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This term, all seminars take place Thursdays at 2pm, room MS.04 (Zeeman Building), unless indicated otherwise. |
![]() Dirk Zeindler, 30.01.2020 |
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. |