Probability and Mathematical Physics Seminar
Probability and the City seminar
Speaker: Bjoern Bringmann (IAS) and Lingfu Zhang (UC Berkeley)
Location: Columbia University, Mathematics Hall, 2990 Broadway
Date: Friday, March 31, 2023, 10:30 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 second meeting of Spring 2023 will feature:

Lingfu Zhang (UC Berkeley)
Title: Random lozenge tiling at cusps and the Pearcey processAbstract: It has been known since CohnKenyonPropp (2000) that uniformly random tiling by lozenges exhibits frozen and disordered regions, which are separated by the 'arctic curve'. For a generic simply connected polygonal domain, the microscopic statistics are widely predicted to be universal, being one of (1) discrete sine process inside the disordered region (2) Airy line ensemble around a smooth point of the curve (3) Pearcey process around a cusp of the curve (4) GUE corner process around a tangent point of the curve. These statistics were proved years ago for special domains, using exact formulas; as for universality, much progress was made more recently. In this talk, I will present a proof of the universality of (3), the remaining open case. Our approach is via a refined comparison between tiling and nonintersecting random walks, for which a new universality result of the Pearcey process is also proved. This is joint work with Jiaoyang Huang and Fan Yang.
 Bjoern Bringmann (IAS)
Title: Invariant Gibbs measures for the threedimensional cubic nonlinear wave equation.
Abstract: In this talk, we prove the invariance of the Gibbs measure for the threedimensional cubic nonlinear wave equation, which is also known as the hyperbolic Φ^4_3model. This result is the hyperbolic counterpart to seminal works on the parabolic Φ^4_3model by Hairer ’14 and HairerMatetski ’18. In the first half of this talk, we illustrate Gibbs measures in the context of Hamiltonian ODEs, which serve as toymodels. We also connect our theorem with classical and recent developments in constructive QFT, dispersive PDEs, and stochastic PDEs. In the second half of this talk, we give a nontechnical overview of the proof. As part of this overview, we first introduce a caloric representation of the Gibbs measure, which leads to an interplay of both parabolic and hyperbolic theories. Then, we discuss our paracontrolled Ansatz and a hidden cancellation between sextic stochastic objects. This is joint work with Y. Deng, A. Nahmod, and H. Yue.