Purdue Probability Seminar, Fall 2021

We study the point process limits of the circular Jacobi beta-ensemble and the real orthogonal beta-ensemble. Using the differential operator framework introduced by Valkó and Virág, we identify the point process scaling limits as spectra of certain stochastic differential operators. This framework also allows us to prove the convergence of the normalized characteristic polynomials of the finite models to certain random analytic functions. In this talk, I will first review the theoretic operator framework and these constructions. Then I will present the operator level convergence and several characterizations of the limiting objects. This talk is based on joint work with Benedek Valkó.

In this talk, I will discuss recent progress in the understanding of the structure in the spectrum of random Schrödinger operators. More specifically, I will introduce the concept of number rigidity in point processes and discuss recent efforts to understand its occurrence in the spectrum of random Schrödinger operators. Based on joint works with Promit Ghosal (MIT), Wenxuan Li (UChicago), and Yuchen Liao (Warwick).

For general dimension \(d\), we identify, with proof, the asymptotic conditional distribution of the number of (Pareto) records broken by an observation given that the observation sets a record. Fix \(d\), and let \({\mathcal K}(d)\) be a random variable with this distribution. We show that the (right) tail of \({\mathcal K}(d)\) satisfies \[ {\mathbb P}({\mathcal K}(d) \geq k) \leq \exp\left[ - \Omega\!\left( k^{(d - 1) / (d^2 - 2)} \right) \right]\mbox{ as $k \to \infty$} \] and \[ {\mathbb P}({\mathcal K}(d) \geq k) \geq \exp\left[ - O\!\left( k^{1 / (d - 1)} \right) \right]\mbox{ as \(k \to \infty\)}. \] When \(d = 2\), the description of \({\mathcal K}(2)\) in terms of a Poisson process agrees with the main result from Fill [Comb. Probab. Comput. 30 (2021) 105--123] that the distribution of \({\mathcal K}(2)\) is Geometric\((1/2)\) with support \(\{0, 1, \ldots\}\). We show that \({\mathbb P}({\mathcal K}(d) \geq 1) = \exp[-\Theta(d)]\) as \(d \to \infty\); in particular, \({\mathcal K}(d) \to 0\) in probability as \(d \to \infty\).

We study Cox/\(G_t\)/\(\infty\) queues driven by Cox processes in a fast oscillatory random environment. Within the framework of stochastic homogenization, we can establish diffusion approximations to the rescaled number-in-system process by proving functional central limit theorems (FCLTs). This framework allows us to obtain both quenched and annealed limits. At the quantitative level, we identify subcritical and supercritical regimes which indicate the relative domi- nance between the randomness in the arrival intensity and that in the service times. In this talk, I will discuss the recent results in our analysis of Cox/\(G_t\)/\(\infty\) queues. Based on joint work with Harsha Honnappa, Samy Tindel, and Aaron Yip.

Non-equilibrium stationary states (NESS) are invariant measures of non-reversible Markov chains. NESS arising from interacting particle systems are objects of interest in the statistical physics community. In two examples (reaction-diffusion on a torus and boundary-driven exclusion on the Sierpinski gasket) we prove central limit theorems. In the first example, the scaling limit of the invariant measure is a superposition of a white noise and a Gaussian free field; in the second example we get a Gaussian random field on the Sierpinski gasket that does not seem to have been studied in the literature yet. The proofs are based on the convergence of some well-chosen martingales constructed from the Markovian dynamics and the hydrodynamic semigroup. Based on joint works with Joe Chen (Colgate University - Hamilton, NY), Chiara Franceschini (IST - Lisbon), Patrícia Gonçalves (IST - Lisbon), Milton Jara (IMPA - Rio de Janeiro) and Rodrigo Marinho (IST - Lisbon).

The loop-erased random walk is a stochastic process modelling a path without self-intersections. Lawler introduced it in the early '80s. Since then, the loop-erased random walk has appeared in connection to other models in statistical mechanics, playing a pivotal role in their study. The first part of the talk will briefly survey our current knowledge on the scaling limits of loop-erased random walks in Z^d. Then we will focus on the three-dimensional case. I will discuss a work in progress on the convergence of the scaling limit of the loop-erased random walk in Z^3 (a collaboration with Xinyi Li and Daisuke Shiraishi).

