On the non-convexity of the time constant in first-passage percolation

Abstract

We give a counterexample to a conjecture of Hammersley and Welsh (1965) about the convexity of the time constant in first-passage percolation, as a functional on the space of distribution functions. The present counterexample only works for first-passage percolation on ℤd for d large. © 1996 Applied Probability Trust.