Workshop: Branching and Interacting Particle Systems
Johannes Gutenberg-Universität Mainz
27th February - 2nd March 2023
Branching random walks are a natural model for various systems of population dynamics and genetics. A key question is understanding the influence on spatially-dependent branching mechanisms of interactions caused, for example, by selection, competition, random environments. It is also important to have mathematical tools to model one or more coexisting spatial populations competing for resources. This workshop aims at bringing together younger colleagues and experts in the field in order to provide a stimulating discussion environment and share the most recent developments on these challenging research topics.
This workshop is part of the DFG Priority Programme SPP 2265: Random Geometric Systems funded by the Deutsche Forschungsgemeinschaft.
Viktor Bezborodov (University of Göttingen)
Elisabetta Candellero (Università degli Studi Roma Tre)
Jiří Černý (University of Basel)
Piotr Dyszewski (Wrocław University)
Alison Etheridge (University of Oxford)
Matthias Hammer (TU Berlin)
Pascal Maillard (Université Toulouse III)
Bastien Mallein (Université Sorbonne Paris Nord)
Pascal Oswald (University of Basel/JGU Mainz)
Sarah Penington (University of Bath)
Matthew Roberts (University of Bath)
Emmanuel Schertzer (University of Vienna)
Alexandre Stauffer (University of Bath)
Zsófia Talyigás (University of Vienna)
Terence Tsui (University of Oxford)
|11:00-11:30||Coffee break||Coffee break||Coffee break|
|13:30-14:15||Registration & Coffee|
|16:00-16:30||Coffee break||Coffee break||Coffee break|
Viktor Bezborodov: Maximal displacement of multidimensional radially symmetric branching random walk.
We consider the maximal displacement of a multidimensional radially symmetric supercritical branching random walk. We show under certain additional assumptions that the maximal displacement can be represented with three terms: a linear term, a logarithmic correction, and a bounded in probability term. A similar result was obtained by Mallein for the branching Brownian motion. We also discuss a general version of the ballot theorem with moving boundary. The latter is an extension of a crossing theorem for random walks by Pemantle and Peres. We also briefly discuss a non-radially symmetric case. The talk is based on joint work with Nina Gantert.
Elisabetta Candellero: On the boundary at infinity for branching random walk.
We introduce an original connection between branching random walk on a graph and the Martin boundary for the underlying random walk. More precisely, we prove that when the graph is transient, supercritical branching random walk converges almost surely (under rescaling) to a random measure on the Martin boundary of the graph. Based on a joint work with T.Hutchcroft (Caltech).
Jiří Černý: Survival and complete convergence for a branching annihilating random walk.
We study a branching-annihilating random walk in which particles evolve on the lattice in discrete generations. Each particle produces a Poissonian number of offspring which independently move to a uniformly chosen site within a fixed distance from their parent's position. Whenever a site is occupied by at least two particles, all the particles at that site are annihilated. This can be thought of as a very strong form of local competition and implies that the system is not monotone.
For certain ranges of the parameters of the model, we show that the system dies out almost surely, or, on the other hand, survives with positive probability. In an even more restricted parameter range, we strengthen the survival results to complete convergence with a non-trivial invariant measure. A central tool in the proof is comparison with oriented percolation on a coarse-grained level, using carefully tuned density profiles which expand in time and are reminiscent of discrete travelling wave solutions.
Based on joint work in progress with Matthias Birkner (Mainz), Alice Callegaro (TU Munich/Mainz), Nina Gantert (TU Munich) and Pascal Oswald (Basel/ Mainz).
Piotr Dyszewski: Branching random walks and stretched exponential tails.
We will investigate a branching random walk where the displacements are independent from the branching mechanism and have a stretched exponential distribution and describe the positions of the particles in the vicinity of the rightmost particle in terms of point process convergence. As a consequence we give a new limit theorem for the position of the rightmost particle. Our methods rely on precise large deviation estimates for sums of i.i.d. random variables with stretched exponential distribution outside the so-called one big jump domain. The talk is based on a joint work with Nina Gantert (TUM) and Thomas Höfelsauer (TUM).
Matthias Hammer: Some Remarks on Ergodicity and Invariant Measure for Subcritical Branching Particle Systems with Immigration.
We consider systems of finitely many particles moving on independent diffusion paths. Each particle branches at a position-dependent rate, producing a random number of offspring –randomly scattered around the parent’s death position– according to a position-dependent law. In addition, new particles immigrate into the system at a constant rate.
We specify a suitable notion of subcritical reproduction under which this branching particle system is ergodic with finite invariant measure on the configuration space. We then discuss some properties of this measure and of the associated intensity measure on the single-particle space. In particular, we give conditions ensuring the existence of continuous and bounded Lebesgue-densities. Interest in these properties is in part motivated by statistical applications.
Based on joint work with Reinhard Hoepfner (Mainz).
Pascal Maillard: (TBD)
Bastien Mallein: Atypical invasion of the reducible multitype branching Brownian motion.
A branching Brownian motion can be described as a particle system on the real line in which particles move independently as Brownian motion, while splitting at rate 1 into two daughter particles. We take interest in a multitype version of this process, in which the diffusion constant of the displacement and the branching rate are both influenced by the type.
