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# Colloquium in probability

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

## Upcoming talks

### 05.12.2022 16:30 Quirin Vogel (TUM): The HYL model and random interlacements

In this talk, we compute the thermodynamic limit of the loop-HYL model, an approximation to the Feynman representation of the hard-core Bose gas. We show that the excess density concentrates on the random interlacements, with a discontinuity at the critical point. Joint work with Matthew Dickson, LMU.
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### 12.12.2022 15:00 Guilherme Reis (TUM - Chair of Probability Theory): Interacting diffusion on random graphs

In this talk we will talk about a class of particle system defined on top of random graphs that has the stochastic Kuramoto model as a particular example. We are interested on limit theorems for the empirical measure of the particles. In other words, we investigate the behavior of a typical particle proving law of large numbers and large deviations results. The graphs we consider include the Erdös-Rényi graph with different levels of sparsity.
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### 16.01.2023 16:30 Umberto De Ambroggio (LMU): Unusually large components in near-critical Erdös-Rényi graphs via ballot theorems

In this talk we describe a probabilistic methodology to derive the precise asymptotic for the probability of observing a maximal component containing more than n^{2/3} vertices in the (near-) critical Erdös-Rényi random graph. Our approach is mostly based on ballot-type estimates for one-dimensional, integer-valued random walks, and improves upon the martingale-based method introduced by Nachmias and Peres in 2009. We also briefly discuss how our method has been adapted to study the same type of problem for (near-) critical percolation on a random d-regular graph, as well as possible future developments.
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## Previous talks

### 21.11.2022 16:30 Tamas Makai (LMU): Degree sequences of random uniform hypergraphs

Consider the probability that a random graph selected uniformly from the set of r-uniform hypergraphs with n vertices and m edges, has a given degree sequence. Previously the value of this probability has been investigated by Kamčev, Liebenau and Wormald, where they examined degree sequences from very sparse to moderately dense hypergraphs when r=o(n^{1/4}) and the variation of the degrees is small, but exceeds the typical degree variation in random hypergraphs. We extend their results, by establishing this result for dense hypergraphs, which hold for any value of r and allow for a greater variation on the degrees.This is joint work with Catherine Greenhill, Mikhail Isaev and Brendan McKay.
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### 14.11.2022 16:30 Alexandra Quitmann (WIAS Berlin): Macroscopic loops in interacting random walk loop soups

We consider a general interacting random walk loop soup that is related to several well-known statistical mechanics models, such as the Spin O(N) model, the double dimer model or the interacting Bose gas. We discuss the system in $\mathbb{Z}^d, d>2$, and present some recent results about the occurrence of macroscopic loops whose length is proportional to the volume of the system as the inverse temperature is large enough.
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### 07.11.2022 16:30 Stefan Grosskinsky (Universität Augsburg): Asymptotics of generalized Pólya-urns with non-linear feedback joint work with Thomas Gottfried

Generalized Pólya urns with non-linear feedback are an established model for the competition in markets. Depending on the feedback function, the model can exhibit monopoly, where a single agent achieves full market share. We examine the asymptotic behaviour with diverging initial market size for a large class of feedback functions, and establish a scaling limit for the evolution of market shares, including a functional central limit theorem. In the monopoly case find a criterion to predict the (in general random) monopolist with high probability under generic initial conditions. Our results reveal an interesting difference between exponentially and more realistic polynomially growing feedback.
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### 25.07.2022 15:00 Dominik Schmid (Universität Princeton): Markov equivalence classes of directed acyclic graphs.

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).
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### 25.07.2022 16:30 Viktor Bezborodov (Universität Göttingen): Continuous-time frog model: linear spread and explosion.

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.
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For talks more than 90 days ago please have a look at the Munich Mathematical Calendar (filter: "Oberseminar Wahrscheinlichkeitstheorie").