### 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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