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

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 $Ξ/\mathbb{E} X$ is small, where $Ξ$ is the maximal edge-traversal time in the percolation path attaining $X$.