A quantum surface (QS) (resp. an imaginary surface (IS)) is
an equivalence class of pairs of simply commenced domains
$D \subsetneq {\mathbb{C } }$ and the Gaussian free fields (GFFs) on $D$
with the free (resp. Dirichlet) boundary condition
induced by the conformal equivalence for random metric spaces.
We define a QS with $N+1$ marked boundary points (MBPs)
and an IS with $N+1$ boundary condition changing points (BCCPs)
on $\partial D$ with $N \in {\mathbb{Z } }_{\geq 1}$, in which
the real (resp. imaginary) part of the sum of ${\mathbb{C } }$-valued
logarithmic (2D Coulomb) potentials arising from the MBPs (resp. BCCPs)
is added to GFF in $D$.
We consider the situation such that the boundary points evolve in time
as a stochastic log-gas on $\partial D$ and
multiple random slits are generated in $D$ by the
multiple Schramm--Loewner evolution (SLE) driven by
that stochastic log-gas.
Then we cut the domain $D$ along the SLE slits, restrict the GFF
on the resulting domain, and pull it back to $D$ following
the reverse flow of the multiple SLE.
We prove that if the log-gases on $\partial D$ follow
the stochastic differential equations well-studied
in random matrix theory ({\it e.g.}, Dyson's Brownian
motion model, the Bru--Wishart processes),
and parameters are properly chosen,
then the coupled systems of GFFs and multiple SLE slits
provide stationary processes.
The obtained random systems are used to solve
interesting geometric problems called
the conformal welding problem and the flow line problem.
The present study extends the previous
results for a QS with two MBPs reported by Sheffield
and for an IS with two BCCPs by Miller and Sheffield.
This is a joint work with Shinji Koshida (Chuo University);
see https://arxiv.org/abs/1903.09925.