Scaling limit of the Aldous-Broder chain on regular graphs: the transient regime.

Prof. Dr. Anita Winter (Universität Duisburg-Essen)

A spanning tree of a finite connected graph G is a connected subgraph of G that contains every vertex and no cycles. A well-known way to generate a uniform spanning tree (UST) is the Aldous-Broder algorithm. This is a stochastic process with values in rooted trees that is driven by a random walk on G and converges as time goes to infinity towards the Brownian Continuum Tree (CRT). On the complete graph it is known that this process has a GH-scaling limit which is referred to as Root Growth with Regrafting dynamics (RGRG). We will show that this is again a universal phenomenon, i.e., it holds for the Aldous-Broder chain on any appropriate sequence of regular graphs in the transient regime. In particular, this holds for the torus in dimensions five and higher.
This is joint work with Osvaldo Angtuncio Hernandez and Gabriel Berzunza Ojeda.

Unusual structures in Gaussian random waves

Prof. Giovanni Peccati (University of Luxembourg)

The term Berry’s random wave describes a canonical model for a Gaussian Laplace eigenfunction on the plane. This universal object naturally arises in connection with several questions in stochastic geometry, especially those concerning the local behaviour of Laplace eigenfunctions—random or deterministic—on chaotic surfaces. After a brief introduction to this model, I will highlight two striking (and still quite mysterious) structures that emerge in high-frequency regimes:
* Randomly scattered scars
* Total disorder processes (if time permits)
The second part of the talk will mainly focus on open problems and conjectures, and will include some personal reflections and anecdotes about doing mathematical research in the era of AI.

Fluctuations of the eigenvalues in the crushed-ice problem

Prof. Dr. Marek Biskup (UCLA)

The eigenvalues of the Laplacian are quantities that underlie many physical phenomena including, for instance, heat conduction or overtones of a drum. In 1974, Kac asked (and in a way answered) the question of how the eigenvalues of the Laplacian change when the underlying domain is perforated by a large number of tiny holes. Soon thereafter, Rauch and Taylor completed Kac’ analysis while linking the problem to a cocktail-party-level question of how fine one should crush the ice cubes to maximize its cooling effect on a drink (read: ambient liquid). Disregarding the analogies, they concluded that the correct quantity to look at is the capacity density of the perforations; scaling the number of perforations by inverse capacity then makes the eigenvalues tend to those of an effective (deterministic) Schrödinger operator as the perforation diameters tend to zero.

In the talk, I will review these findings relying at first on the classical connection to Wiener sausage which earned this problem much independent attention by probabilists over several decades. I will then proceed to discuss how one can capture the fluctuations of the eigenvalues and prove a Central Limit Theorem for the eigenvalues that are simple in the aforementioned limit. The method of proof in this part is quite different, relying largely on martingale representation and rank-one type of perturbations. For centering by expected eigenvalues the CLT holds in all dimensions 2 and above. For centering by limiting eigenvalues one has to restrict to dimensions less than 6 as non-trivial corrections arise in other cases. The talk is based on an upcoming joint work with Ryoki Fukushima (University of Tsukuba).

The cutoff phenomenon for random walks

Prof. Dr. Perla Sousi (University of Cambridge)
I will talk about the mixing time which is the time it takes for a random walk to reach equilibrium. My focus will be on the cutoff phenomenon observed when the transition to equilibrium happens abruptly in time. I will survey the developments in the last 30 years and present a recent universality result for graphs with a random matching that was obtained in collaboration with J. Hermon and A. Sly.