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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 thislemma may be applied to several increasing set-valued processes. Our mainresult concerns a model of first passage percolation on a finite graph, wherethe 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 $\Xi/\mathbb{E} X$ issmall, where $\Xi$ is the maximal edge-traversal time in the percolation pathattaining $X$.

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