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

Adv. Stud. Pure Math., 2021: 315-340 (2021) DOI: 10.2969/aspm/08710315


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.


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

Digital Object Identifier: 10.2969/aspm/08710315

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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