We consider geometric last-passage percolation, in which the vertices of Z^2 are assigned i.i.d. Geometric weights with a fixed parameter. The last-passage time between two points is the largest total weight of an up-right path between the points. Up-right paths which achieve this maximum are called geodesics. We show that with probability one, the only bi-infinite geodesic paths are horizontal and vertical lines. We also highlight two claims that, if verified, would extend our proof to a more general class of distributions. Based on joint work with C. Janjigian and F. Rassoul-Agha.

Cooperative motion random walks form a family of random walks where each step is dependent upon the distribution of the walk itself. Movement is promoted at locations of high probability and dampened in locations of low probability. These processes are a generalization of the hipster random walk introduced by Addario-Berry et. al. in 2020. We study the process through a recursive equation satisfied by its CDF, allowing the evolution of the walk to be related to a finite difference scheme. I will discuss this relationship and how PDEs can be used to describe the distributional convergence of asymmetric and symmetric cooperative motion. This talk is based on joint work with Louigi Addario-Berry and Jessica Lin.

In this talk, I present a recent work in which we investigate models from the theory of regularity structures as features in machine learning tasks. A model is a polynomial function of a space-time signal designed to well-approximate solutions to partial differential equations (PDEs). Models can be seen as multi-dimensional generalisations of signatures of paths; this work therefore aims to extend the use of signatures in data science beyond the context of time-ordered data. I will introduce a flexible definition of a model feature vector and two algorithms which combine these features with supervised linear regression. I will also present several numerical experiments in which we use these algorithms to predict solutions to parabolic and hyperbolic PDEs with a given forcing and boundary conditions. Interestingly, in the hyperbolic case, the prediction power relies heavily on whether the boundary conditions are appropriately included in the model. Based on joint work with Andris Gerasimovics and Hendrik Weber.

We consider a random walk in an i.i.d. uniformly elliptic random environment in the integer lattice with no drift. It is known that the quenched heat kernel of this RWRE does not have deterministic Gaussian bounds. In this talk we will present lower and upper bounds for the heat kernel which are Gaussian up to a random multiplicative factor which has exponential moment. This improves the L^p integrability of the bounds obtained by Fabes and Stroock in deterministic environments. Joint work with H.V.Tran (Madison).

In This talk, we consider a rough differential equation driven by a Markovian rough path. We demonstrate that if the vector fields satisfy the parabolic Hörmander's condition, then the solution admits a smooth density with a Gaussian type upper bound, given that the generator of the driving noise satisfy certain non-degenerate conditions. The main new ingredient is the study of non-degenerate property of the Jacobian process of the driving noise. We will also discuss small ball estimates and Norris’s type lemma for diffusion.

Estimation/prediction with "contaminated" data is a classical problem considered in Huber's seminal work in the 60's. This set-up regained a lot of attention recently in the modern literature of robust statistics and machine learning. A modern take on the statistical optimality of "robust" estimators concerns with the "best" convergence rate on the sample size, effective dimension, contamination fraction and, sometimes, the failure probability restricted to computationally tractable estimators. In this talk we consider the least-squares regression problem with a linear subgaussian class, including the setting where a fraction of the label sample is contaminated by adversarial outliers. Our contributions to this problem span different perspectives. First, we study outlier-robust high-dimensional least-squares when the parameter is sparse/low-rank and show tractable optimality adaptively to sparsity/low-rankness and the contamination fraction. Second, the literature in high-dimensional least squares typically assumes the noise is independent of the features. We present a statistical theory which allows for heterogenous feature-dependent noise without extra structural assumptions. A third concern is optimality wrt failure probability. Most rates for this problem are optimal only "on average" but suboptimal when concerned with the failure probability. We present new optimal rates for an estimator whose tuning is adaptive to the failure probability having optimal rates uniformly on any confidence level. Our estimator is based on a new 'Sorted Huber loss" which we show by numerical experiments can significantly outperform the classical Huber loss. Fourthly, we consider the problem of trace-regression with matrix decomposition. While natural in high-dimensional statistics and compressed sensing, to the best of our knowledge no optimality theory has been presented before. Our statistical theory crucially relies on new concentration inequalities for the Multiplier Process and Product Processes derived via Talagrand-like generic chaining techniques. We also crucially need Chevet's inequality to show optimality of a robust estimator with label contamination.