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): Eyal Lubetzky, Paul Bourgade, Hugo Falconet, and the probability group
Past Events

Thursday, December 22, 202211AM, Warren Weaver Hall 1302
Conformal covariance of connection probabilities in 2d critical percolation
Federico Camia, NYU Abu DhabiSynopsis:
We discuss the scaling limit of various connection probabilities in twodimensional critical percolation.

Friday, December 9, 202212:05PM, Warren Weaver Hall 109
Upper tail large deviations for chemical distance in supercritical percolation
Barbara Dembin, ETH ZurichSynopsis:
We consider supercritical bond percolation onZ^d and study the chemical distance, i.e., the graph distance on the infinite cluster. It is wellknown that there exists a deterministic constant μ(x) such that the chemical distance D(0,nx) between two connected points 0 and nx grows like nμ(x). We prove the existence of the rate function for the upper tail large deviation event {D(0,nx)>nμ(x)(1+ϵ),0↔nx} for d>=3.
Joint work with Shuta Nakajima.

Friday, December 9, 202211:10AM, Warren Weaver Hall 109
Wilson loop expectations in finite gauge group lattice gauge theories
Sky Cao, IASSynopsis:
Lattice gauge theories with finite gauge groups are statistical mechanical models, very much akin to the Ising model, but with some twists. In recent years there have been various works studying the behavior of Wilson loop observables (which are the basic observables associated with these models) in various regimes. In my talk I will survey these results and give some proof ideas.

Friday, December 2, 202211:10AM, Warren Weaver Hall 1302
Concentration of Equilibria and Relative Instability in the Asymmetric pSpin Model
Pax Kivimae, CourantSynopsis:
We study the number of equilibria (stationary points) in a nonrelaxational analog of the gradient flow for the spherical pspin model. The asymptotics for the expected number of equilibria in this model were recently computed by Fyodorov, followed by a similar computation for the expected number of stable equilibria by Garcia. These results suggest a critical threshold in terms of the strength of the nonrelaxational term, above which the system would transition from having an abundance of stable equilibria, to having none at all. This mirrors the transition from relative to absolute instability found in the generalized MayWigner model.
We confirm this picture by showing that for $p>9$ the number of equilibria, as well as the number of stable equilibria, concentrate around their respective averages, generalizing recent results of Subag and Zeitouni in the relaxational case. This confirms the existence of the above transition, as well as shows that the fraction of equilibria which are stable coincides with its annealed variant.

Friday, November 11, 202211:10AM, Warren Weaver Hall 1302
Large deviation estimates for Selberg’s central limit theorem and applications
Emma Bailey, CUNY Graduate CenterSynopsis:
Abstract:
Selberg’s central limit theorem gives that the logarithm of the Riemann zeta function taken at a uniformly drawn height in $[T, 2T]$ behaves as a complex centered Gaussian random variable with variance $\log\log T$. A natural question is to investigate how far the Gaussian decay persists. We present results on the right tail for the real part of the logarithm, where the absolute value of zeta is `unusually large’, on the scale of the exponential of the variance. Our proof employs a recursive scheme of Arguin, Bourgade and Radziwi{\l}{\l} to inductively work with the logarithm of zeta, interpreted as a random walk. The result is in agreement with the corresponding (known) random matrix result, under the usual dictionary, and has a number of nice corollaries. This work is joint with LouisPierre Arguin.

Friday, November 4, 202212:10PM, Warren Weaver Hall 1302
The maximum of logcorrelated Gaussian fields in random environments
Florian Schweiger, Weizmann Institute of ScienceSynopsis:
In recent years there has been a lot of progress in studying the extrema of logarithmically correlated random fields. In the talk I will review some of the results in the area, and then discuss an extension to logcorrelated Gaussian fields in an environment that is itself random. One key example is the twodimensional Gaussian free field on a supercritical bond percolation cluster. In order to study such models, one needs to combine tools from quantitative stochastic homogenization and classical probabilistic estimates for branching structures.
Based on joint work with Ofer Zeitouni.

