May 31st 2023

# Speed Results on the N-particle Branching Random Walk

## Zsófia Talyigás

In this talk Zsófia will discuss her results on the speed of the particle cloud in the N-BRW in the case when the jump distribution has stretched exponential tails.

## Link to Join MS Teams Talk

**This talk will be broadcast at 15:00 BST 31st May on MS Teams only.**

Meeting ID: 393 527 089 878

Passcode: dsm7py

## Abstract

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.

## About Zsófia

Zsófia Talyigás is a postdoc within the University of Vienna working with Emmanuel Schertzer. Previously, she completed her PhD in the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa) at the University of Bath supervised by Sarah Penington and Matthew Roberts. Her research is in probability theory: she is interested in branching processes with selection; that is, in models that are related to the evolution of a population under natural selection. The most interesting questions in this topic concern the speed of evolution, the genealogical structure and the fitness profile of the population.

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