Hitting and Trapping Times on Branched Structures

TL;DR

This paper derives a formula for calculating hitting times in branched structures, specifically combs, and explores their implications for reaction-diffusion processes.

Contribution

It provides a closed-form formula for hitting times in generic branched structures and applies it to combs to analyze mean-first passage times.

Findings

Derived a closed-form formula for hitting times between nodes.

Calculated mean-first passage times for comb structures.

Discussed applications in reaction-diffusion systems.

Abstract

In this work we consider a simple random walk embedded in a generic branched structure and we find a close-form formula to calculate the hitting time $H\left(i,f\right)$ between two arbitrary nodes $i$ and $j$. We then use this formula to obtain the set of hitting times $\left\{ H\left(i,f\right)\right\} $ for combs and their expectation values, namely the mean-first passage time $\left( \mbox{MFPT}_{f} \right)$, where the average is performed over the initial node while the final node $f$ is given, and the global mean-first passage time $\left( \mbox{GMFPT} \right)$, where the average is performed over both the initial and the final node. Finally, we discuss applications in the context of reaction-diffusion problems.