A $N$-branching random walk with random selection

TL;DR

This paper analyzes a solvable model of branching random walk with random selection, focusing on population evolution, asymptotic speed, and genealogical structure in a system with N individuals and probabilistic offspring placement.

Contribution

It introduces an exactly solvable model of N-branching random walk with random selection, providing explicit calculations of asymptotic speed and genealogical behavior.

Findings

Explicit formula for asymptotic speed of the population

Characterization of genealogical structure over time

Insights into the effects of random selection on population dynamics

Abstract

We consider an exactly solvable model of branching random walk with random selection, which describes the evolution of a population with $N$ individuals on the real line. At each time step, every individual reproduces independently, and its offspring are positioned around its current locations. Among all children, $N$ individuals are sampled at random without replacement to form the next generation, such that an individual at position $x$ is chosen with probability proportional to $\mathrm{e}^{\beta x}$. We compute the asymptotic speed and the genealogical behavior of the system.