How to generate the tip of branching random walks evolved to large times

TL;DR

This paper introduces an efficient algorithm for simulating the extremal region of one-dimensional branching random walks at large times, focusing computational effort on the tip where relevant phenomena occur.

Contribution

The authors develop a simple, linear-complexity algorithm to simulate the tip of branching random walks, enabling analysis at large times where traditional methods are computationally infeasible.

Findings

Algorithm efficiently simulates the tip region at large times

Allows arbitrary positioning of the rightmost particle beyond typical range

Demonstrates effectiveness by evaluating a previously intractable observable

Abstract

In a branching process, the number of particles increases exponentially with time, which makes numerical simulations for large times difficult. In many applications, however, only the region close to the extremal particles is relevant (the "tip"). We present a simple algorithm which allows to simulate a branching random walk in one dimension, keeping only the particles that arrive within some distance of the rightmost particle at a predefined time $T$. The complexity of the algorithm grows linearly with $T$. We can furthermore choose to require that the realizations have their rightmost particle arbitrarily far on the right from its typical position. We illustrate our algorithm by evaluating an observable for which no other practical method is known.