Monte Carlo study of the tip region of branching random walks evolved to large times

TL;DR

This study uses Monte Carlo simulations to analyze the tip region of branching random walks at large times, revealing how particle density distributions behave under different constraints and supporting recent theoretical predictions.

Contribution

It introduces an efficient Monte Carlo method to study the tip region of branching Brownian motion at large times, providing numerical evidence for theoretical conjectures.

Findings

Mean and typical particle numbers grow exponentially with elta x when unconstrained.

Typical particle number is suppressed by a factor ^{-\u03b6elta x^{2/3}} under certain constraints.

Numerical results support recent analytical predictions in the infinite-time limit.

Abstract

We implement a discretization of the one-dimensional branching Brownian motion in the form of a Monte Carlo event generator, designed to efficiently produce ensembles of realizations in which the rightmost lead particle at the final time $T$ is constrained to have a position $X$ larger than some predefined value $X_{\text{min}}$. The latter may be chosen arbitrarily far from the expectation value of $X$, and the evolution time after which observables on the particle density near the lead particle are measured may be as large as $T\sim 10^4$. We then calculate numerically the probability distribution $p_n(\Delta x)$ of the number $n$ of particles in the interval $[X-\Delta x,X]$ as a function of $\Delta x$. When $X_{\text{min}}$ is significantly smaller than the expectation value of the position of the rightmost lead particle, i.e. when $X$ is effectively unconstrained, we check that both the mean and the typical values of $n$ grow exponentially with $\Delta x$, up to a linear prefactor and to finite-$T$ corrections. When $X_{\text{min}}$ is picked far ahead of the latter but within a region extending over a size of order $\sqrt{T}$ to its right, the mean value of the particle number still grows exponentially with $\Delta x$, but its typical value is lower by a multiplicative factor consistent with $e^{-\zeta\Delta x^{2/3}}$, where $\zeta$ is a number of order unity. These numerical results bring strong support to recent analytical calculations and conjectures in the infinite-time limit.