Fall 2023
Monday 4.15  5.15 pm
Room 2147
Scheduled virtual talks will be held on Zoom, Monday 4:155:15 pm.
A link to a Zoom classroom will appear here!!
Schedule

September 11
Yier Lin
University of ChicagoMultipoint Lyapunov Exponents of the Stochastic Heat Equation
Abstract: We study the Stochastic Heat Equation with multiplicative spacetime white noise. Extensive research has already been conducted on the onepoint Lyapunov exponents of this equation. In this talk, I will present how we can compute the multipoint Lyapunov exponents by leveraging a combination of integrability and probability. As a byproduct, we also solve a quadratic optimization problem.

September 18
Lingfu Zhang
UC BerkeleyGeodesics in LastPassage Percolation under Large Deviations
Abstract: In KPZ universality, an important family of models arises from 2D lastpassage percolation (LPP): in a 2D i.i.d. random field, one considers the geodesic connecting two vertices, which is defined as the upright path maximizing its weight, i.e., the sum/integral of the random field along it. A characteristic KPZ behavior is the 2/3 geodesic fluctuation exponent, which has been proven for some LPPs with exactly solvable structures. A topic of much recent interest is such models under upper and lowertail large deviations, i.e., when the geodesic weight is atypically large or small. In prior works, it was established that the geodesic exponent changes to 1/2 (more localized) and 1 (delocalized) respectively. In this talk, I will describe a further refined picture: the geodesic converges to a Brownian bridge under the upper tail, and a uniformly chosen function from a oneparameter family under the lower tail. I will also discuss the proofs, using a combination of algebraic, geometric, and probabilistic arguments.
This is based on two forthcoming works, one joint with Shirshendu Ganguly and Milind Hegde, and the other with Shirshendu Ganguly.

September 25
Amol Aggarwal
Columbia UniversityA Characterization for the Airy Line Ensemble
Abstract: The Airy line ensemble is a universal scaling limit that is believed (and in some cases proven) to govern the fluctuations of many probabilistic systems, such as random surfaces, interacting particle systems, and stochastic interfaces. It is an example of a "Brownian line ensemble," which informally means that it is an infinite, ordered sequence of random continuous curves that look like nonintersecting Brownian motions. In this talk we survey recent results characterizing the Airy line ensemble as the unique Brownian line ensemble whose top curve decays parabolically, and we explain why this result is useful for proving convergence theorems for various discrete stochastic models. This is based on joint work with Jiaoyang Huang.

October 2
Mehtaab Sawhney
MITThe limiting spectral distribution of iid matrices
Abstract: Let A be an n by n matrix with iid Ber(d/n) entries. We show that the empirical measure of the eigenvalues converges, in probability, to a deterministic distribution. The proof involves incrementally exposing the randomness of the underlying matrix and studying the evolution of the singular values. The talk will focus on the simpler model case where we give a substantially simplified proof of the sparse circular law of Rudelson and Tikhomirov. Joint w. Ashwin Sah, Julian Sahasrabudhe

October 9
Indigenous Peoples' Day

October 16
Ramon von Handel
PrincetonGroups, embeddings, and random walks
Abstract: A finite group with a given set of generators may naturally be viewed as a metric space endowed with the word metric. Understanding its geometry is a basic question of geometric group theory. One may ask, for example, when this metric space can be embedded with low distortion into a normed space (in other words, "can the word metric be approximately described a norm"?) My aim in this talk is to explain a connection between this problem and the properties of random walks on groups. Even the simplest possible example where this connection is fully understood, the discrete cube, settled a longstanding problem of Enflo in the geometry of Banach spaces (joint work with Ivanisvili and Volberg). In other groups, however, this program gives rise to natural questions about the behavior of random walks that do not appear to have been studied in the probability literature. I aim to explain ongoing work with Mira Gordin on the symmetric group, and some challenges we encounter in understanding more general situations.

Octobe 23
Hao Shen
UW MadisonInvariant measure and universality of the 2D YangMills Langevin dynamic
Abstract: In [CCHS20] by Chandra, Chevyrev, Hairer and S., a Langevin dynamic for 2D YangMills (YM) was constructed on 2D torus. In this talk we discuss some new results based on a joint paper with Chevyrev [CS23]. We prove that the 2D YM measure is invariant for the Langevin dynamic constructed in [CCHS20]. Our argument relies on a combination of regularity structures, lattice gaugefixing, and Bourgain’s method for invariant measures. In particular, we prove a universality result which states that for a wide class of lattice YM gauge theories, their corresponding Langevin dynamics converge to the same continuum dynamic constructed in [CCHS20]. An important step is a proof of uniqueness for the mass renormalisation of the gaugecovariant continuum Langevin dynamic, which allows us to identify the limit of discrete approximations. As corollaries we obtain a gaugefixed decomposition of the YM measure into a Gaussian free field and an almost Lipschitz remainder, and a proof of universality for the 2D YM measure under a wide class of discrete approximations.

