Random walk and electric currents in networks

Source abstract

ABSTRACT Let G be a locally finite connected graph and c be a positive real-valued function defined on its edges. Let D (ξ) denote the sum of the values of c on the edges incident with a vertex ξ. A particle starts at some vertex α and performs an infinite random walk in which (i) the ξ j are vertices of G , (ii), λ j . is an edge joining ξ j –1 to ξ j ( j = 1, 2, 3, …), (iii) if λ is any edge incident with ξ j , then Let υ be a set of vertices of G such that the complementary set of vertices is finite and includes α. A geometrical characterization is given of the probability (τ, say) that the particle will visit some element of υ without first returning to α. An essentially equivalent problem is obtained by regarding G as an electrical network and c (λ) as the conductance of an edge λ; the current flowing through the network from α to υ when an external agency maintains α at potential I and all elements of υ at potential 0 is found to be τ D (α). A necessary and sufficient condition (of a geometrical character) for the particle to be certain to return to α. is obtained; and, as an application, a new proof is given of a conjecture of Gillis (3) regarding centrally biased random walk on an n –dimensional lattice.