Mathematics

The Loewner energy for simple closed curves first arises from the small-parameter large deviations of Schramm-Loewner evolution (SLE), a family of random fractal curves modeling interfaces in 2D statistical mechanics. In a certain way, this energy measures the roundness of a Jordan curve, and we show that it is finite if and only if the curve is a Weil-Petersson quasicircle. This intriguing class of curves has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, spectral theory, and string theory, and has been studied since the eighties. The myriad of perspectives on this class of curves is both luxurious and mysterious. I will overview the links between Loewner energy and SLE, Weil-Petersson quasicircles, and other branches of math it touches on. I will also highlight how ideas from probability theory inspire new results on Weil-Petersson quasicircles and discuss further directions.

Random field Ising model is a canonical example to study the effect of disorder on long range order. In 70’s, Imry-Ma predicted that in the presence of weak disorder, the long-range order persists at low temperatures in three dimensions and above but disappears in two dimensions. In this talk, I will review mathematical development surrounding this prediction, and I will focus on recent progress on exponential decay (joint with Jiaming Xia) and on correlation length in two dimensions (joint with Mateo Wirth). In addition, I will describe a recent general inequality for the Ising model (joint with Jian Song and Rongfeng Sun) which has implications for the random field Ising model.

(Joint work with Perla Sousi.)

A sequence of Markov chains is said to exhibit cutoff if it exhibits an abrupt convergence to equilibrium. Loosely speaking, this means that the epsilon mixing time (i.e. the first time the worst-case total-variation distance from equilibrium is at most epsilon) is asymptotically independent of epsilon. An emerging paradigm is that cutoff is related to entropic concentration. In the case of random walks on random graphs G=(V,E), the entropic time can be defined as the time at which the entropy of random walk on some auxiliary graph (often the Benjamini-Schramm limit) is log |V|. Previous works established cutoff at the entropic time in the case of the configuration model. We consider a random graph model in which a random graph G'=(V,E') is obtained from a given graph G=(V,E) with an even number of vertices by picking a random perfect matching of the vertices, and adding an edge between each pair of matched vertices. We prove cutoff at the entropic time, provided G is of bounded degree and its connected components are of size at least 3. Previous works were restricted to the case that the random graph is locally tree-like.

Liouville quantum gravity (LQG) is a natural model for a random fractal surface with origins in conformal field theory and string theory. A powerful tool in the study of LQG is conformal welding, where multiple LQG surfaces are combined into a single LQG surface. The interfaces between the original LQG surfaces are typically described by variants of the random fractal curves known as Schramm-Loewner evolutions (SLE). We will present a few recent conformal welding results for LQG surfaces and applications to the integrability of SLE and LQG partition functions. Based on joint work with Ang and Sun and with Lehmkuehler.

It is an established result in the field of analysis of diffusion processes on fractals, that the transition density of the diffusion typically satisfies analogs of Gaussian bounds which involve a space-time scaling exponent β greater than two and thereby are called SUB-Gaussian bounds. The exponent β, called the walk dimension of the diffusion, could be considered as representing “how close the geometry of the fractal is to being smooth”. It has been observed by Kigami in [Math. Ann. 340 (2008), 781-804] that, in the case of the standard two-dimensional Sierpinski gasket, one can decrease this exponent to two (so that Gaussian bounds hold) by suitable changes of the metric and the measure while keeping the associated Dirichlet form (the quadratic energy functional) the same. Then it is natural to ask how general this phenomenon is for diffusions.

This talk is aimed at presenting (partial) answers to this question. More specifically, the talk will present the following results:

(1) For any symmetric diffusion on a locally compact separable metric measure space in which any bounded set is relatively compact, the infimum over all possible values of the exponent β after “suitable” changes of the metric and the measure is ALWAYS two unless it is infinite. (We call this infimum the conformal walk dimension of the diffusion).

(2) The infimum as in (1) above is NOT attained, in the case of the Brownian motion on the standard (two-dimensional) Sierpinski carpet (as well as that on the standard three-and higher-dimensional Sierpinski gaskets).

This talk is based on joint works with Mathav Murugan (UBC). The results are given in arXiv:2008.12836, except for the non-attainment result for the Sierpinski carpet in (2) above, which is in progress.