Colloquium in probability and other talks in summer term 2022

Monday, 2^th May 2022, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)
Sarai Hernandez-Torres (Technion, Haifa)
Title: The chemical distance in random interlacements in the low-intensity regime
Abstract: Random interlacements is a Poissonian soup of doubly-infinite random walk trajectories on Z^d, with a parameter u > 0 controlling the intensity of the Poisson point process. In a natural way, the model defines a percolation on the edges of Z^d with long-range correlations. We consider the time constant associated to the chemical distance in random interlacements at low intensity u > 0. It is conjectured that the time constant times u^{1/2} converges to the Euclidean norm, as u ↓ 0. In dimensions d ≥ 5, we prove a sharp upper bound and an almost sharp lower bound for the time constant as the intensity decays to zero. Joint work with Eviatar Procaccia and Ron Rosenthal.

Monday, 23^th May 2022, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)
Pierre Tarres (TUM)
Title: *-Edge-reinforced random walk and *-vertex-reinforced jump processes: limit measure and random Schrödinger representation
Abstract: TBA

Monday, 4^th July 2022, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)
Simone Floreani (TU Delft)
Title: Hydrodynamics for the partial exclusion process in random environment
Abstract: In this talk, I present a partial exclusion process in random environment, a system of random walks where the random environment is obtained by assigning a maximal occupancy to each site of the Euclidean lattice. This maximal occupancy is allowed to randomly vary among sites, and partial exclusion occurs. Under the assumption of ergodicity under translation and uniform ellipticity of the environment, we prove that the quenched hydrodynamic limit is a heat equation with a homogenized diffusion matrix. The first part of the talk is based on a joint work with Frank Redig (TU Delft) and Federico Sau (IST Austria).Finally, I will discuss some recent progresses in the understanding of what happens when removing the uniform ellipticity assumption. After recalling some results on the Bouchaud’s trap model, I will show that, when assuming that the maximal occupancies are heavy tailed and i.i.d., the hydrodynamic limit is the fractional-kinetics equation.The second part of the talk is based on an ongoing project with Alberto Chiarini (University of Padova) and Frank Redig (TU Delft).

Monday, 11^th July 2022, 15:00, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)
Thomas Richthammer (Universität Paderborn)
Title: Spatial Monotonicity Properties of Bernoulli Percolation
Abstract: We consider Bernoulli percolation on a graph G =(V,E). Interpreting some chosen reference vertex o in V as the origin of an infection, the percolation cluster of o corresponds to the set of all infected vertices.  It is very natural to expect that the probability for a vertex v in V to be infected should (in some sense) be decreasing in the distance of v to o. One possible rigorous formulation of this property is the famous bunkbed conjecture, which dates back to the 80s and still remains wide open. It seems that this kind of spatial monotonicity property of percolation in general is difficult to obtain. Here we present several new results relying on symmetry considerations or a Markov chain approach. Some of these results are joint work with Philipp König.
Im Anschluss daran:
Monday, 11^th July 2022, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)
Olga Aryasova (Universität Jena)
Title: On exponential almost sure synchronization of a one-dimensional diffusion with nonregular drift
Abstract: We study the asymptotic behaviour of a real-valued diffusion whose non-regular
drift is given as a sum of a dissipative term and a bounded measurable one. We
prove that two trajectories of that diffusion converge almost surely to one another at
an exponential explicit rate as soon as the dissipative coefficient is large enough. A
similar result in Lp is obtained.
Institute of Geophysics National Academy of Sciences of Ukraine, Kyiv.
Now Friedrich–Schiller–Universität Jena

Monday, 18^th July 2022, 16:30, LMU, room B252, Theresienstr. 39, Munich
Ruizhe Sun (LMU)
Title: Epidemics on Random Intersection Graphs
Abstract: In this thesis, we develop a Reed-Frost model based on random intersection
graphs. Our interest is the size of the set of ultimately recovered individuals as
population size grows to infinity. Several branching processes will be constructed
as approximating processes to serve this purpose. Eventually, benefiting from
the clique-based structure provided by the random intersection graph, we will
discuss the exact distribution of the quantity of interest in both small and large
outbreaks.
Im Anschluss daran:
Monday, 18^th July 2022, 17:30, LMU, room B252, Theresienstr. 39, Munich
Andreas Fleischmann (LMU)
Title: Existenz und Eindeutigkiet von räumlichen
Geburts- und Todesprozessen (MSc presentation)
Abstract: Im Rahmen dieses Vortrags erarbeiten wir zwei verschiedene Herangehensweisen für das Klären von Existenz- und Eindeutigkeitsfragen von räumlichen Geburts- und Todesprozessen. Dies umfasst zum einen die Möglichkeit, den Geburts- und Todesprozess als Sprungprozess aufzufassen. Dieser Ansatz geht auf Preston 1975 zurück und erfordert das eingehende Studium des Phänomens der Explosion und Kopplungen von Sprungrozessen. Zum Anderen stellen wir eine Möglichkeit vor, den räumlichen Geburts- und Todesprozess als eine Art Projektion aus einem Poisson-Prozess aufzufassen. Dies ermöglicht die Konstruktion und Untersuchung von Prozessen mit mehr als endlich vielen Geburten in endlicher Zeit. Dieser Ansatz geht auf Kurtz 1980, Garcia 1995, 2006, Bezborodov 2019 und andere zurück.

Monday, 25^th July 2022, 15:00, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)
Dominik Schmid (Universität Princeton)
Title: Markov equivalence classes of directed acyclic graphs.
Abstract: Can we reconstruct a directed acyclic graph having only access to its v-structures, encoding conditional independence between the sites, but without knowing its edge directions? In this talk, we study the probability to have a unique way of such a reconstruction when the directed acyclic graph G is chosen uniformly at random on a fixed number of sites. More generally, we study the size of its Markov equivalence class, containing all graphs with the same edge set as G when forgetting the edge directions, and having the same v-structures. This talk is based on ongoing work with Allan Sly (Princeton University).
Im Anschluss daran:
Monday, 25^th July 2022, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)
Viktor Bezborodov (Universität Göttingen)
Title: Continuous-time frog model: linear spread and explosion.
Abstract: In this talk we consider a continuous-time frog model on Z^d. As the discrete-time random walk is a.s. bounded for every fixed time, the original discrete-time frog model grows linearly with time no matter how heavy-tailed the distribution of the number of sleeping frogs per site is. This is no longer the case for the continuous-time model, and we discuss conditions on the initial distribution μ (mu) of number of sleeping particles per site ensuring linear growth, faster than linear growth, or explosion. The proof technique is based on a comparison with certain percolation-type models such as totally asymmetric discrete Boolean percolation or greedy lattice animals. We also discuss how these techniques can be applied to similar stochastic growth models.

Monday, 26^th September 2022, 16:30, LMU, room B252, Theresienstr. 39, Munich
Bogdan-Nicolae Nan (LMU)
Title: TBA (MSc presentation)