Purdue Probability Seminar, Spring 2026

Time: Wednesdays from 1:30-2:30pm (EST), unless otherwise noted.

Location:

SCHM 316

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We study the effect of a time-independent white noise disorder on a Brownian particle in four dimensions. This problem can be reformulated as a scaling-critical, singular SPDE. By tuning the noise strength appropriately, we obtain an exact analysis of the perturbative terms arising in the corresponding Green's function.

This yields the first systematic approach towards renormalising scaling-critical SPDEs. To place our result in context, we will also compare with the case of space-time disorder, highlighting both parallels and key differences. This is joint work with Tommaso Rosati.

A multiplicative matrix Brownian motion is a natural type of stochastic process with values in GL(N,C), the group of invertible NxN matrices. We can approximate the Brownian motion by multiplying together a large number of independent random matrices that are small in the multiplicative sense (close to the identity). There are then two interesting limits one can take: either as the number of steps in the random walk tends to infinity or as the size of the matrices tends to infinity.

In this talk, I will mostly focus on what happens for a fixed number of steps as the size N of the matrices tends to infinity. The limit can be described as a "free" multiplicative random walk, in which we multiply together freely independent elements of the form "identity plus circular". We then analyze this product using the "linearization" method using a novel freeness result for block elements.

The talk will be self-contained and will have lots of pictures and animations!

Bond percolation in the 3-dimensional integer lattice is closely related to plaquette percolation on the same lattice, which is built from a random subset of 2-dimensional squares instead of edges. The topology of the surfaces formed by the plaquettes can be understood via a variation of bond percolation called entanglement percolation, which allows connections by linked loops in addition to paths. We will also discuss a truncation question for entanglement percolation and give partial results for general enhancement models. This talk will aim to be self-contained, and in particular will assume no topological background. Based on joint work with Benjamin Schweinhart and David Sivakoff.

In this talk, I will introduce the notion of (rough) additive functions: distributional differential forms for which line integration is well-defined. These additive functions are reminiscent of rough paths theory and make tools such as controlled rough paths and the (discrete) sewing lemma applicable in this setting. These additive functions have proven useful in (probabilistic) Yang–Mills theory. More precisely, I will discuss several applications: defining gauge-invariant observables, defining singular objects arising in a specific SPDE, and studying the properties of discrete covariant Laplacian with a rough connection such as its resolvent and the associated Gaussian free field. This talk is based on joint work with Ilya Chevyrev and Tom Klose, and on joint work with Ajay Chandra.

"On-scale" stretched exponential estimates for fluctuations of models within the Kardar-Parisi-Zhang universality class have played a particularly important role in mathematical work seeking to make physically motivated heuristic arguments about random growth models rigorous.

This talk will explain a simple coupling argument which recovers a moment generating function identity originally discovered by Rains using integrable probability techniques in the context of last-passage percolation. This coupling approach has proven to generalize significantly since it was first introduced about five years ago. The talk will then discuss how identities of this type can be used to derive the type of stretched exponential bounds mentioned above and when these bounds are sharp. The discussion will highlight some recent and in-progress improvements and extensions of older work.

Based on joint work with Elnur Emrah and Timo Seppalainen.

The Chen-Fliess series (or signature) has become a powerful feature set for analysing streamed data, and an important recent direction links signature features with kernel methods in statistical learning.

I will review recent results showing how tools from free probability allow signature kernels to arise as universal limits of randomised controlled differential equations (CDEs). This will be illustrated via Cartan developments into matrix Lie groups, where the limiting kernel depends on the group: for the unitary case, I will show how satisfies a new kernel trick given by a quadratic functional equation determined by Schwinger-Dyson relations for families of free semicircular random variables.

If time permits, I will also outline new extensions showing that both the signature kernel and the Schwinger-Dyson kernel arise as solutions to two-parameter rough differential equations, a viewpoint that yields natural well-posedness, stability results, and principled numerical schemes with explicit complexity guarantees.