David
Dereudre
University of Lille
Interacting Random Systems  📅
Institute
Head
Alexander Zass
Usual time
Wednesdays, 11:30 (CET)
Usual venue
Weierstrass Lecture Room (WIAS-405-406)
Number of talks
26
May 2026
Julian
Kern
FU Berlin
Quan
Shi
AMSS Chinese Academy of Sciences
Two phase transitions for catalytic branching Markov chains
Abstract.
Consider a continuous-time branching Markov chain $(Z_t,t\ge 0)$ on a locally finite graph $G$ rooted at $\mathbf{o}$. Each particle moves according to an irreducible Markov process $\xi$ and branches at a rate that depends on their location: the branching rate is $\lambda_{rt}\ge 0$ at the root and $\lambda\ge 0$ elsewhere. The offspring distribution is supercritical with mean $m>1$, has no extinction and finite second moment. We characterize the recurrence/transience phase transition for this catalytic branching Markov chain. Furthermore, under suitable assumptions we prove a second phase transition concerning the asymptotic behaviour of the relative empirical density, $(Z_t(G))^{-1} Z_t$, where $Z_t$ is the empirical measure of the particles and $Z_t(G)$ is the total population size. If $(m-1)(\lambda_0-\lambda)> \gamma_{esc}$, where $\gamma_{esc}$ is the escape probability that $\xi$ never returns to the root, then $(Z_t(G))^{-1} Z_t$ converges almost surely to a deterministic probability measure. If $(m-1)(\lambda_0-\lambda)\in (0, \gamma_{esc}]$, then $(Z_t(G))^{-1} Z_t$ converges almost surely to zero. When the graph is the integer lattice $G=\mathbb{Z}^d$ and $\xi$ is the simple random walk, our results confirm several conjectures of Mailler and Schapira [Ann. Appl. Probab. 2026], which studied this model via a different approach. Based on a joint work in progress with Xinxin Chen (Bejing Normal Unversity), Nina Gantert (Technical University of Munich) and Haojie Hou (Beijing Institute of Technology).
April 2026
Elena
Magnanini
WIAS Berlin
Non-Standard Fluctuations of the edge density in the edge-triangle Model
Abstract.
Exponential Random Graph Models extend the classical (dense) Erdős–Rényi random graph by incorporating higher-order structural features through a Hamiltonian formalism inspired by statistical mechanics. Among these, the edge-triangle model is a basic but non trivial example, where the competition between edge and triangle terms leads to a rich phase structure. In this talk we will give an overview of known limit theorems and concentration results for subgraph densities, with particular emphasis on an open problem: a non-standard CLT for the edge density. The conjecture will be supported by a mean-field analysis, which allows for explicit computations and suggests a different fluctuation scaling, while also providing some insight into related open questions. This talk is based on joint works with A. Bianchi, F. Collet, and G. Passuello.
Willem
van Zuijlen
WIAS Berlin
Anderson Hamiltonians with correlated Gaussian potentials
Abstract.
Anderson Hamiltonians, which are random Schrödinger operators, model the evolution of electrons or quantum states in a disordered system. Philip Warren Anderson (1958) showed that if there is too much disorder in the system, instead of seeing a diffusive behaviour for the electron, they get trapped. A similar localisation effect takes place in the parabolic Anderson model, which is the parabolic problem or Cauchy problem related to the Anderson Hamiltonian. The spectral properties of the Hamiltonian determine this localisation behaviour. Many such models have been studied, but often with a potential field that is i.i.d. We study these Hamiltonians with a correlated Gaussian potential and consider its eigenvalue order statistics. <br>This is joint work with Giuseppe Cannizzaro and Cyril Labbé.
March 2026
Anh Duc
Vu
WIAS Berlin
Generalised graphical representations
Abstract.
Numerous dynamic spatial models feature interactions between particles that are based on waiting times; the most prominent examples being first passage percolation, where a fluid spreads from a source, and the contact process, where an infection spreads on a graph with the possibility of recovery. However, waiting times are unnatural for certain models. One can argue that people have a daily routine and meet each other on a relatively fixed schedule, rather than a random waiting time after having met someone else. With the help of graphical representations, we develop a framework that moves away from waiting times towards so called contact times. As an application, we present results on a first contact percolation model, where people follow a random daily meeting routine, as well as contact processes with periodic recoveries.
