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Degenerate random environments
- Source :
- Random Structures & Algorithms. 45:111-137
- Publication Year :
- 2012
- Publisher :
- Wiley, 2012.
-
Abstract
- We consider connectivity properties of certain i.i.d. random environments on i¾?d, 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
- Subjects :
- Random graph
Discrete mathematics
Percolation critical exponents
Random field
Applied Mathematics
General Mathematics
Probability (math.PR)
010102 general mathematics
Random function
Random element
01 natural sciences
Computer Graphics and Computer-Aided Design
Directed percolation
Combinatorics
010104 statistics & probability
FOS: Mathematics
Random compact set
Continuum percolation theory
0101 mathematics
Mathematics - Probability
Software
Mathematics
Subjects
Details
- ISSN :
- 10429832
- Volume :
- 45
- Database :
- OpenAIRE
- Journal :
- Random Structures & Algorithms
- Accession number :
- edsair.doi.dedup.....06146b0d4554bc398c1449d87f1a57b1