October 30
Pu Yu
MITWelding of Liouville quantum gravity disks and the nonsimple multiple SLE
Abstract: Liouville quantum gravity is a natural model for a random fractal surface, and SLE curves describe the scaling limit of interfaces from many 2D lattice models. The multiple SLE and their partition functions have been well studied for \kappa between 0 and 6, and their existence remain open for \kappa in (6,8). In this talk, we will explain how this problem is settled from the welding of nonsimple LQG disks. Based on joint work with Ang, Holden and Sun.

November 6
Jacopo Borga
StanfordOn the geometry of uniform meandric systems
Abstract: In 1912, Henri Poincaré asked the following simple question: “In how many different ways can a simple loop in the plane, called a meander, cross a line a specified number of times?” Despite many efforts, this question remains open after over a century.
In this talk, I will focus on meandric systems, which are coupled collections of meanders. I will present (1) a conjecture which describes the largescale geometry of a uniform meandric system and (2) several rigorous results which are consistent with this conjecture.
Based on joint work with Ewain Gwynne and Minjae Park.

November 13
Zhongyang Li
University of ConnecticutPlanar Site Percolation via Tree Embeddings
Abstract: I will show that for any infinite, connected, planar graph G properly embedded into the plane with a minimal vertex degree of at least 7, the i.i.d. Bernoulli(p) site percolation on G almost surely (a.s.) has infinitely many infinite 1clusters for any $p \in (p_c^{site},1−p_c^{site})$. Moreover, $p_c^{site}$ < $1/2$, so the above interval is nonempty. This confirms a conjecture by Benjamini and Schramm in 1996. The proof is based on a novel construction of embedded trees on such graphs.

November 20
Bjoern Bringmann
IAS/PrincetonA paracontrolled approach to the stochastic YangMills equation in two dimensions
Abstract: We discuss the stochastic YangMills heat equation on the twodimensional torus. The local wellposedness and gaugecovariance of this system was first proven by Chandra, Chevyrev, Hairer, and Shen using regularity structures. We present an alternative proof, which instead relies on paracontrolled calculus. For this talk, no prior knowledge of singular SPDEs is required.
This is joint work with Sky Cao.

November 27
The dimer model in 3D
Abstract: A dimer tiling of Z^d is a collection of edges such that every vertex is covered exactly once. In 2000, Cohn, Kenyon, and Propp showed that 2D dimer tilings satisfy a large deviations principle. In joint work with Nishant Chandgotia and Scott Sheffield, we prove an analogous large deviations principle for dimers in 3D. A lot of the results for dimers in two dimensions use tools and exact formulas (e.g. the height function representation of a tiling or the Kasteleyn determinant formula) that are specific to dimension 2. I will explain how to formulate the large deviations principle in 3D, show simulations, try to give some intuition for why three dimensions is different from two. Time permitting, I will explain some of the ways that we use a smaller set of tools (e.g. Hall’s matching theorem or a double dimer swapping operation) in our arguments.

December 4
Mariya Shcherbina
Institute for Low Temperature Physics of National Ukrainian Ac. Sci.Super symmetry approach to the non hermitian random matrices: deformed Ginibre ensemble
Abstract: We consider a complex Ginibre ensemble of random matrices with a deformation $H=H_0+A$, where $H_0$ is a Gaussian complex Ginibre matrix and $A$ is a rather general deformation matrix. The analysis of such ensemble is motivated by many problems of random matrix theory and its applications. We use the Grassmann integration methods to obtain integral representation of spectral correlation functions of the first and the second order and then discuss the analysis of these representations with a saddle point method. Applications of such an analysis to the problems of local regime of the deformed Ginibre ensemble will be discussed.

December 11
Michael Magee
Durham University & IASRandom unitary representations
Abstract: I'll discuss the notion of random unitary representations of discrete infinite groups. These give rise to simply stated, new, and challenging questions in random matrix theory. I'll focus on representations of free groups and surface groups, and highlight two of my contributions there.
In the first, joint with Puder, we give an exact formula for the expected value of the trace of a U(n)valued word map  or Wilson loop  where the word/loop is in a free group, in terms of deep topological properties of the word/loop. In the second, we prove that for elements of surface groups, the expected value of the normalized trace of the SU(n)valued Wilson loop converges to zero as n tends to infinity if the loop is nonnullhomotopic. This extends a classic result of Voiculescu from free groups to fundamental groups of closed orientable surfaces, and is also connected to quantum YangMills theories.