# Seminar "Interacting Random Systems"

The IRS seminar is the group seminar of Research Group 5: Interacting random systems and Leibniz Group 6: Probabilistic methods for dynamic communication networks.

It is currently offered in a hybrid format -- online and in person. The zoom link will be sent shortly before the time of the talk.
Please contact the organisers if you are not included in the mailing list and would like to follow the talk or be kept up to date with the seminar. You can also access the calendar online or download it here.

 Usual Time: Wednesdays, 11:30 (CET) Place: Weierstrass Institute for Applied Analysis and Stochastics Mohrenstraße 39, 10117 Berlin Weierstrass Lecture Room (WIAS-405-406) and Online (zoom) Organiser: Alexander Zass

### Upcoming

Thursday, 05.09.2024, 12:00 (WIAS-405-406)
Martijn Gösgens (Eindhoven University of Technology)
The Erdős-Rényi random graph conditioned on every component being a clique

We consider an Erdős-Rényi random graph, conditioned on the rare event that all connected components are fully connected. Such graphs can be considered as partitions of vertices into cliques. Hence, this conditional distribution defines a distribution over partitions. The motivation for this model comes from community detection, where one wishes to partition graph vertices into groups that are better connected internally than externally. Using tools from analytic combinatorics, we prove limit theorems for several graph observables of this conditional distribution: the number of cliques; the number of edges; and the degree distribution. We consider several regimes of the connection probability $p$ as the number of vertices n diverges. For $p=\frac{1}{2}$, the conditioning yields the uniform distribution over set partitions, which is well-studied, but has not been studied as a graph distribution before. For $p<\frac{1}{2}$, we show that the number of cliques is of the order $n/\sqrt{\log n}$, while for $p>\frac{1}{2}$, we prove that the graph consists of a single clique with high probability. This shows that there is a phase transition at $p=\frac{1}{2}$. We additionally study the near-critical regime $p\downarrow\frac{1}{2}$, as well as the sparse regime $p\downarrow 0$.

#### Next talks

Past seminars
• 2024

• 05.06.2024, Gerónimo Uribe Bravo (UNAM):A pathwise approach to time-change
• 22.05.2024, Emmanuel Jacob (ENS Lyon):Local weak limit of dynamical random graphs
• 15.05.2024, Adrián González Casanova (UNAM and UC Berkeley):A random model of Poissonian interacting trajectories inspired by the Lenski experiment
• 08.05.2024, Tejas Iyer (WIAS Berlin):Persistent hubs in generalised preferential attachment trees and certain branching processes
• 24.04.2024, Julia Hörrmann (WIAS Berlin):High-dimensional random tessellations, deep neural networks and image classification in a security relevant setting
• 10.04.2024, Heide Langhammer (WIAS Berlin): A spatial coagulation process
• 20.02.2024, Utkir Rozikov (Uzbekistan Institute of Mathematics): Gibbs measures of Potts model on trees
• 20.02.2024, Jia-Jie Zhu (WIAS Berlin): Kernelization, approximation, and entropy-dissipation of gradient flows: from Wasserstein to Fisher–Rao
• 14.02.2024, Christian Mönch (JGU Mainz): Inhomogeneous long-range percolation: recent results
• 07.02.2024, Sabine Jansen (LMU München): Infinite-dimensional Lie algebras, Lévy random fields and stochastic processes
• 01.02.2024, Jan Niklas Latz (Czech Academy of Sciences, Prague): Monotone interacting particle systems: survival and the upper invariant law
• 24.01.2024, Alessandra Faggionato (La Sapienza University of Rome): Transport in weighted Delaunay triangulations
• 10.01.2024, Wolfgang König (WIAS and TU Berlin): A PhD project on dormancy in population dynamics in random environment
• 2023

