On the continuity of the time constant of first-passage percolation

Abstract

LetUbe the distribution function of the non-negative passage time of an individual edge of the square lattice, and leta0nbe the minimal passage time from (0, 0) to (n, 0). The processa0n/nconverges in probability asn→ ∞to a finite constantμ(U) called the time constant. It is proven thatμ(Uk)→ μ(U) wheneverUkconverges weakly toU.