Large Deviations of First Passage Times of Branching Random Walks in $\mathbb{R}^d$: Asymptotics and Algorithms

TL;DR

This paper studies the probabilities of rare events in the first passage times of branching random walks in multi-dimensional space, providing asymptotic analysis and an efficient algorithm for computing these probabilities.

Contribution

It introduces a novel asymptotic analysis of large deviations for FPT in branching random walks and develops a polynomial-time algorithm for their computation.

Findings

Derived asymptotic formulas for large deviation probabilities.

Developed an asymptotically optimal polynomial-time algorithm.

Validated the algorithm's accuracy through numerical experiments.

Abstract

We investigate the large deviation probabilities of first passage times (FPT) of discrete-time supercritical non-lattice branching random walks (BRWs) in $\mathbb{R}^d$ where $d\geq 1$. The FPT refers to the first time the BRW enters a ball of radius one with a distance $x$ from the origin, conditioned upon the process's survival. Furthermore, we apply the spine decomposition technique to construct an asymptotically optimal polynomial-time algorithm for computing the lower large deviation probabilities of the FPT. The accuracy of our algorithm is also verified numerically. Our analysis not only provides a deeper theoretical understanding of these stochastic processes but also offers new insights into the microstructural features that are key to characterizing the strength of polymers.