First-passage percolation on the square lattice. I
- 1 March 1977
- journal article
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 9 (1) , 38-54
- https://doi.org/10.2307/1425815
Abstract
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 linex = n; (ii) a proof that the time constant is zero when the atom at zero of the underlying distribution exceedsC, 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.Keywords
This publication has 6 references indexed in Scilit:
- Remarks on renewal theory for percolation processesJournal of Applied Probability, 1976
- The Ergodic Theory of Subadditive Stochastic ProcessesJournal of the Royal Statistical Society Series B: Statistical Methodology, 1968
- First-Passage PercolationJournal of the Royal Statistical Society Series B: Statistical Methodology, 1966
- Exact Critical Percolation Probabilities for Site and Bond Problems in Two DimensionsJournal of Mathematical Physics, 1964
- A lower bound for the critical probability in a certain percolation processMathematical Proceedings of the Cambridge Philosophical Society, 1960
- Excluded-Volume Problem and the Ising Model of FerromagnetismPhysical Review B, 1959