This term, all seminars take place Thursdays at 2pm, room MS.04 (Zeeman Building), unless indicated otherwise.



Schedule for

Dirk Zeindler, 30.01.2020

Seminars

09.01.2025


16.01.2025


23.01.2025


30.01.2025


06.02.2025 Gabriel Berzunza Ojeda (University of Liverpool)
Convergence of the Aldous-Broder Markov chain on high dimensional regular graphs
The Brownian continuum random tree emerges as a fundamental limit shape for various discrete tree models. It arises as the scaling limit of, for instance, the uniform spanning tree on the complete graph with N vertices or on the torus of size-length N, for d at least 5 (Peres and Revelle (2005)). Furthermore, the study of uniform spanning trees on connected graphs exhibits deep connections to several other key areas within probability theory. These include loop-erased random walks, potential theory, conformally invariant scaling limits, domino tilings, the Abelian sandpile model, and the Sznitman interlacement process. The Aldous-Broder Markov chain is a simple algorithm for generating random spanning trees. This Markov chain, defined on a graph G=(V,E), evolves through a sequence of rooted trees, each with a subset of $V$ as its vertex set. In Evans, Pitman and Winter (2006), it was shown that the suitable rescaled Aldous-Broder Markov chain on a complete graph converges to the so-called root growth with regrafting process (RGRG process) weakly with respect to the Gromov-Hausdorff topology. In this talk, we study the convergence of the rescaled Aldous-Broder Markov chain on high-dimensional regular graphs towards the RGRG process. Joint work with Osvaldo Angtuncio-Hernandez (Centro de Investigacion en Matematicas (CIMAT)) and Anita Winter (Universitat Duisburg-Essen).
13.02.2025


20.02.2025 Peter Gracar (University of Leeds)
Lipschitz cutset for fractal graphs and applications to the spread of infections
For Bernoulli supercritical percolation on the $d$-dimensional lattice it is well understood that the infinite component exists “everywhere”. In fact, it can be shown that this component contains as a subset a Lipschitz connected hyper-surface that can be built along any of the $d-1$ possible canonical hyperplanes of $mathbb{Z}^d$. In this talk, we will explain how one can construct a set satisfying similar properties on the Sierpiński gasket and then show how a multi-scale construction can be used to get its existence even for particle dependent percolation.

More precisely, we will consider the fractal Sierpiński gasket or carpet graph in dimension $dgeq 2$, denoted by $G$. At time $0$, we place a Poisson point process of particles onto the graph and let them perform independent simple random walks, which in this setting exhibit sub-diffusive behaviour. We will generalise the concept of particle process dependent Lipschitz percolation to the (coarse graining of the) space-time graph $G imes mathbb{R}$, where the opened/closed state of space-time cells is measurable with respect to the particle process inside the cell. We will discuss an application of this generalised framework through the following: if particles can spread an infection when they share a site of $G$, and if they recover independently at some rate $gamma>0$, then if $gamma$ is sufficiently small, the infection started with a single infected particle survives indefinitely with positive probability.
27.02.2025 Henrik Ueberschär (Sorbonne Université)
Multifractality for periodic solutions of certain PDE
Many dynamical systems are in a state of transition between two regimes. Examples are firing patterns of neurons, disordered quantum systems or pseudo-integrable systems. A common feature which is often observed for critical states of such systems is a multifractal self-similarity in a certain scaling regime which cannot be captured by a single fractal exponent but only by a spectrum of fractal exponents.
I will discuss a proof of multifractality of solutions for certain stationary Schrödinger equations with a singular potential on the square torus (joint with Jon Keating). Towards the end of the talk, I will allude to some new work on multifractal scaling and solutions to nonlinear PDE in fluid dynamics on cubic tori.
Contact: Nikos
06.03.2025


13.03.2025 Roman Kotecký (University of Warwick / CTS Prague)
Surface tension for the Widom-Rowlinson model in continuum
The Widom-Rowlinson model of interacting particles in continuum is one of few cases where phase transition on the "liquid/gas" coexistence line is rigorously understood.
I will discuss how "liquid" surface layer fluctuations lead to a derivation of low temperature asymptotics of the surface tension with an entropic term featuring a fractional power of the temperature. The analysis relies on a careful examination of the surface tension limit and its expression in terms of the spectral radius of an appropriately chosen transfer operator.
The topic is a vital ingredient in a series of papers under preparation, jointly with Frank den Hollander, Sabine Jansen, and Elena Pulvirenti, devoted to metastability for the Widom-Rowlinson model.