Prob/Stat Seminar

Stochastic reaction-diffusion equations are important models in mathematics and in applied sciences such as spatial population genetics and ecology. These equations describe a quantity (density/concentration of an entity) that evolves over space and time, taking into account random fluctuations. However, for many reaction terms and noises, the solution notion of these equations is still missing in dimension two or above, hindering the study of spatial effect on stochastic dynamics through these equations.

In this talk, I will discuss a new approach, namely, to study these equations on metric graphs (and fractals) that flexibly parametrize the underlying space. This enables us to not only bypass the ill-posedness issue of these equations in higher dimensions, but also assess the impact of space and stochasticity on the coexistence and the genealogies of interacting populations. We will focus on the computation of the probability of extinction, the quasi-stationary distribution, the asymptotic speed and other long-time behaviors for stochastic reaction-diffusion equations of Fisher-KPP type.

This talk is really three talks. In the first part, our main focus is an algorithm used to approximately solve an optimal transport problem that often arises in economic applications: how do we minimise the cost of producers and consumers, when the cost is a concave function of the distance? In the second part, we analyze how feedback can be used to improve the perfromance of certain card guessing games, closely related to well-known statistical problems such as bias-detection in clinical trials and hypothesis testing. In the third part, motivated by a novel notion of curvature on graphs, we provide some close-to-exact formula for the hitting times in Erdös-Rényi random graphs. These stories are held together by a common theme: sometimes, greediness and dependencies are not that harmful. This is based on joint works with J. He, S. Steinerberger and R. Tripathi.

Sufficient dimension reduction (SDR) is an effective way to detect nonlinear relationships between response variables and covariates by reducing the dimensionality of covariates without information loss. The principal fitted component (PFC) model is a way to implement SDR using some class of basis functions, however, the PFC model is not efficient when there are many irrelevant or noisy covariates. There have been a few studies on the selection of variables in the PFC model via penalized regression or sequential likelihood ratio test. A novel variable selection technique in the PFC model has been proposed by incorporating a recent development in multiple testing such as mirror statistics and random data splitting. It is highlighted how a mirror statistic is constructed in the PFC model using the idea of projection of coefficients to the other space generated from data splitting. The proposed method is superior to some existing methods in terms of false discovery rate (FDR) control and applicability to high-dimensional cases. In particular, the proposed method outperforms other methods as the number of covariates tends to be getting larger, which would be appealing in high dimensional data analysis.

Co-evolving network models, wherein dynamics such as random walks on the network influence the evolution of the network structure, which in turn influences the dynamics, are of interest in a range of domains. While much of the literature in this area is currently supported by numerics, providing evidence for fascinating conjectures and phase transitions, proving rigorous results has been quite challenging. The two aims of this talk are:

(i) Propose a general class of co-evolving network models driven by local exploration, started from a single vertex called the root. New vertices enter the system, explore the current network via randomly sampling a vertex and then exploring the graph for a random number of steps in the direction of the root. Specific choices of the exploration step distribution lead to preferential attachment with global attachment functionals such as PageRank scores. We show that the model shows non-trivial phase transition phenomena, including condensation of a fixed density of edges about the root, as well as phase transitions of functionals such as the PageRank distribution.

(ii) Develop mathematical machinery for the rigorous analysis of such models, including local weak convergence, infinite dimensional urn models, related multi-type branching processes and corresponding Perron-Frobenius theory, branching random walks, and in particular relating tail exponents of various functionals to the scaling exponents of quasi-stationary distributions of associated random walks. (Based on joint work with Sayan Banerjee and Shankar Bhamidi.)

Concentration inequalities for real-valued functions are well understood in many settings and are classical probabilistic tools in theory and applications -- however, much less is known about concentration phenomena for vector-valued functions. We present a novel vector-valued concentration inequality for the uniform measure on the symmetric group. Furthermore, we discuss the implications of this result regarding the distortion of embeddings of the symmetric group into Banach spaces, a question which is of interest in metric geometry and algorithmic applications. We build on prior work of Ivanisvili, van Handel, and Volberg (2020) who proved a vector-valued inequality on the discrete hypercube, resolving a question of Enflo in the metric theory of Banach spaces. This talk is based on joint work with Ramon van Handel.

A Liouville quantum gravity (LQG) surface is a "canonical" random two-dimensional Riemannian manifold that is conjectured to be the scaling limit of a wide variety of random planar graph models. LQG was formulated initially as a random measure space and, more recently, as a random metric space. In this talk, I will explain how the LQG measure can be recovered as the Minkowski content measure with respect to the LQG metric, thereby providing a direct connection between the two formulations for the first time. Our primary tool is the mating-of-trees theory of Duplantier, Miller, and Sheffield, which says that an LQG surface is an infinitely divisible metric measure space when explored by an independent space-filling Schramm–Loewner evolution (SLE) curve. This is joint work with Ewain Gwynne (University of Chicago).