I am co-organizing a virtual Probability Seminar with Wei-Kuo Chen and Arnab Sen at University of Minnesota for
the Fall 2022 semester. Please email me to get the Zoom link for the seminar series.
The seminar is held on Fridays at 3:30pm (ET), unless otherwise noted below.
| 09/16/2022 | Gourab Ray (Math & Stat, U Victoria) Special Time: 2:30-3:30pm (ET) |
| Title: Conformal invariance of dimers on Riemann surfaces, loops soups, and cycle rooted forests. | |
| Abstract: We prove conformal invariance of dimers on general Temperleyan graphs embedded on Riemann surfaces of finite topological type, satisfying a certain invariance principle for simple random walk. I will describe how this problem naturally leads to a scaling limit problem of a two sided loop erased random walk, and how we solve this problem by developing a natural extension of the connection between loop soups and Uniform spanning trees to the setting of multiply connected surfaces. I will also describe certain robust estimates of the mass of large loops in a loop soup on a general graph, which seem to be new even in the simply connected setting. Joint work with N. Berestycki and B. Laslier. | |
| 09/23/2022 | Will Perkins (UIC/GaTech) |
| Title: Potential-weighted connective constants and uniqueness of Gibbs measures. | |
| Abstract: Classical gases (or Gibbs point processes) are models of gases or fluids, with particles interacting in the continuum via a density against background Poisson processes. A prominent example is the hard sphere model. The major questions about these models are about phase transitions, and while these models have been studied in mathematics for over 70 years, some of the most fundamental questions remain open. After giving some background on these models, I will describe a new method for proving absence of phase transition (uniqueness of infinite volume Gibbs measures) in the low-density regime of processes interacting via a repulsive pair potential. The method involves defining a new quantity, the "potential-weighted connective constant", and is motivated by techniques from computer science. Joint work with Marcus Michelen. | |
| 09/30/2022 | Perla Sousi (Cambridge) Special Time: 12:00-1:00pm (ET) |
| Title: Phase transition for the late points of random walk. | |
| Abstract: Let X be a simple random walk in Znd with d≥3 and let tcov be the expected time it takes for X to visit all vertices of the torus. In joint work with Prévost and Rodriguez we study the set Lα of points that have not been visited by time αtcov and prove that it exhibits a phase transition: there exists α* so that for all α > α* and all ε>0 there exists a coupling between Lα and two i.i.d. Bernoulli sets B± on the torus with parameters n-(a±ε)d with the property that B-⊆ Lα⊆ B+ with probability tending to 1 as n→∞. When α ≤ α*, we prove that there is no such coupling. | |
| 10/07/2022 | Nima Anari (Computer Science, Stanford) |
| Title: Fast sampling via domain sparsification. | |
| Abstract: I will describe a framework called domain sparsification that allows us to speed up sampling from various combinatorial distributions after a preprocessing step. I will highlight the following application and time-permitting describe others: for sampling random spanning trees from a graph G=(V, E), we show how a nearly-linear-time (in |E|) preprocessing step allows us to sample spanning trees in time nearly-linear in |V|. I will give the necessary background (involving high-dimensional-expanders and methods based on this theory for analyzing Markov chain mixing), and give some elements of the proof for domain sparsification. Based on joint works with: Michal Derezinski, Yang P. Liu, Thuy-Duong Vuong, Elizabeth Yang | |
| 10/14/2022 | Subhabrata Sen (Statistics, Harvard) |
| Title: TAP equations for orthogonally invariant spin glasses at high temperature | |
| Abstract: We consider an Ising spin-glass model with coupling matrix J = ODOT, where O ∈ Rnxn is a Haar uniform orthogonal matrix, and D is diagonal. Parisi and Potters (1995) conjectured that the magnetization in this model satisfies a system of Thouless-Anderson-Palmer (TAP) equations at high-temperature. In this talk, we will discuss a proof of these TAP equations (in an L2 sense) at sufficiently high temperature. Our approach is based on proving the convergence of an appropriate Approximate Message Passing (AMP) algorithm to the magnetization vector. This is based on joint work with Zhou Fan (Yale) and Yufan Li (Harvard). | |
| 10/28/2022 | Ohad Feldheim (Math, Hebrew University of Jerusalem) |
| Special Time: 11:00am-12:00pm (ET) | |
| Title: A phase transition in zero count probability for Stationary Gaussian Processes. | |
| Abstract: Consider a centered real stationary Gaussian process f(t), that is, a random real function with multinormal marginals whose distribution is invariant under translations. This is the most commonly used model for random noise (random signals, ocean surface fluctuations, etc’). The expected number of zeroes of such a process in an interval [0,T] is bT, where b can be computed by the celebrated Kac-Rice formula. Here we study the probability of a significant deviation of this random variable, known as overcrowding and undercrowding of zeroes, given by having at least (b+d)T zeroes, or at most (b-d)T respectively. We show that when the support of the spectrum of the process is separated from infinity (respectively from zero) there is a sharp phase transition in d in the overcrowding probability (respectively in undercrowding), at a value proportional to the edge of the support of the spectrum. At this point the probability of the rare event in question changes abruptly from exponential to sub-Gaussian. The methods used to show this result involve tools from the theory of Gaussian processes and complex analysis. The topic will be fully introduced and no prior familiarity with SGPs is assumed. Joint work with Naomi Feldheim and Lakshmi Priya. | |
