Seminar Schedule:
The seminar meets on Tuesdays from 4:15 to 5:15 pm EST.
Abstract: In joint work with Kannan Soundararajan, we consider the behavior of random multiplicative functions when summed over subsets of the integers in [1, N], and give several examples of sets where such sums satisfy a central limit theorem. In contrast, as we know from the work of Harper, the partial sums over all integers in [1, N] do not satisfy a central limit theorem. The key quantity involved in the study is the “multiplicative energy” of a set, which is an object of interest to combinatorialists.
Abstract: We prove that the fluctuations of the eigenvalues converge to the Gaussian Free Field (GFF) on the unit disk. These fluctuations appear on a non-natural 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.
Abstract: In this talk, I will discuss the application of replica-exchange Langevin diffusion (LD) for two tasks: solving non-convex optimization and sampling from multimodal target distributions. For the non-convex optimization problem, we use replica-exchange to facilitate the collaboration between gradient descent (LD with zero temperature) and LD. We show that this algorithm converges to the global minimum linearly with high probability, assuming the objective function is strongly convex in a neighborhood of the unique global minimum. By replacing gradients with stochastic gradients, and adding a proper threshold to the exchange mechanism, our algorithm can also be used in the online setting. For the Markov chain Monte Carlo problem, the convergence rate of LD to stationarity can be significantly reduced if the target distribution has multiple isolated modes. Replica exchange allows us to add another LD sampling a high-temperature version of the target density to facilitate faster convergence. When the target density is a mixture of log-concave densities, we quantify the spectral gap of replica-exchange LD and show that the algorithm with a properly chosen temperature and exchange intensity can achieve constant or even better convergence rates. We further quantify the benefit of replica exchange for multiple LDs sampling at different temperatures.
Abstract: In first-passage percolation, we place i.i.d. nonnegative weights on the edges of the d-dimensional cubic lattice Z^d and study the induced metric T = T(x,y). Letting B(s) be the set of vertices with distance at most s from the origin (the random ball of radius s), it is known that as s grows, B(s) tends to a deterministic convex shape. In recent works with J. Hanson, J. Gold, W.-K. Lam, and X. Shen, we have looked at geometric and topological questions regarding the ball B(s) for finite but large times. I will describe some of these results, including estimates for the length of the boundary of B(s) and the size and number of “holes” (finite components of the complement) of B(s). There are still many unsolved questions.
Abstract: The t-PNG model is a one-parameter deformation of the polynuclear growth model that was recently introduced by Aggarwal, Borodin, and Wheeler, who studied its fluctuations using integrable probability methods. In this talk, we will discuss how to use techniques from interacting particle systems to prove a strong law of large numbers for this model. To do so, we will introduce a new colored version of the model that allows us to apply Liggett’s subadditive ergodic theorem to obtain the hydrodynamic limit. The t-PNG model also has a close connection to the stochastic six vertex model. We will discuss this connection and explain how we can prove similar results for the stochastic six vertex model. This talk is based on joint work with Yier Lin.
Abstract: Random walks in cooling random environments are a model of random walks in dynamic random environments where the random environment is re-sampled at a fixed sequence of times (called the cooling sequence) and the environment remains constant between these re-sampling times. We study the limiting distributions of the walk in the case when distribution on environments is such that a walk in a fixed environment has a κ-stable limiting distribution for some κ ∈ (1,2). It was previously conjectured that for cooling maps whose gaps between re-sampling times grow polynomially that the model should exhibit a phase transition from Gaussian limits to κ-stable depending on the exponent of the polynomial growth of the re-sampling gaps. We confirm this conjecture, identifying the precise exponent at which the phase transition occurs and proving that at the critical exponent the limiting distribution is a generalized tempered κ-stable distribution. The proofs require us to prove some previously unknown facts about one-dimensional random walks in random environments which are of independent interest.
This is based on joint work with Luca Avena and Conrado da Costa.
Abstract: I will discuss statistical inference problems on edge-correlated stochastic block models. We determine the information-theoretic threshold for exact recovery of the latent vertex correspondence between two correlated block models, a task known as graph matching. As an application, we show how one can exactly recover the latent communities using multiple correlated graphs in parameter regimes where it is information-theoretically impossible to do so using just a single graph. Furthermore, we obtain the precise threshold for exact community recovery using multiple correlated graphs, which captures the interplay between the community recovery and graph matching tasks. This is based on joint work with Julia Gaudio and Anirudh Sridhar.
Abstract: Given a finite poset that is not completely ordered, is it always possible to find two elements x and y, such that the probability that x is less than y in the random linear extension of the poset, is bounded away from 0 and 1? Kahn-Saks gave an affirmative answer and showed that this probability falls between 3/11 (0.273) and 8/11 (0.727). The currently best known bound is 0.276 and 0.724 by Brightwell-Felsner-Trotter, and it is believed that the optimal bound should be 1/3 and 2/3, also known as the 1/3-2/3 Conjecture. Most notably, log-concave and cross product inequalities played the central role in deriving both bounds. In this talk we will discuss various generalizations of these results together with related open problems. This talk is joint work with Igor Pak and Greta Panova, and is intended for the general audience.
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