• 13.12.2023, Beatriz Salvador (IST Lisbon): From duality to correlations: Application to the partial exclusion process $SEP(\alpha)$
• 07.12.2023, Giovanni Conforti (École Polytechnique, Paris): Invariant integrated convexity profiles for Hamilton-Jacobi-Bellman equations and applications
• 29.11.2023, Robert Patterson (WIAS Berlin): Gradient flows and the exclusion process
• 15.11.2023, Elena Magnanini (WIAS Berlin): Gelation and hydrodynamic limits in cluster coagulation processes
• 01.11.2023, Wolfgang König (WIAS and TU Berlin): A PhD project on dense Erdős--Rényi graphs
• 18.10.2023, Simone Floreani (Bonn University): Universal properties of non-equilibrium steady states of boundary-driven symmetric systems
• 11.10.2023, Bas Lodewijks (Augsburg University): Long-range first-passage percolation on the complete graph
• 04.10.2023, Subhro Ghosh (National University of Singapore): Rigidity phenomena in strongly correlated random point fields and the emergence of forbidden regions
• 04.10.2023, Tiffany Y. Y. Lo (Uppsala University): The expected degree distribution in duplication divergence models
• 27.09.2023, Michiel Renger (TU Munich): Collisions in the exclusion process
• 12.07.2023, Andreas Klippel (TU Darmstadt): Comparison of the random loop model to percolation and infinite loops in the random link model
• 06.07.2023, Naotaka Kajino (Kyoto University): Two-sided bounds on tail probabilities of the number of collisions of random walks
• 06.07.2023, Yuki Tokushige (University of Manchester): Scaling Limits of SRWs on the Long-Range Percolation Cluster
• 28.06.2023, Céline Kerriou (Universität zu Köln): Condensation in scale-free geometric graphs with excess edges
• 21.06.2023, Sanjoy Kumar Jhawar (WIAS Berlin): Moderate deviations in Poisson navigations
• 07.06.2023, Günter Last (Karlsruher Institut für Technologie): A Palm approach to tail processes and tail measures
• 24.05.2023, Renato Soares dos Santos (UFMG, Brasil): Ballistic random walk on the zero-range process
• 17.05.2023, Clément Erignoux (INRIA Lille): Symmetric and asymmetric hydrodynamics for the facilitated exclusion process via mapping
• 16.05.2023, Christian Hirsch (Aarhus University): On the topology of higher-order age-dependent random connection model
• 11-12-15.05.2023, Julian Kern (WIAS Berlin): Mini course: A unified approach to non-gradient systems (notes here)
• 09.05.2023, Julian Kern (WIAS Berlin): The crux of the non-gradient method (notes here)
• 03.05.2023, Luisa Andreis (Politecnico di Milano): Rare events for sparse random graphs and some new large deviation results
• 26.04.2023, Heide Langhammer (WIAS Berlin): LDP for the particle statistics in a spatial coagulation process
• 19.04.2023, Tejas Iyer (WIAS Berlin): Properties of recursive trees with independent fitnesses
• 15.03.2023, Christof Külske (Ruhr Universität Bochum): New states with non-singleton localization sets for gradient models on trees
• 01.03.2023, Partha Pratim Ghosh (TU Braunschweig): A Last Progeny Modified Branching Random Walk
• 01.03.2023, Élie de Panafieu (Nokia Bell Labs, France): Active clustering of a set using pairwise comparisons
• 08.02.2023, Dave Jacobi (TU Berlin): Super-Brownian motion with dormancy
• 01.02.2023, Alice Callegaro (TU München/JGU Mainz): Survival and complete convergence for a branching annihilating random walk
• 25.01.2023, Alexander Drewitz (Universität zu Köln): Percolation, long-range correlations and critical exponents on transient graphs
• 11.01.2023, Chiranjib Mukherjee (WWU Münster): Homogenization of Hamilton--Jacobi--Bellman equations on continuum percolation clusters
• 2022

• 07.12.2022, Daniel Heydecker (MPI Leipzig): From LDPs of the Kac Process to the H-Theorem
• 30.11.2022, Julian Kern (WIAS Berlin): A Fleming-Viot model with varying population size
• 16.11.2022, Jonas Köppl (WIAS Berlin): Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties
• 19.10.2022, Nicolas Forien (Sapienza Università di Roma): Sleepy frogs playing ping-pong on an overcrowded torus: The supercritical phase of Activated Random Walks
• 12.10.2022, Arne Grauer (Universität zu Köln): Short paths in scale-free geometric random graphs
• 28.09.2022, Willem van Zuijlen (WIAS Berlin): Weakly self-avoiding walk in a random potential
• 13.07.2022, Quirin Vogel (NYU Shanghai): The variational principle and the Bose gas
• 22.06.2022, Lukas Lüchtrath (Universität zu Köln): The various phases of long-range inhomogeneous percolation
• 01.06.2022, Francesca Cottini (University of Milano-Bicocca): Gaussian Limits for Subcritical Chaos
• 18.05.2022, Richard Kraaij (TU Delft): Large deviations for weakly coupled slow-fast systems via the comparison principle of an associated Hamilton-Jacobi equation
• 04.05.2022, Chengcheng Ling (TU Berlin): Singular SDEs and PDEs
• 20.04.2022, Martin Heida (WIAS Berlin): Measure theoretic aspects of stochastic homogenization
• 06.04.2022, Simone Baldassarri (University of Florence): Critical Droplets and sharp asymptotics for Kawasaki dynamics with strongly anisotropic interactions
• 23.03.2022, Florian Nie (TU Berlin): The stochastic F-KPP Equation and on/off branching coalescing Brownian Motion
• 23.02.2022, Ana Djurdjevac (Zuse Institute Berlin): Approximation of the Dean-Kawasaki equation
• 09.02.2022, Tobias Paul (HU Berlin): Modelling interactions of mutation, dormancy and transfer
• 02.02.2022, Quirin Vogel (NYU Shanghai): Infinite loops - The limit of the Feynman representation of the Bose gas (online talk)
• 12.01.2022, Wolfgang König (WIAS Berlin): The free energy of a box version of the interacting Bose gas
• 2021

• 17.12.2021, Daniel Ueltschi (University of Warwick): Loop models and the universal distribution of the loop lengths
• 08.12.2021, Elena Magnanini (WIAS Berlin): Limit theorems for the edge density in exponential random graphs
• 24.11.2021, Sanjoy Kumar Jhawar (WIAS Berlin): Percolation in enhanced random connection models
• 10.11.2021, Anh Duc Vu (WIAS Berlin): Percolation theory and effective conductivity
• 27.10.2021, András Tóbiás (TU Berlin): Virus dynamics in the presence of contact-mediated host dormancy
• 13.10.2021, Tejas Iyer (WIAS Berlin): Degrees of fixed vertices and power law degree distributions in preferential attachment trees with neighbourhood influence
• 29.09.2021, Michiel Renger (WIAS Berlin): Open problem: large deviations of transport maps