How large should be the redundant numbers of copy to make a rare event probable

TL;DR

This paper develops criteria to determine how many copies of a searcher are needed to make rare events likely, using splitting probabilities and geometric properties, with applications to cell biology.

Contribution

It introduces explicit criteria for estimating the number of copies needed for rare events, based on splitting probabilities and domain geometry, with analytical and simulation results.

Findings

Explicit computation of splitting probabilities in domains with killing regions

Optimal trajectories avoid the killing region for large number of copies

Applications demonstrated in cell biology contexts

Abstract

The redundancy principle provides the framework to study how rare events are made possible with probability 1 in accelerated time, by making many copies of similar random searchers. But what is $n$ large? To estimate large $n$ with respect to the geometrical properties of a domain and the dynamics, we present here a criteria based on splitting probabilities between a small fraction of the exploration space associated to an activation process and other absorbing regions where trajectories can be terminated. We obtain explicit computations especially when there is a killing region located inside the domain that we compare with stochastic simulations. We present also examples of extreme trajectories with killing in dimension 2. For a large $n$, the optimal trajectories avoid penetrating inside the killing region. Finally we discuss some applications to cell biology.