Purdue Probability Seminar, Spring 2025

A new estimator is proposed for estimating the tail exponent of a heavy-tailed distribution. This estimator, referred to as the layered Hill estimator, is a generalization of the traditional Hill estimator, building upon a layered structure formed by clusters of extreme values. We argue that the layered Hill estimator provides a robust alternative to the traditional approach, exhibiting desirable asymptotic properties such as consistency and asymptotic normality for the tail exponent. Both theoretical analysis and simulation studies demonstrate that the layered Hill estimator shows significantly better and more robust performance, particularly when a portion of the extreme data is missing. This is joint work with Takashi Owada (Purdue).

Consider random polynomials of the form $\sum_{i=0}^n a_ix^i$ where the coefficients $a_i$ are independent with mean zero and finite second moments. We discuss how the probability that such a polynomial (when $n$ is even) has no zeros decays as $n^{-b+o(1)}$, where $b$ is a universal constant independent of the distributions of $a_i$. This resolves a conjecture by Poonen and Stoll. Notable progress on this conjecture was made by Dembo, Poonen, Shao, and Zeitouni, who demonstrated the same result when coefficients are identically and independently distributed (i.i.d.) with all higher moments being finite. Extending their results to the finite moment case remained an open problem until now. Our approach, in contrast to previous works in this area, relies on the combinatorics of multiscale analysis of random polynomials. This work is based on joint collaboration with Sumit Mukherjee.

In this talk I will start by motivating the fundamental importance of singular stochastic partial differential equations in (i) our understanding of the large-scale behaviour of dynamic random systems and (ii) developing a rigorous approach to quantum field theory. I will describe the key mathematical difficulties these equations pose, and sketch how combining analytic, probabilistic, and algebraic arguments have allowed mathematicians to overcome these difficulties and develop a powerful new PDE theory. I will also touch on some more recent developments in this area, namely applications to gauge theory and non-commutative probability theory.

Strain diversity varies significantly among diseases. Some diseases exhibit extreme long-term diversity (such as pneumococcus and malaria), while others remain single-strain despite high mutation rates (such as measles and chickenpox). The persistence of strain diversity is influenced by a complex interplay of strain innovation, pathogen life history, transmission dynamics, and immune responses. Our goal is to identify the key factors that drive, maintain, or prevent long-term diversity. We developed Multi-Strain Eco-Evo Dynamics (MultiSEED), a theoretical framework to predict the long-term strain diversity observed in common diseases by integrating an n-strain status-based SIR (Susceptible-Infected-Recovered) model with continuous-time stochastic processes at the ecological and evolutionary time scales. The numerical framework enables fast calculation of the expected number of transient and long-persisting strains when supplied with epidemiological and genetic measures. Our results show that host population size largely determines the magnitude of strain diversity, while parasite innovation rates have a minor impact. The combination of basic reproduction number (R0) and resource recruitment rates determines the strain dynamics regime: while the parameter range of flu produces a constant strain replacement regime, characteristics of strep pneumonia and malaria ensure a co-existence regime of many strains.

The motivation comes from the question on how to identify a sub-Riemannian structure from the transition density of a diffusion (heat kernel) on a metric measure space. In this talk we focus on the identification of the compact $3$-dimensional sub-Riemannian model space, which is also known as the $3$-sphere equipped with a Hopf fibration structure $S^1\to S^3\to \mathbb{CP}^1$. Namely, given a metric measure space equipped with a heat kernel that takes the form of the sub-elliptic heat kernel on $S^3$, we show that this space is (bundle-)isometric to the Hopf-fibration. This is a joint work with Masha Gordina.

I will discuss the concentration and fluctuation of \(\Phi^4\) Gibbs type measures from the perspectives of statistical physics, quantum field theory, and probability theory. The focus will be on the low temperature behavior and the thermodynamic limit of these probability measures, with particular attention to fluctuations around the soliton manifold.

We prove that a suitably rescaled and renormalized height function of a weakly asymmetric simple exclusion process on a circle converges to the Cole-Hopf solution of the KPZ equation. This is an analogue of the celebrated result by Bertini and Giacomin from 1997 for the exclusion process on a circle with any particle density. The main goal of our work is to analyze the interacting particle system using the framework of regularity structures without applying the Gaertner transformation, a discrete version of the Cole-Hopf transformation that linearizes the KPZ equation. This is joint work with Ruojun Huang and Hendrik Weber.

Let $B^{H}$ be a fractional Brownian motion in $\mathbb{R}^{d}$ of Hurst index $H\in\left(0,1\right)$. Our first objective in this study is to determine, for $E\subset (0,1]$ and $F\subset\mathbb{R}^d$ being arbitrary nonrandom compact sets, the essential supremum of the Hausdorff dimension of $B^H(E)\cap F$, known as the codimension formula, as well as for related processes. The explicit formula is given in terms of the Hausdorff dimension of $E \times F$ under the condition $\dim_{MA}(E)\leq Hd$ where $\dim_{MA}(E)$ is the modified Assouad dimension of $E$. Our work is a natural continuation of the investigations carried out by Xiao and Khoshnevisan in the case of Brownian motion and also provides an answer to the open problem P2 given there.

We consider the rate of convergence of the central limit theorem (CLT) for martingales. For martingales with a wide range of integrability, we will quantify the rate of convergence of the CLT via Wasserstein distances of order r, 1 \(\leq\) r \(\leq\) 3. Our bounds are in terms of Lyapunov's coefficients and the \(L^{r/2}\) fluctuation of the total conditional variances. Our Wasserstein-1 bound is optimal up to a multiplicative constant.