I am co-organizing a virtual Probability Seminar with Wei-Kuo Chen and Arnab Sen at University of Minnesota for
the Fall 2024 semester. Please email me to get the Zoom link for the seminar series.
The seminar is held on Fridays at 2:30pm (ET), unless otherwise noted below.
09/14/2024 | Qiang Wu (U Minnesota) |
Title: Joint parameter estimations for spin glasses. | |
Abstract: Spin glasses are disordered statistical physics system with both ferromagnetic and anti-ferromagnetic spin interactions. The Gibbs measure belongs to the exponential family with parameters, such as inverse temperature β>0 and external field h in R. Given a sample from the Gibbs measure of a spin glass model, we study the problem of estimating system parameters. In 2007, Chatterjee first proved that under reasonable conditions, for spin glass models with h=0, the maximum pseudo-likelihood estimator for β is √N-consistent. However, the approach has been restricted to the single parameter estimation setting. The joint estimation of (β,h) for spin glasses has remained open. In this talk, I will present a joint work with Wei-Kuo Chen, Arnab Sen, which shows that under some easily verifiable conditions, the bi-variate maximum pseudo-likelihood estimator is indeed jointly √N-consistent for a large collection of spin glasses, including the Sherrington-Kirkpatrick model and its diluted variants. | |
09/20/2024 | Kesav Krishnan (U Victoria) |
Title: Uniqueness and CLT for the ground state Disordered Monomer-Dimer Model on Z^{d} | |
Abstract: The Monomer-Dimer model is a Gibbs measure defined on the space of matchings (not necessarily perfect) on a graph. This talk will concerned the disordered version of this problem with additional environmental randomness, the weights of the matchings are themselves random variables. The ground state of the model is a measure concentrated on matchings of minimal energy, in other words the Gibbs measure at zero temperature. In this talk, I will introduce the model on the discrete torus. In joint work with Gourab Ray, we establish the uniqueness in limit of the ground state as the lattice size tends to infinity. This involves certain ergodic theoretic tools such as the Burton Keane argument. We use this uniqueness to establish a central limit theorem for the ground state energy by applying Chatterjee’s powerful method of normal approximation. | |
09/27/2024 | Jorge Garza Vargas (Caltech) |
Title: A new approach to strong convergence of random matrices. | |
Abstract: Friedman's celebrated 2004 result states that, as the number of vertices goes to infinity, random d-regular graphs are (with high probability) nearly optimal expanders, meaning that the top non-trivial eigenvalue of their (random) adjacency matrix converges in probability to 2 sqrt(d-1). Since expanders are of great interest in number theory and computer science, Friedman's paper (which was ~100 pages) has attracted a lot of attention in the last two decades and more efficient proofs of his result (which yield vast generalizations) have been found. However, all the approaches to Friedman's theorem and its extensions relied on very delicate and sophisticated combinatorial considerations, making it hard to apply those ideas to other settings of interest. In this talk I will discuss a fundamentally new (analytic) approach to Friedman's theorem which yields an elementary proof that can be written in just a few pages. Our approach also allows us to establish strong convergence (i.e. sharp norm estimates) for much more general models of tuples of random matrices (random regular graphs corresponding to the particular case of adding independent random permutations). These results can be used to show that certain infinite objects admit very strong finite dimensional approximations, which has important implications in operator algebras, spectral geometry, and differential geometry. This is joint work with Chi-Fang Chen, Joel Tropp, and Ramon van Handel. | |
10/04/2024 | Curtis Grant (Northwestern) |
Title: Metastability in Heavy Tailed Spin Glass Dynamic. | |
Abstract: Metastability is a phenomenon widely believed to hold for Markov chains associated to spin glass models. Roughly, for a complex energy landscape, one should have a decomposition into wells, and the corresponding dynamics tracking which well the original Markov chain is in should be asymptotically Markovian. We will provide an overview of known results for spin glass dynamics, and then explain recent results with Reza Gheissari on metastability for heavy-tailed spin glasses. | |
10/18/2024 | Haotian Gu (Duke) |
Title: On maximum of Poissonian log-correlated fields. | |
Abstract: In recent years there has been intense ineterest in extreme values of logarithmically correlated fields (LCFs), in connection with problems on Gaussian multiplicative chaos, random matrices, branching random walks, reaction-diffusion PDE, and L-functions in analytic number theory. The sharpest results are for Gaussian or nearly-Gaussian fields. On the other hand, characteristic polynomials of sparse random matrices give rise to LCFs with Poissonian tails. In an earlier work on permutation matrices, Cook and Zeitouni obtained the leading order of the maximum. I will discuss new refined results on the maximum for a related class of random trigonometric polynomials with Poissonian tails. We find the sub-leading order behavior is significantly different from the ubiquitous "Bramson correction" term for Gaussian LCFs, and can be modeled by a branching random walk with a randomly time-varying offspring distribution. Based on joint work with Nicholas Cook (Duke). | |
10/25/2024 | Thuy-Duong Vuong (Berkeley) |
Title: Improved mixing time for Markov chains on statistical physics model. | |
Abstract: In this talk, I will discuss recent progress on bounding the mixing time of Markov chains to sample from spin systems from statistical physics. For example, for the Sherrington-Kirkpatrick (SK) Ising model, we show the optimal quasi-linear mixing time up to the current-best temperature threshold of β= 0.295 [AKV24], improving upon the previous best bound of β = 0.25 [AJKPV21, CE22]. The recent spectral/entropic independence framework [ALO’20, CLV21-22, CFYZ21-22, AJKPV21-24, CE22] controls the mixing time of Markov chains using spectral bounds on the covariance of the stationary distribution. Our new result for the SK Ising model relies upon a new bound for this covariance matrix by generalizing Oppenheim’s trickle-down to linear-tilt localization schemes. Based on joint work with Nima Anari, Vishesh Jain, Frederic Koehler, and Huy Tuan Pham. | |
11/1/2024 | Xusheng Zhang (Oxford) |
Title: Mean-field random-cluster dynamics from high-entropy initializations. | |
Abstract: A common challenge in using Markov chain for sampling from high-dimensional distributions is multimodality, where the chain may get trapped far from stationarity. However, this issue often applies only to worst-case initializations and can be mitigated by using high-entropy initializations, such as product or weakly correlated distributions. From such starting points, the dynamics can escape saddle points and spread mass correctly across dominant modes. In this talk, I will discuss our results on convergence from high-entropy initializations for the random-cluster models on the complete graph. We focus on the Chayes–Machta or the Swendsen–Wang dynamics for the random-cluster model showing that these chains mix rapidly from specific product measures, even though they mix exponentially slowly from worst-case initializations. The analogous results hold for the Glauber dynamics on the Potts model. Our proofs involve approximating high-dimensional dynamics with 1-dimensional random processes and analyzing their escape from saddle points. | |
11/15/2024 | Yujin H. Kim (NYU) |
Title: TBA | |
Abstract: TBA | |
11/22/2024 | Ben Mckenna (Georgia Tech) |
Title: TBA | |
Abstract: TBA | |