Alexander
Bobenko
TU Berlin
February 2026
Marta
Dai Pra
HU Berlin
Multi-type logistic branching processes with selection: frequency process and genealogy for large carrying capacities
Abstract.
We present a model for growth in a multi-species population. We consider two types evolving as a logistic branching process with mutation, where one of the types has a selective advantage. We first study the frequency of the disadvantageous type and show that, once the population approaches the carrying capacity, its evolution converges to a Gillespie-Wright-Fisher diffusion process. We then study the dynamics backward in time: we fix a time horizon at which the population is at carrying capacity and we study the ancestral relations of a sample of individuals. We prove that, provided that the advantageous and disadvantageous branching measures are ordered, this ancestral line process converges to the moment dual of the limiting diffusion. This talk is based on joint work with Julian Kern.
Marta
Dai Pra
HU Berlin
Elias
Zimmermann
Universität Leipzig
Quantitative mixing properties for Gibbs fields on bounded degree graphs
Abstract.
Mixing properties are an important object of study within the theory of Gibbs measures. In this talk I will focus on two quantitative mixing properties: exponential $\phi$-mixing (also called weak mixing) and exponential $\psi$-mixing (also called ratio weak mixing). These mixing conditions are well studied for Gibbs fields on square lattices. For instance, the unique Gibbs state of the Ising model above the critical temperature exhibits both types of mixing. In this talk we shall consider random fields on arbitrary graphs with bounded degree. While exponential $\phi$-mixing is known to hold whenever Dobrushin's uniqueness condition is satisfied, exponential $\psi$-mixing has been studied very little in this general setting. The main result I'm going to present shows that for Markovian random fields $\phi$-mixing with sufficiently high exponential decay rate implies $\psi$-mixing with slower, but still exponential decay rate. This generalizes a result of Alexander for square lattices and allows to formulate a general criterion for exponential $\psi$-mixing of Gibbs measures in terms of the corresponding Dobrushin constant. I will illustrate how this criterion applies to Ising and Potts models on regular trees.
January 2026
Yannic
Steenbeck
TU Braunschweig
Reversible birth-and-death dynamics in continuum: Free-energy dissipation and attractor properties
Abstract.
The theory of Gibbs point processes, describing interacting systems of particles in continuous space, is by now a classical topic in probability and statistical mechanics and for various questions, somewhat satisfactory results have been obtained. On the lattice, these results on the equilibrium behavior of large systems of interacting particles are complemented by a detailed study of associated dynamics. In contrast, very little is known about dynamical aspects in continuum, in particular out-of-equilibrium dynamics and convergence to equilibrium. In this talk, I will review basic terms of the probabilistic theory of statistical mechanics and present our first investigation of the long-time behavior of birth-and-death dynamics in continuum space. Our main results there confirm the physical intuition that the free energy is non-increasing in time and moreover governs the long-term behavior of large systems of interacting particles.
Marcel
Ortgiese
University of Bath
Dynamic accessibility percolation
Abstract.
Accessibility percolation is a simple model in evolutionary biology describing how a population driven by the evolutionary forces of selection and mutation explores a fitness landscape. Mathematically, the fitness landscape is modeled by attaching random weights to the vertices of a graph. Then, accessible percolation asks whether there are paths of increasing fitness of a certain length. I will review some of the progress in this area and then consider the question of what happens if the fitness landscape changes over time. In particular, I will focus on the case when the underlying graph is a regular tree. Depending on the ratio of depth to width of the tree, we will see different scaling regimes for the time it takes to see an increasing path. Some of the proofs rely on adapting techniques from the area of noise sensitivity for Boolean functions.
December 2025
Anh Duc
Vu
WIAS Berlin
November 2025
Anh Duc
Vu
WIAS Berlin
François
Baccelli
ENS and INRIA Paris
On Dynamical Poisson–Voronoi Tessellations
Abstract.
