Schedule

• September 11

  Yier Lin
  University of Chicago

  Multi-point Lyapunov Exponents of the Stochastic Heat Equation

  Abstract: We study the Stochastic Heat Equation with multiplicative space-time white noise. Extensive research has already been conducted on the one-point Lyapunov exponents of this equation. In this talk, I will present how we can compute the multi-point 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 Berkeley

  Geodesics in Last-Passage Percolation under Large Deviations

  Abstract: In KPZ universality, an important family of models arises from 2D last-passage percolation (LPP): in a 2D i.i.d. random field, one considers the geodesic connecting two vertices, which is defined as the up-right 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 lower-tail 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 one-parameter 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 University

  A 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 non-intersecting 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
  MIT

  The 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
  Princeton

  Groups, 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 long-standing 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 Madison

  Invariant measure and universality of the 2D Yang-Mills Langevin dynamic

  Abstract: In [CCHS20] by Chandra, Chevyrev, Hairer and S., a Langevin dynamic for 2D Yang-Mills (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 gauge-fixing, 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 gauge-covariant continuum Langevin dynamic, which allows us to identify the limit of discrete approximations. As corollaries we obtain a gauge-fixed 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
  MIT

  Welding of Liouville quantum gravity disks and the non-simple 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 non-simple LQG disks. Based on joint work with Ang, Holden and Sun.

• November 6

  Jacopo Borga
  Stanford

  On 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 large-scale 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 Connecticut

  Planar 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 1-clusters for any $p \in (p_c^{site},1−p_c^{site})$. Moreover, $p_c^{site}$ < $1/2$, so the above interval is non-empty. 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/Princeton

  A para-controlled approach to the stochastic Yang-Mills equation in two dimensions

  Abstract: We discuss the stochastic Yang-Mills heat equation on the two-dimensional torus. The local well-posedness and gauge-covariance of this system was first proven by Chandra, Chevyrev, Hairer, and Shen using regularity structures. We present an alternative proof, which instead relies on para-controlled calculus. For this talk, no prior knowledge of singular SPDEs is required.

  This is joint work with Sky Cao.

• November 27

  Catherine Wolfram
  MIT

  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 & IAS

  Random 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 non-null-homotopic. This extends a classic result of Voiculescu from free groups to fundamental groups of closed orientable surfaces, and is also connected to quantum Yang-Mills theories.