Graph homomorphisms through random walks

Random Walks and Graph Homomorphisms

Graph Homomorphisms and Phase Transitions Bell Laboratories 2c-379 Lucent Technologies 700 Mountain Ave

We model physical systems with \hard constraints" by the space Hom(G;H) of homomorphisms from a locally nite graph G to a xed nite constraint graph H. For any assignment of positive real activities to the nodes of H, there is at least one Gibbs measure on Hom(G;H); when G is in nite, there may be more than one. When G is a regular tree, the simple, invariant Gibbs measures on Hom(G;H) correspon...

Random Walks on Hypergraphs with Applications to Disease-Gene Prioritization

Typically, gene interaction networks are expressed as graphs, storing pairwise interactions between genes. Because of the vast amount of literature on statistical graph inference, this is a useful representation in practice. However, such a pairwise representation ignores more complex features of gene interactions, such as gene regulation and assembly. In this thesis, we propose a hypergraph mo...

Electric Networks and Commute Time

The equivalence of random walks on weighted graphs with reversible Markov chains has long been known. Another such correspondence exists between electric networks and these random walks. Results in the language of random walks have analogues in the language of electric networks and visa versa. We outline this correspondence and describe how to translate several of the major terms of electric ne...

Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks

Fuchsian groups (acting as isometries of the hyperbolic plane) occur naturally in geometry, combinatorial group theory, and other contexts. We use character-theoretic and probabilistic methods to study the spaces of homomorphisms from Fuchsian groups to symmetric groups. We obtain a wide variety of applications, ranging from counting branched coverings of Riemann surfaces, to subgroup growth an...