Probability and Mathematical Physics Seminar
Probability and the City seminar
Speaker: Reza Gheissari (Northwestern) and Xin Sun (UPenn)
Location: Warren Weaver Hall 1302
Date: Friday, March 3, 2023, 11:10 a.m.
Synopsis:
The Probability and the City seminar is a joint meeting of the probability seminars at Courant and at Columbia, held twice every semester (hosted once at each institution).
The first meeting of Spring 2023 will feature:
- Reza Gheissari (Northwestern University)
Title: Cutoff in the Glauber dynamics for the Gaussian free field
Abstract:
The Gaussian free field (GFF) is a canonical model of random surfaces, generalizing the Brownian bridge to two dimensions. It arises naturally as the stationary solution to the stochastic heat equation with additive noise (SHE), and together the SHE and GFF are expected to be the universal scaling limit of many random surface evolutions arising in lattice statistical physics. We consider the mixing time (time to converge to stationarity, when started out of equilibrium) for the pre-limiting object, the discrete Gaussian free field (DGFF) evolving under the Glauber dynamics. We establish that on a box of side-length $n$ in $\mathbb Z^2$, the Glauber dynamics for the DGFF exhibits the cutoff phenomenon, mixing exactly at time $\frac{2}{\pi^2} n^2 \log n$. Based on joint work with S. Ganguly.
- Xin Sun (University of Pennsylvania)
Title: Random surface, planar lattice model, and conformal field theory
Abstract:
Liouville quantum gravity (LQG) is a theory of random surfaces that originated from string theory. Schramm Loewner evolution (SLE) is a family of random planar curves describing scaling limits of many 2D lattice models at their criticality. Before the rigorous study via LQG and SLE in probability, random surfaces and scaling limits of lattice models have been studied via another approach in theoretical physics called conformal field theory (CFT) since the 1980s. In this talk, I will demonstrate how a combination of ideas from LQG/SLE and CFT can be used to rigorously prove several long standing predictions in physics on random surfaces and planar lattice models, including the law of the random modulus of the scaling limit of uniform triangulation of the annular topology, and the crossing formula for critical planar percolation on an annulus. I will then present some conjectures which further illustrate the deep and rich interaction between LQG/SLE and CFT. Based on joint works with Ang, Holden, Remy, Xu, and Zhuang.