Imaginary geometry [Miller-Sheffield '16] provides a coupling between Schramm-Loewner evolutions (SLE) and Gaussian free field (GFF), and even can be combined with Liouville quantum gravity (LQG) through mating of trees [Duplantier-Miller-Sheffield '18]. This talk gives a brief survey on this program. Then we suggest a construction of north-going flow line in imaginary geometry from alternating west and east-going flow lines, using an excursion theory for planar Brownian motions. This leads a convergence of multiple trees in peanosphere which has been employed in specific settings [Gwynne-Holden-Sun '16, Li-Sun-Watson '17]. Joint work with E. Gwynne, N. Holden, X. Sun, and S. Watson.

The Swendsen-Wang dynamics is an MCMC sampler of the Ising/Potts model, which recolors many vertices at once, as opposed to the classical single-site Glauber dynamics. Although widely used in practice due to efficiency, the mixing time of the Swendsen-Wang dynamics is far from being well-understood, mainly because of its non-local behavior.  In this talk, we prove cutoff phenomenon for the Swendsen-Wang dynamics on the lattice at high enough temperatures, meaning that the Markov chain exhibits a sharp transition from “unmixed” to “well-mixed.” The proof combines two earlier methods of proving cutoff, the update support [Lubetzky-Sly ’13] and information percolation [Lubetzky-Sly ’16], to establish cutoff in a non-local dynamics. Joint work with Allan Sly.

In this presentation, we shall analyze random processes exhibiting metastable /tunneling behaviors among several metastable valleys. Such behaviors can be described by a Markov chain after a suitable rescaling. We will focus on three models: random walks in a potential field, condensing zero-range processes, and metastable diffusion processes.