Seminar

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