Consider a dynamical network model featuring mobile stations on the Euclidean plane. The initial locations of the stations are given by a homogeneous Poisson point process. The stations are all moving at a constant speed and in a random direction. Consider fixed observers located in the Euclidean plane, which are served by the mobile stations. Each observer stays connected to the nearest station at any given point of time. Since the stations are moving, an observer disconnects and connects with different stations over time, by always selecting whichever station is the closest. This gives rise to a dynamical version of the Poisson–Voronoi tessellation. The focus of this talk is on the sequence of "handover" events of a typical observer, which are the epochs when its association changes. This defines a point process on the time axis, the "handover point process". We show that this point process is stationary and we determine its main properties, in particular its intensity and the joint distribution of its inter-event times. We also analyze the handover Palm distributions of several variables of practical interest. This includes the distance to the closest mobile station and the point process of all other mobile stations at handover epochs. The analysis is conducted both in the single-speed and in the multi-speed scenarios. It leads to the identification of the three dimensional state variables that "Markovize" the association dynamics. The analysis is based on a specific system of non-compact particles. The motivations are in the modeling of low or medium orbit satellite wireless communication networks. The model studied here is a planar "caricature" of this problem, which is initially defined on the sphere. Joint work with Sanjoy Kumar Jhawar.
Sophia-Marie
Mellis
Universität Bielefeld
Genealogies in multitype populations: branching processes and structured coalescents
Abstract.
This talk focuses on the interplay between type and ancestry in two different multitype population models. In the first part, we briefly discuss the long-term behavior of critical multitype branching processes conditioned on survival, both with respect to the forward and the ancestral processes. Despite substantial differences in forward-time behavior and required techniques, their ancestral processes retain key structural similarities to the supercritical case. The main part of the talk then focuses on structured populations divided into $d$ colonies, where individuals migrate at rates proportional to a global scaling parameter $K$. We sample $N(K)$ individuals evenly across colonies and trace their ancestral lineages backward in time. Within each colony, coalescence occurs at a constant rate as in the Kingman coalescent. We encode the system's state as a $d$-dimensional vector of empirical measures, recording both current lineage locations and the colonies of their sampled descendants. Our focus is on how the sample size affects the asymptotic behavior of this process as $K \to \infty$ (representing fast migration), distinguishing two regimes: the critical-sampling regime ($N(K) \sim K$) and the large-sampling regime ($N(K) \gg K$). After suitable time-space rescaling, we prove convergence to $d$-dimensional coagulation equations in both sampling regimes. In the critical regime, the solution admits a representation via a multitype birth-death process; in the large-sample regime, via the entrance law of a multitype Feller diffusion.
October 2025
Hanna
Stange
Universität Münster
Non-local transport distance for point processes
Abstract.
We introduce a non-local transport distance on the space of point processes and analyse the induced geometry. We show — among other things — that the Ornstein–Uhlenbeck semigroup is the gradient flow of the specific relative entropy, functional inequalities like a Talagrand inequality, and exponential convergence of the Ornstein–Uhlenbeck flow to the Poisson point process. Towards the end, we discuss ongoing work on the extension of the framework adapted to Papangelou point processes. Based on joint work with Martin Huesmann and Matthias Erbar.
Helia
Shafigh
WIAS Berlin
Responsive dormancy of a spatial population among a moving trap
Abstract.
In this talk, we study a spatial model for dormancy in a random environment via a two-type branching random walk in continuous-time, where individuals switch between dormant and active states depending on the current state of a fluctuating environment (responsive switching). The branching mechanism is governed by the same random environment, which is here taken to be a simple symmetric random walk. We will interpret the presence of this random walk as a trap which attempts to kill the individuals whenever it meets them. The responsive switching between the active and dormant state is defined so that active individuals become dormant only when a trap is present at their location and remain active otherwise. Conversely, dormant individuals can only wake up once the environment becomes trap-free again. We quantify the influence of dormancy on population survival by analyzing the long-time asymptotics of the expected population size. The starting point for our mathematical considerations and proofs is the Parabolic Anderson Model via the Feynman-Kac formula. In particular, we investigate the quantitative role of dormancy by extending the Parabolic Anderson Model to a two-type random walk framework. Joint work with Leo Tyrpak.
Tejas
Iyer
WIAS Berlin
When a branching process grows like its mean
Abstract.
