# Oberseminar Wahrscheinlichkeitstheorie und andere Vorträge im Sommersemester 2021

**Organisers**: Nina Gantert (TUM), Noam Berger (TUM), Markus Heydenreich (LMU), Franz Merkl (LMU), Silke Rolles (TUM), Konstantinos Panagiotou (LMU), Sabine Jansen (LMU),

**Talks**:

**Monday**, 19^{th} April 2021, 16:00, (using zoom)

Alice Callegaro (TUM)

Title: A spatially-dependent fragmentation process

Abstract: We define a spatially-dependent fragmentation process, which involves rectangles breaking up into progressively smaller pieces at rates that depend on their shape. Long, thin rectangles are more likely to break quickly, and are also more likely to split along their longest side. We are interested in how the system evolves over time: how many fragments are there of different shapes and sizes, and how did they reach that state? Our theorem gives an almost sure growth rate along paths, which does not match the growth rate in expectation - there are paths where the expected number of fragments of that shape and size is exponentially large, but in reality no such fragments exist at large times almost surely.

**Monday**, 26^{th} April 2021, 16:00, (using zoom)

Gidi Amir (Bar-Ilan University)

Title: The firefighter problem on infinite groups and graphs

Abstract: In the Firefighter model, a fire erupts on some finite set X_0 and in every time step all vertices adjacent to the fire catch fire as well (burning vertices continue to burn indefinitely) . At turn n we are allowed to protect f(n) vertices so that they never catch fire. The firefighter problem, also known as the fire-containment problem, asks how large should f(n) be so that we will eventually contain any initial fire. We are mainly interested in the asymptotic behaviour of f in relation with the geometry of the graph, focusing on Cayley graphs. Dyer, Martinez-Pedroza and Thorne proved that the growth rate of f is quasi-isometry invariant. Develin and Hartke proved upper bounds on the containment function for Z^d and conjectured that the correct bounds is cn^{d-2}. In joint work with Rangel Baldasso and Gady Kozma we prove this conjecture and show that this actually holds for any polynomial growth group. We will also survey other results on the problem for larger groups and their relation to the growth rate branching numbers. In the 2nd part of the talk we will introduce another related problem - "fire-retainment" that asks for saving only a positive portion of the graph. This turns out to be much more complicated. We give a full answer for the polynomial growth case and some interesting examples for larger groups. This part is based on joint work with Rangel Baldasso, Maria Gerasimova and Gady Kozma.

**Monday**, 10^{th} May 2021, 16:00, (using zoom)

Pierre Tarrès (NYU Shanghai)

Title: The *-Edge Reinforced random walk, bayesian statistics and statistical physics

Abstract: We will introduce recent non-reversible generalizations of the Edge-Reinforced Random Walk and its motivation in Bayesian statistics for variable order Markov Chains. The process is again partially exchangeable in the sense of Diaconis and Freedman (1982), and its mixing measure can be explicitly computed. It can also be associated to a continuous process called the *-Vertex Reinforced Random Walk, which itself is in general not exchangeable. If time allows, we will also discuss some properties of that process.Based on joint work with S. Bacallado and C. Sabo.

**Monday**, 17^{th} May 2021, 16:00, (using zoom)

Yuki Tokushige (TUM)

Title: Differentiability of the speed of biased random walks on Galton-Watson trees

Abstract: We prove that the speed of a $\lambda$-biased random walk on a supercritical Galton-Watson tree (with/without leaves) is differentiable for $\lambda$ such that the walk is ballistic and obeys a central limit theorem. We also give an expression of the derivative using a certain 2-dimensional Gaussian random variable, which naturally arise as limits of functionals of a biased random walk. The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres. In particular, an important role is played by moment estimates of regeneration times, which are locally uniform in $\lambda$. This talk is based on a joint work with Adam Bowditch (University College Dublin).

