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.


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 process

    Abstract: It has been known since Cohn-Kenyon-Propp (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 non-intersecting 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 three-dimensional cubic nonlinear wave equation.

    Abstract: In this talk, we prove the invariance of the Gibbs measure for the three-dimensional cubic nonlinear wave equation, which is also known as the hyperbolic Φ^4_3-model. This result is the hyperbolic counterpart to seminal works on the parabolic Φ^4_3-model by Hairer ’14 and Hairer-Matetski ’18. In the first half of this talk, we illustrate Gibbs measures in the context of Hamiltonian ODEs, which serve as toy-models. 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 non-technical 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 para-controlled Ansatz and a hidden cancellation between sextic stochastic objects. This is joint work with Y. Deng, A. Nahmod, and H. Yue.