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.
Categories: math.PR cond-mat.stat-mech math-ph math.MP