| 11/04/2022 | Elliot Paquette (Math & Stat, McGill) |
| Title: The homogenization of SGD in high dimensions. | |
| Abstract: Stochastic gradient descent (SGD) is one of the most, if not the most, influential optimization algorithms in use today. It is the subject of extensive empirical and theoretical research, principally in justifying its performance at minimizing very large (high dimensional) nonlinear optimization. This talk is about the precise high-dimensional limit behavior (specifically generalization and training dynamics) of SGD in a highdimensional least squares problems. High dimensionality is enforced by a family of resolvent conditions on the data matrix, and data-target pair, which can be viewed as a type of eigenvector delocalization. We show that the trajectory of SGD is quantitively close to the solution of a stochastic differential equation, which we call homogenized SGD, and whose behavior is explicitly solvable using renewal theory and the spectrum of the data. Based on joint works with Courtney Paquette and Kiwon Lee (McGill), and Fabian Pedregosa, Jeffrey Pennington and Ben Adlam (Google Brain). | |
| 11/11/2022 | Yuxin Zhou (Math, Northwestern) |
| Title: The Spherical p+s Spin Glass At Zero Temperature. | |
| Abstract: We consider the spherical p+s spin glass model and determine the structure of its Parisi measure at zero temperature. We show that depending on the values of p and s, four scenarios may emerge, including the existence of 1-FRSB and 2-FRSB phases as predicted by Crisanti and Leuzzi. Furthermore, we obtain a complete characterization of the support of the Parisi measure for all possible values of the parameters p, s and λ. This is based on a joint work with Antonio Auffinger. | |
| 11/18/2022 | Qiang Wu (Math, UIUC) |
| Title: Cluster expansion approach to mean field spin glass models. | |
| Abstract: Cluster expansion approach is a powerful combinatorial scheme originated in mathematical physics, which has been used to establish many rigorous results in classical spin systems, such as Ising model, hardcore model etc. However, in disordered systems, this technique is not well-developed. In particular, for mean field spin glass models, the only known results are due to Aizenman-Lebowitz-Ruelle for Sherrington-Kirkpatrick (SK) model in 1987 and Kosters for diluted SK model in 2006. Both works are restricted to zero external field and 2-spin interactions. It was believed that this approach does not work if an external field is present in the system. In this talk, we will show that under some “weak external field”, this approach still works but with a new cluster structure. Besides that, in a separate work, we extend this idea to general mixed p-spin models. It enables us to establish fluctuation results in mixed p-spin models up to the critical inverse temperature. A by-product of our results is an explicit characterization of critical inverse temperature for general mean field spin glass models. Time permits, we will discuss some further implications of our results, such as the difference of pure even and odd p-spin models, and classification of mean field spin glass models, and extension to multi-species mixed p-spin models etc. This is based on joint works with Partha S. Dey. | |
| 12/02/2022 | Shuta Nakajima (Meiji U) |
| Title: On the upper tail large deviation rate function for chemical distance in supercritical percolation. | |
| Abstract: We consider supercritical bond percolation on Zd and study the chemical distance, i.e., the graph distance on the infinite cluster. It is well-known 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). Garet and Marchand prove that the probability of the upper tail large deviation event {D(0, nx)>n μ(x)(1+ε), 0↔nx} decays exponentially with respect to n. In this talk, we discuss the existence of the rate function for the upper tail large deviation when d ≥3 and ε>0 is small enough. Moreover, we show that for any ε>0, the upper tail large deviation event is created by space-time cut-points (points that all paths from 0 to nx must cross after a given time) that forces the geodesics to "lose time" by going in a non-optimal direction or by wiggling a lot. This enables us to express the rate function in terms of the rate function for a space-time cut-point. This talk is based on joint work with Barbara Dembin. | |
| 12/09/2022 | Changji Xu (CMSA, Harvard) |
| Title: Spectral gap estimates for mixed p-spin models at high temperature. | |
| Abstract: We consider general mixed p-spin mean field spin glass models and provide a method to prove that the spectral gap of the Dirichlet form corresponding to Glauber dynamics is of order one at sufficiently high temperature. Our proof is based on an iteration scheme relating the spectral gap of the N-spin system to that of suitably conditioned subsystems. | |