Georgia Institute of Technology College of Sciences

The incipient infinite cluster was first proposed by physicists in the 1970s as a canonical example of a two-dimensional medium on which random walk is subdiffusive. It is the measure obtained in critical percolation by conditioning on the existence of an infinite cluster, which is a probability zero event. Kesten presented the first rigorous two-dimensional construction of this object as a weak limit of the one-arm event. In high dimensions, van der Hofstad and Jarai constructed the IIC as a weak limit of the two-point connection using the lace expansion. Our work presents a new high-dimensional construction which is "robust", establishing that the weak limit is independent of the choice of conditioning. The main tools used are Kesten's original two-dimensional construction combined with Kozma and Nachmias' regularity method. Our robustness allows for several applications, such as the explicit computation of the limiting distribution of the chemical distance, which forms the content of our upcoming project. This is joint work with Shirshendu Chatterjee, Jack Hanson, and Philippe Sosoe. The preprint can be found at https://arxiv.org/abs/2502.10882.

In this talk, I will present the results of a collaboration with Benjamin McKenna on the injective norm of large random Gaussian tensors and uniform random quantum states, and describe some of the context underlying this work. The injective norm is a natural generalization to tensors of the operator norm of a matrix and appears in multiple fields. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, known as geometric entanglement. In our preprint, we provide a high-probability upper bound on the injective norm of real and complex Gaussian random tensors, which corresponds to a lower bound on the geometric entanglement of random quantum states, and to a bound on the ground-state energy of a particular multispecies spherical spin glass model.

Suppose that particles are randomly distributed in $R^d$, and they are subject to identical stochastic motion independently of each other. The Smoluchowski process describes fluctuations of the number of particles in an observation region over time. The goal is to infer on particle displacement process from such count data. We discuss probabilistic properties of the Smoluchowski processes and consider related statistical problems for two different models of the particle displacement process: the undeviated uniform motion (when a particle moves with random constant velocity along a straight line) and the Brownian motion displacement. In these settings we develop estimators with provable accuracy guarantees.

Data imbalance is a fundamental challenge in data analysis, where minority classes account for a small fraction of the training data compared to majority classes. Many existing techniques attempt to compensate for the underrepresentation of minority classes, which are often critical in applications such as rare disease detection and anomaly detection. Notably, in empirical deep learning, the large model size exacerbates the issue. However, despite extensive empirical heuristics, the statistical foundations of these methods remain underdeveloped, which poses an issue to the reliability of these machine learning models.

In this talk, I will examine imbalanced classification problems in high dimensions, focusing on support vector machine (SVMs) and logistic regression. I will introduce a "truncation" phenomenon---which we verifed across single-cell tabular data, image data, and text data---where overfitting in high dimensions distorts the distribution of logits on training data. I will provide a theoretical foundation by characterizing the asymptotic distribution via a variational formulation. This analysis formalizes the intuition that overfitting disproportionately harms minority classes and reveals how margin rebalancing---a widely used deep learning heuristic---mitigates data imbalance. As a consequence, the theory offers both qualitative and quantitative insights into generalization errors and uncertainty measures such as calibration.

This talk is based on a joint work with Jingyang Lyu (3rd-year Stats PhD student) and Kangjie Zhou (Columbia Statistics): arXiv:2502.11323.