Probability and Mathematical Physics Seminar
This seminar covers a wide range of topics in pure and applied probability and in mathematical physics. Unless otherwise noted, the talks take place on Fridays, 11:10am–12pm, in room WWH 1302. See directions to the Courant Institute.
To be kept informed, you can subscribe to the Probability Seminar Mailing list.
Twice every semester, we will have a joint Columbia–Courant meeting (hosted once at each institution) as the Probability and the City seminar.
Seminar Organizer(s): The Probability group
Upcoming Events

Friday, October 7, 202211:10AM, Warren Weaver Hall 1302
TBA
Lorenzo Taggi, Sapienza Università di Roma 
Friday, October 14, 20223PM, Warren Weaver Hall 1302
Covering systems of congruences
Bob Hough, Stony BrookSynopsis:
A distinct covering system of congruences is a list of congruences \[ a_i \bmod m_i, \qquad i = 1, 2, ..., k \] whose union is the integers. Erd\H{o}s asked if the least modulus $m_1$ of a distinct covering system of congruences can be arbitrarily large (the minimum modulus problem for covering systems, $1000) and if there exist distinct covering systems of congruences all of whose moduli are odd (the odd problem for covering systems, $25). I'll discuss my proof of a negative answer to the minimum modulus problem, and a quantitative refinement with Pace Nielsen that proves that any distinct covering system of congruences has a modulus divisible by either 2 or 3. The proofs use the probabilistic method and in particular use a sequence of pseudorandom probability measures adapted to the covering process. Time permitting, I may briefly discuss a reformulation of our method due to Balister, Bollob\'{a}s, Morris, Sahasrabudhe and Tiba which solves a conjecture of Shinzel (any distinct covering system of congruences has one modulus that divides another) and gives a negative answer to the squarefree version of the odd problem.

Friday, October 28, 202211:10AM, Warren Weaver Hall 1302
Probability and the City seminar
Evita Nestoridi and Tom TrogdonSynopsis:
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 Fall 2022 will feature: Evita Nestoridi (Princeton)
 Tom Trogdon (University of Washington)

Friday, December 2, 202211:10AM, Warren Weaver Hall 1302
TBA
Pax Kivimae, Courant 
Friday, December 9, 202211:10AM, Warren Weaver Hall 1302
TBA
Sky Cao, IAS
Past Events

Friday, September 30, 202211:10AM, Warren Weaver Hall 1302
A central limit theorem for square ice
Wei Wu, NYU ShanghaiSynopsis:
An important open question is to show that the height function associated with the square ice model (i.e., planar six vertex model with uniform weights), or equivalently the uniform graph homeomorphisms, converges to a continuum Gaussian free field In the scaling limit, I will review some recent results about this model, including that the single point height function, upon renormalization, converges to a Gaussian random variable.

Friday, September 23, 202211:10AM, Warren Weaver Hall 1302
How do the eigenvalues of a large random matrix behave?
Giorgio Cipolloni, PrincetonSynopsis:
We prove that the fluctuations of the eigenvalues converge to the Gaussian Free Field (GFF) on the unit disk. These fluctuations appear on a nonnatural scale, due to strong correlations between the eigenvalues.Then, motivated by the long time behaviour of the ODE \dot{u}=Xu, we give a precise estimate on the eigenvalue with the largest real part and on the spectral radius of X. 
Friday, September 16, 202210AM, Columbia University, Mathematics Hall, 2990 Broadway 520
Probability and the City seminar
Nina Holden and Ron PeledSynopsis:
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 Fall 2022 will feature: Nina Holden (Courant Institute, NYU)
Conformal welding in Liouville quantum gravity: recent results and applications
Abstract:
Liouville quantum gravity (LQG) is a natural model for a random fractal surface with origin in the physics literature. 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 SchrammLoewner evolutions (SLE). We will present a few recent conformal welding results for LQG surfaces and their applications, which range from SLE and LQG to planar maps and random permutations. Based on joint works with Ang and Sun, with Lehmkuehler, and with Borga, Sun and Yu.
 Ron Peled (TelAviv University)
Random packings and liquid crystals
Abstract:
Let T be a subset of R^d, such as a ball, a cube or a cylinder, and consider all possibilities for packing translates of T, perhaps with its rotations, in some bounded domain in R^d. What does a typical packing of this sort look like? One mathematical formalization of this question is to fix the density of the packing and sample uniformly among all possible packings with this density. Discrete versions of the question may be formulated on lattice graphs.
The question arises naturally in the sciences, where T may be thought of as a molecule and its packing is related to the spatial arrangement of molecules of a material under given conditions. In some cases, the material forms a liquid crystal  states of matter which are, in a sense, between liquids and crystals.
I will review ideas from this topic, mentioning some of the predictions and the mathematical progress. Time permitting, I will elaborate on a recent result, joint with Daniel Hadas, on the structure of highdensity packings of 2x2 squares with centers on the square lattice.
The talk is meant to be accessible to a general mathematical audience.
 Nina Holden (Courant Institute, NYU)

Friday, September 9, 202211:10AM, Warren Weaver Hall 1302
Liouville quantum gravity from random matrix dynamics
Hugo Falconet, Courant InstituteSynopsis:
The Liouville quantum gravity measure is a properly renormalized exponential of the 2d GFF. In this talk, I will explain how it appears as a limit of natural random matrix dynamics: if (U_t) is a Brownian motion on the unitary group at equilibrium, then the measures $det(U_t  e^{i theta}^gamma dt dtheta$ converge to the 2d LQG measure with parameter $gamma$, in the limit of large dimension. This extends results from Webb, Nikula and Saksman for fixed time. The proof relies on a new method for FisherHartwig asymptotics of Toeplitz determinants with real symbols, which extends to multitime settings. In particular, I will explain how to obtain multitime loop equations by stochastic analysis on Lie groups.
This is based on a joint work with Paul Bourgade.