When the process is irreducible (e.g. when particles of type 1 can give birth to particles of type 2, but not reciprocally), an anomalous spreading phenomenon may occur, in which the speed of the multitype process is strictly larger than the speed of each "pure" process. We take interest in the asymptotic behavior of extremal particles in this setting, showing the convergence in law of the extremal process centered around the median of the maximal displacement.
Pascal Oswald: On the tightness of the maximum of branching Brownian motion in a spatially random environment.
We consider one-dimensional branching Brownian motion in a spatially random branching environment (BBMRE). It is well known that the distribution function of the maximal displacement solves a non-linear PDE known as the (randomised) Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation, with suitable Heaviside initial condition. In the homogeneous setting solutions to the F-KPP equation converge towards a traveling wave solution. A direct consequence of this is that the distribution of the maximally displaced particle of (homogeneous) branching Brownian motion recentered at its median is tight. In contrast, in the presence of a spatially random environment the situation becomes more intricate: solutions to the randomised F-KPP equation do not admit traveling waves and the transition fronts of these solutions are in general not bounded uniformly in time. Using a mixture of probabilistic and analytic arguments, we show that the distribution of the maximal particle of BBMRE recentered at its median is tight for almost every realisation of the environment.
Based on joint work with Jiří Černý (University of Basel) and Alexander Drewitz (Universität zu Köln).
Sarah Penington: Branching random walk with non-local competition.
We study a particle system in which particles reproduce, move randomly in space, and compete with each other. We prove global survival and determine the asymptotic spread of the population, when the norm of the competition kernel is sufficiently small. In contrast to most previous work, we allow the competition kernel to have an arbitrary, or even infinite range, whence the term 'non-local competition'. This makes the particle system non-monotone and of infinite-range dependence, meaning that the usual comparison arguments break down and have to be replaced by a more hands-on approach.
Based on joint work with Pascal Maillard.
Matthew Roberts: A Gaussian particle distribution for branching Brownian motion with an inhomogeneous branching rate.
Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to one-dimensional Brownian motion with drift, each particle may either split into two or die, and the difference between the birth and death rates is a linear function of the position of the particle. We show that, under certain assumptions, after a sufficiently long time, the empirical distribution of the positions of the particles is approximately Gaussian. This provides mathematically rigorous justification for results in the biology literature indicating that the distribution of the fitness levels of individuals in a population over time evolves like a Gaussian travelling wave.
Emmanuel Schertzer: (TBD)
Alexandre Stauffer: Non-equilibrium multi-scale analysis and coexistence in competing first-passage percolation
We consider a natural random growth process with competition on ℤd called first-passage percolation in a hostile environment, that consists of two first-passage percolation processes FPP1 and FPPλ that compete for the occupancy of sites. Initially FPP1 occupies the origin and spreads through the edges of ℤd at rate 1, while FPPλ is initialised at sites called seeds that are distributed according to a product of Bernoulli measures of parameter p. A seed remains dormant until FPP1 or FPPλ attempts to occupy it, after which it spreads through the edges of ℤd at rate λ. We will discuss the results known for this model and present a recent proof that the two types can coexist (concurrently produce an infinite cluster) on ℤd. We remark that, though counterintuitive, the above model is not monotone in the sense that adding a seed of FPPλ could favor FPP1. A central contribution of our work is the development of a novel multi-scale analysis to analyze this model, which we call a multi-scale analysis with non-equilibrium feedback and which we believe could help analyze other models with non-equilibrium dynamics and lack of monotonicity.
Zsófia Talyigás: Asymptotic speed of the N-particle branching random walk with stretched exponential jump distribution.
The N-particle branching random walk (N-BRW) is a discrete time branching particle system with selection. In the N-BRW, N particles have locations on the real line at all times. At each time step, each of the N particles has two offspring. Each of the 2N offspring particles performs a jump from the location of its parent, independently from the other jumps according to some fixed jump distribution. Then among the 2N offspring particles, only the N rightmost particles survive to form the next generation.
In this talk I will discuss our results on the speed of the particle cloud in the N-BRW in the case when the jump distribution has stretched exponential tails. The "light-tailed" case (when the jump distribution has some exponential moments) was studied by Bérard and Gouéré, and the polynomial-tailed case was investigated in the work of Bérard and Maillard. As these two cases are significantly different from each other, we aimed to fill the gap by studying the intermediate stretched exponential case. We describe the first order and give lower and upper bounds on the second order of the asymptotic speed as the number of particles N goes to infinity.
This is joint work with Sarah Penington.
Terence Tsui: Modelling cell-free DNA fragmentomics with a class of fragmentation process.
In this talk, I will first introduce what cell-free DNAs are and explain why their fragment length profile is significant to oncologists. I will then define a class of Markov processes, namely the FRagmentation with IMmigration and Exit (FRIME) processes, and demonstrate how they can effectively approximate cfDNA fragment profiles with Monte Carlo simulations. Finally, I will end this talk with theoretical results on the stationarity measures of FRIME processes, and show how they justify our numerical simulation results. This talk will be of interest to computational biologists, statisticians and probabilists working on branching and fragmenting particle systems.
Institut für Mathematik
The workshop will take place in Room 04-432, located on the 4th floor.
The next room in the same corridor have been booked for discussion.
A map of the campus and directions can be found here.
Matthias Birkner (JGU Mainz), birkner(at)mathematik.uni-mainz.de
Alice Callegaro (JGU Mainz/TU Munich), alice.callegaro(at)tum.de
Nina Gantert (TU Munich), nina.gantert(at)tum.de