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

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A simple lemma bounds s.d.(T)/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 s.d.(X)/E X is small if and only if Ξ/E X is small, where Ξ is the maximal edge-traversal time in the percolation path attaining X.