Purdue Probability Seminar, Spring 2026

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.

In this talk we consider the open one-dimensional KPZ equation on the interval \([0,L]\) with Neumann boundary conditions. For \(L\sim t^{\alpha}\) and stationary initial conditions, we obtain matching upper and lower bounds on the variance of the height function for \(\alpha\in[0,2/3]\) for different choices of the boundary parameters. Additionally, for fixed \(L\) and an arbitrary probability measure as initial conditions, we show Gaussian fluctuations for the height function as \(t\to\infty\). Joint work with Sayan Das and Antonios Zitridis.

I will discuss how Wilson's renormalization group can be used to construct and study probabilistic formulations of quantum field theories. Afterwards I'll turn to work in preparation, building on approaches developed by Ma and then Gawędzki and Kupiainen, on a model of a vector valued random field with quartic self-interaction when the number of components N is large (but finite).

This is joint work with Léonard Ferdinand

We study the fluctuations of subgraph counts in hyperbolic random geometric graphs on the d-dimensional Poincaré ball in the heterogeneous, heavy-tailed degree regime. In a hyperbolic random geometric graph whose vertices are given by a Poisson point process on a growing hyperbolic ball, we consider two basic families of subgraphs: star shape counts and clique counts, and we analyze their global counts and maxima over the vertex set. Working in the parameter regime where a small number of vertices close to the center of the Poincaré ball carry very large degrees and act as hubs, we establish joint functional limit theorems for suitably normalized star shape and clique count processes together with the associated maxima processes. The limits are given by a two-dimensional dependent process whose components are a stable Lévy process and an extremal Fréchet process, reflecting the fact that a small number of hubs dominates both the total number of local subgraphs and their extremes. As an application, we derive fluctuation results for the global clustering coefficient, showing that its asymptotic behavior is described by the ratio of the components of a bivariate Lévy process with perfectly dependent stable jumps.

The study of \(\mathbb{Q}_p^d\)-valued stochastic processes is an active area of research in \(p\)-adic mathematical physics and related fields. While in the \(\mathbb{R}^d\) setting, lots is known about scaling limits of random walks, only recently have we shown that a family of \(p\)-adic Levy processes associated to analogues of the heat equation is a scaling limit of random walks. By using the additional structure of a \(p\)-adic field, this family includes processes with a restricted class of anisotropies. We enlarge this class by constructing a one-parameter family of random walks on \(\mathbb{Q}_p^2\) and determining their scaling limits. This talk is based on joint work with David Weisbart.

Directed polymer models describe the behavior of a long chain molecule in a random medium, where the path of the polymer is influenced by a competition between energy gained from the environment and the natural entropy of the chain. This class of models has deep connections to statistical physics, and their mathematical study relies on a combination of classical probability methods and stochastic analysis.

In this talk I will start with a general introduction to directed polymer models, giving all the necessary background and notation from scratch. I will then describe a framework for studying continuous polymers in higher dimensions, driven by a broad class of Gaussian random environments. A central object in the analysis is a linear stochastic partial differential equation whose solution encodes the statistical weight of the polymer paths.

The main results I will discuss include (if I have enough time): structural properties of the partition function such as stationarity and a natural flow property; a sharp criterion distinguishing two qualitatively different regimes, depending on whether the polymer path is essentially free or strongly influenced by the environment; and a diffusive limit theorem at high temperature in dimension three and above, showing that the polymer behaves like Brownian motion in the weak disorder regime. This work extends a classical one-dimensional theory by Alberts-Khanin-Quastel to a broader higher-dimensional setting.