Supported in part by the Marsden fund, administered by RSNZ.
Degenerate random environments
Article first published online: 28 NOV 2012
© 2012 Wiley Periodicals, Inc.
Random Structures & Algorithms
Volume 45, Issue 1, pages 111–137, August 2014
How to Cite
Holmes, M. and Salisbury, T. S. (2014), Degenerate random environments. Random Struct. Alg., 45: 111–137. doi: 10.1002/rsa.20473
- Issue published online: 27 JUN 2014
- Article first published online: 28 NOV 2012
- Manuscript Accepted: 3 AUG 2012
- Manuscript Revised: 26 APR 2012
- Manuscript Received: 13 JUL 2010
- random environment;
- phase transition;
We consider connectivity properties of certain i.i.d. random environments on , where at each location some steps may not be available. Site percolation and oriented percolation are examples of such environments. In these models, one of the quantities most often studied is the (random) set of vertices that can be reached from the origin by following a connected path. More generally, for the models we consider, multiple different types of connectivity are of interest, including: the set of vertices that can be reached from the origin; the set of vertices from which the origin can be reached; the intersection of the two. As with percolation models, many of the models we consider admit, or are expected to admit phase transitions. Among the main results of the paper is a proof of the existence of phase transitions for some two-dimensional models that are non-monotone in their underlying parameter, and an improved bound on the critical value for oriented site percolation on the triangular lattice. The connectivity of the random directed graphs provides a foundation for understanding the asymptotic properties of random walks in these random environments, which we study in a second paper. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 111–137, 2014