022

Digital Object Identifier: 10.2969/aspm/08710315

Subjects:
Primary: 60B20 , 60D05 , 60J67 , 82C22

Keywords: Bru–Wishart process , Dyson model , Gaussian free fields , Imaginary surface and imaginary geometry , multiple SLE , Schramm–Loewner evolution , Stochastic log-gases

Rights: Copyright © 2021 Mathematical Society of Japan

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