R. T. Smythe

First-passage percolation on the square lattice

We consider several problems in the theory of first-passage percolation on the two-dimensional integer lattice. Our results include: (i) a mean ergodic theorem for the first-passage time from (0,0) to the line x = n ; (ii) a proof that the time constant is zero when the atom at zero of the underlying distribution exceeds C , the critical percolation probability for the square lattice; (iii) a proof of the a.s. existence of routes for the unrestricted first-passage processes; (iv) a.s. and mean ergodic theorems for a class of reach processes; (v) continuity results for the time constant as a functional of the underlying distribution.

Lower bounds for the critical probability in percolation models with oriented bonds

In completely or partially oriented percolation models, a conceptually simple method, using barriers to enclose all open paths from the origin, improves the best previous lower bounds for the critical percolation probabilities.