Friday, November 4, 202211:10AM, Warren Weaver Hall 1302
Optimal Gradientbased Algorithms for Nonconcave Bandit Optimization
Qi Lei, NYUSynopsis:
Bandit problems with linear or concave reward have been extensively studied, but relatively few works have studied bandits with nonconcave reward. In this talk, we consider a large family of bandit problems where the unknown underlying reward function is nonconcave, including the lowrank generalized linear bandit problems and twolayer neural network with polynomial activation bandit problem. For the lowrank generalized linear bandit problem, we provide a minimaxoptimal algorithm in the dimension, refuting both conjectures in (Lu et al. 2021) and (Jun et al. 2019). Our algorithms are based on a unified zerothorder optimization paradigm that applies in great generality and attains optimal rates in several structured polynomial settings (in the dimension). We further demonstrate the applicability of our algorithms in RL in the generative model setting, resulting in improved sample complexity over prior approaches. Finally, we show that the standard optimistic algorithms (e.g., UCB) are suboptimal by dimension factors. In the neural net setting (with polynomial activation functions) with noiseless reward, we provide a bandit algorithm with sample complexity equal to the intrinsic algebraic dimension. Again, we show that optimistic approaches have worse sample complexity, polynomial in the extrinsic dimension (which could be exponentially worse in the polynomial degree).

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)

Monday, October 24, 20229AM, Kimmel Center 802
Random media & large deviations (day 4)
S. Serfaty, E. Paquette, R. Atar, E. Subag, A. DemboSynopsis:
Conference in honor of Ofer Zeitouni's 60th birthday
9:00 AM Sylvia Serfaty
9:50 AM Break
10:00 AM Elliot Paquette
10:50 AM Break
11:10 AM Rami Atar
12:00 PM Lunch break
2:00 PM Eliran Subag
2:50 PM Break
3:10 PM Amir Dembo 
Sunday, October 23, 20229AM, Warren Weaver Hall 109
Random media & large deviations (day 3)
A. Montanari, P. Deift, E. Kosygina / J. Peterson, S. Armstrong, S. Olla, F. AugeriSynopsis:
Conference in honor of Ofer Zeitouni's 60th birthday
9:00 AM Andrea Montanari
9:50 AM Break
10:00 AM Percy Deift
10:50 AM Break
11:10 AM Elena Kosygina / Jon Peterson
12:00 PM Lunch break
2:00 PM Scott Armstrong
2:50 PM Break
3:10 PM Stefano Olla
4:00 PM Break
4:10 PM Fanny Augeri 
Saturday, October 22, 20229AM, Warren Weaver Hall 109
Random media & large deviations (day 2)
A. Guionnet, E. Bolthausen, T. Kumagai, L. Ryzhik, K. Ramanan, M. RudelsonSynopsis:
Conference in honor of Ofer Zeitouni's 60th birthday
9:00 AM Alice Guionnet
9:50 AM Break
10:00 AM Erwin Bolthausen
10:50 AM Break
11:10 AM Takashi Kumagai
12:00 PM Lunch break
2:00 PM Lenya Ryzhik
2:50 PM Break
3:10 PM Kavita Ramanan
4:00 PM Break
4:10 PM Mark Rudelson 
Friday, October 21, 20229AM, Warren Weaver Hall 109
Random media & large deviations (day 1)
G. Ben Arous, S.R.S. Varadhan, I. Corwin, G. Kozma, N. Gantert, N. SunSynopsis:
Conference in honor of Ofer Zeitouni's 60th birthday
9:00 AM Gerard Ben Arous
9:50 AM Break
10:00 AM S.R.Srinivasa Varadhan
10:50 AM Break
11:10 AM Ivan Corwin
12:00 PM Lunch break
2:00 PM Gady Kozma
2:50 PM Break
3:10 PM Nina Gantert
4:00 PM Break
4:10 PM Nike Sun 
Tuesday, October 18, 20223:30PM, Warren Weaver Hall 1302
Macroscopic Loops in the Spin O(N), double dimer and related models
Lorenzo Taggi, Sapienza Università di RomaSynopsis:
We consider a general system of interacting random loops which includes several models of interest, such as the spin O(N) model, the double dimer model, random lattice permutations, and is related to the loop O(N) model and to the interacting Bose gas in discrete space. We present an overview on these models, introduce some of the main open questions about the size and the geometry of the loops, and present some recent results about the occurrence of macroscopic loops in dimensions d > 2 as the inverse temperature is large enough. More precisely, we consider arbitrary large boxes and prove that, given any two vertices whose distance is proportional to the diameter of the box, the probability of observing a loop visiting both is uniformly positive.

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, 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.