On Random Graph Homomorphisms Into Z

Loading...
Thumbnail Image
Penn collection
Statistics Papers
Degree type
Discipline
Subject
Statistics and Probability
Funder
Grant number
License
Copyright date
Distributor
Related resources
Author
Benjamini, Itai
Häggström, Olle
Mossel, Elchanan
Contributor
Abstract

Given a bipartite connected finite graph G=(V, E) and a vertex v0∈V, we consider a uniform probability measure on the set of graph homomorphisms f: V→Z satisfying f(v0)=0. This measure can be viewed as a G-indexed random walk on Z, generalizing both the usual time-indexed random walk and tree-indexed random walk. Several general inequalities for the G-indexed random walk are derived, including an upper bound on fluctuations implying that the distance d(f(u), f(v)) between f(u) and f(v) is stochastically dominated by the distance to 0 of a simple random walk on Z having run for d(u, v) steps. Various special cases are studied. For instance, when G is an n-level regular tree with all vertices on the last level wired to an additional single vertex, we show that the expected range of the walk is O(log n). This result can also be rephrased as a statement about conditional branching random walk. To prove it, a power-type Pascal triangle is introduced and exploited.

Advisor
Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Publication date
2000-01-01
Journal title
Journal of Combinatorial Theory, Series B
Volume number
Issue number
Publisher
Publisher DOI
Journal Issue
Comments
Recommended citation
Collection