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.
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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.
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