Weak Concentration for First Passage Percolation Times on Graphs and General Increasing Set-valued Processes

TL;DR

This paper establishes a simple bound on the relative standard deviation of first passage percolation times in graphs, linking it to the maximum edge traversal time, with implications for understanding variability in such stochastic processes.

Contribution

It introduces a lemma bounding the normalized standard deviation of hitting times in Markov chains and applies it to analyze variability in first passage percolation on graphs with exponential edge weights.

Findings

Normalized standard deviation is small when maximum edge traversal time is small.

The bound applies to arbitrary vertex pairs in finite graphs.

Provides insight into the variability of percolation times based on edge traversal times.

Abstract

A simple lemma bounds $\mathrm{s.d.}(T)/\mathbb{E} T$ for hitting times $T$ in Markov chains with a certain strong monotonicity property. We show how this lemma may be applied to several increasing set-valued processes. Our main result concerns a model of first passage percolation on a finite graph, where the traversal times of edges are independent Exponentials with arbitrary rates. Consider the percolation time $X$ between two arbitrary vertices. We prove that $\mathrm{s.d.}(X)/\mathbb{E} X$ is small if and only if $\Xi/\mathbb{E} X$ is small, where $\Xi$ is the maximal edge-traversal time in the percolation path attaining $X$.