A classical question in the study of discrete-time branching processes is determining when the normalized population size admits a non-degenerate limit. In the Bienaymé–Galton–Watson setting, where each individual produces a random number $X$ of offspring and reproduction is identically distributed across generations, the well-known Kesten–Stigum theorem states that such a limit exists if and only if $\mathbb{E}[X \log X] < \infty$. In this talk, we extend this result to branching processes in varying environments, where the offspring distribution may change from generation to generation, and discuss some ongoing work.
September 2025
CĂ©sar
Zarco Romero
WIAS Berlin
Energy increment on abstract measure spaces
Abstract.
A relevant concept in the proof of the weak regularity lemma (as in the original version) is that of the energy of a sigma algebra. We shall first introduce the notion of a semiring together with some examples. We then revisit the classical argument in the proof of the weak regularity lemma in its most abstract sense and from a probabilistic point of view. We shall finally mention a noncommutative martingale convexity inequality from Ricard and Xu (2014) to argue a variant of it on $L^p$ spaces. Applications concern, for instance, large dense graphs, arithmetic progressions, and hypergraphs.
July 2025
Johannes
Bäumler
UCLA
The truncation problem for long-range percolation
Abstract.
In long-range percolation on the integer lattice, for each pair of points \{ x, y \}, there is an open edge between these points with probability depending on the Euclidean distance between the points, independent of all other edges. When are the long edges necessary for the existence of an infinite cluster? The truncation problem asks whether one can remove all long enough edges while still retaining an infinite open cluster. We discuss this question in the non-summable regime in dimensions d≥3. Here we show that the truncation problem has an affirmative answer.
Jonas
Köppl
WIAS Berlin
Two edges suffice: the planar lattice two-neighbor graph percolates
Abstract.
The $k$-neighbor graph is a directed percolation model on the hypercubic lattice $\mathbb{Z}^d$ in which each vertex independently picks exactly $k$ of its $2d$ nearest neighbors at random, and we open directed edges towards those. We prove that the $2$-neighbor graph percolates on $\mathbb{Z}^2$, i.e., that the origin is connected to infinity with positive probability. The proof rests on duality, an exploration algorithm, a comparison to i.i.d. bond percolation under constraints as well as enhancement arguments. As a byproduct, we show that i.i.d. bond percolation with forbidden local patterns has a strictly larger percolation threshold than $1/2$. Additionally, our main result provides further evidence that, in low dimensions, less variability is beneficial for percolation.
June 2025
D.R. Michiel
Renger
TU MĂĽnchen
Collisions in the exclusion process II
Abstract.
(Ongoing work together with Mario Ayala (TUM)) The simple symmetric exclusion process (SSEP) and a system of independent random walkers both converge to the same diffusion equation. To understand the difference, we count the number of times a SSEP particle would have jumped if the particles were independent but does not jump because the target site is occupied. This number can also be interpreted as the number of times two SSEP particles 'collide'. We study the hydrodynamic limit of various related variables in different scaling regimes (by playing with the lattice spacing and particle numbers).
Michiel
Renger
TU MĂĽnchen
Elena
Pulvirenti
TU Delft
Metastability for the Curie–Weiss–Potts model with unbounded random interactions
Abstract.
I will first introduce the model, i.e. a disordered version of the mean-field q-spin Potts model, where the interaction coefficients between spins are general independent random variables. These random variables are chosen to have fixed mean (for simplicity taken to be 1), well defined log-moment generating function and finite variance. I will then present quantitative estimates of metastability in the regime of large number of particles at fixed temperature, when the system evolves according to a Glauber dynamics. This means that the spin configuration is viewed as a Markov chain where spins flip according to Metropolis rates at a fixed inverse temperature. Our main result identifies conditions ensuring that, with high probability, the system behaves like the corresponding system where the random couplings are replaced by their averages. More precisely, we prove that the metastability of the former system is implied with high probability by the metastability of the latter. Moreover, we consider relevant metastable hitting times of the two systems and find the asymptotic tail behaviour and the moments of their ratio. Our proofs use the potential-theoretic approach to metastability in combination with concentration inequalities. Based on a joint work in collaboration with Johan Dubbeldam, Vicente Lenz and Martin Slowik.
May 2025
Adrian
Röllin
University of Leeds
Centered Subgraph Counts in Dense Random Graphs