**Monday**, 31^{th} May 2021, 16:00, (using zoom)

Stanislav Volkov (Lund University)

Title: Interacting Pólya urns with removals as linear competition process

Abstract: A linear competition process is a continuous time Markov chain defined as follows. The process has N (N\ge 1) non-negative integer components. Each component increases with the linear birth rate or decreases with a rate given by some linear function of other components. A zero value is an absorbing state for each component: should a component become zero ("extinct"), it would stay zero for good. For all possible interactions we show that a.s. eventually only a (possibly, random) subset of non-interacting components can survive. A similar result holds for the relevant generalized Pólya urn model with removals.(Based on a joint work with Serguei Popov and Vadim Shcherbakov.)

**Monday**, 7^{th} June 2021, 16:00, (using zoom)

Tom Hutchcroft (Universität Cambridge)

Title: Supercritical percolation on finite transitive graphs

Abstract: In Bernoulli bond percolation, each edge of some graph are chosen to be either deleted or retained independently at random with retention probability p. For many large finite graphs, there is a phase transition such that if p is sufficiently large then there exists a *giant* cluster whose volume is proportional to that of the graph with high probability. We prove that in this phase the giant cluster must be unique with high probability: this was previously known only for tori and expander graphs via methods specific to those cases. Joint work with Philip Easo.

**Monday**, 14^{th} June 2021, 16:00, (using zoom)

Nicolas Blank (LMU; MSc presentation)

Title: Distance Estimates in the Random Connection Model

Abstract: We introduce the weight-dependent random connection model, which is a general class of geometric random graphs. The vertices are given by a marked Poisson process on Euclidean space, and the probability of an edge between two marked Poisson points is given through a connectivity function. We consider a specific choice of connectivity function and derive a random graph model that corresponds to a continuum version of the scale-free percolation model introduced by Deijfen, van der Hofstad, and Hooghiemstra (2013). We sketch how one can transfer results about the degree distribution and about the graph distances from the scale-free percolation model to the random connection model.

**Monday**, 21^{th} June 2021, 16:00, (using zoom)

Rémy Poudevigne-Auboiron (University of Cambridge)

Title: Monotonicity and phase transition for the edge-reinforced random walk

Abstract: The linearly edge reinforced random walk (ERRW) was introduced in 1986 by Coppersmith and Diaconis and is one of the first example of reinforced random walks. Recently a link has been found between this model, the vertex reinforced jump process and a random spin model. Because of these links it was possible to show that in dimension 3 and above, the ERRW is recurrent for large reinforcement and transient for small ones and thus exhibits a phase transition. We will present the links between those models and show that the model has some monotonicity (the larger the reinforcements the more recurrent it is) and that its phase transition is unique.

**Monday**, 28^{th} June 2021, 16:00, (using zoom)

Eugene Lytvynov (Swansea University)

Title: Orthogonal polynomials of Lévy white noise and umbral calculus

Abstract: Classical umbral calculus is the theory of Sheffer

polynomial sequences, which are characterised by the exponential form

of their generating function. Meixner in 1934 found all Sheffer

sequences that are orthogonal with respect to a probability measure on

the real line. The class of such probability measures consists of

Gaussian, Poisson, gamma, negative binomial and Meixner distributions.

Note that all these measures are infinitely divisible, hence they give

rise to a corresponding Lévy process. Let $\mathcal D$ denote the

space of all smooth functions on the real line with compact support,

and let $\mathcal D'$ be its dual space, i.e., the space of all

generalized functions on the real line. We will introduce the notion

of a polynomial sequence on $\mathcal D'$ and a Sheffer sequence on

$\mathcal D'$. A Lévy white noise measure is a probability measure on

$\mathcal D?$ which is the law of a generalised stochastic process

obtained as the (generalised) derivative of a Lévy process. We will

find the class of all Lévy white noises for which there exists an

orthogonal polynomial sequence on $\mathcal D'$. This class will be in

one-to-one correspondence with the Meixner class of probability

measures on the real line, and the corresponding orthogonal

polynomial sequences on $\mathcal D'$ are all Sheffer sequences.

Extending Grabiner's result related to the one-dimensional umbral

calculus, we will construct a class of spaces of entire functions on

the complexification of $\mathcal D'$ that is spanned by Sheffer

polynomial sequences. This will, in particular, extend the well-known

characterisation of the Hida test space of Gaussian white noise as a

space of entire functions.

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