%0 Journal Article %1 nashwilliams1959random %A Nash-Williams, C. St J. A. %D 1959 %J Mathematical Proceedings of the Cambridge Philosophical Society %K electrical_networks random_walk random_walks_on_graphs resistance_distance %N 02 %P 181--194 %R 10.1017/S0305004100033879 %T Random walk and electric currents in networks %U http://journals.cambridge.org/article_S0305004100033879 %V 55 %X 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 S0305004100033879_eqnU001 in which (i) the ξj are vertices of G, (ii), λj. is an edge joining ξj–1 to ξj (j = 1, 2, 3, …), (iii) if λ is any edge incident with ξj, then S0305004100033879_eqnU002 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 τD(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.

@article{nashwilliams1959random, 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 S0305004100033879_eqnU001 in which (i) the ξj are vertices of G, (ii), λj. is an edge joining ξj–1 to ξj (j = 1, 2, 3, …), (iii) if λ is any edge incident with ξj, then S0305004100033879_eqnU002 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 τD(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.}, added-at = {2014-08-25T18:12:11.000+0200}, author = {Nash-Williams, C. St J. A.}, biburl = {https://www.bibsonomy.org/bibtex/26c07a7f745e6d467e8b4aca645d21b0d/peter.ralph}, doi = {10.1017/S0305004100033879}, interhash = {3b78dbc24ddfdf42c614e044014da680}, intrahash = {6c07a7f745e6d467e8b4aca645d21b0d}, issn = {1469-8064}, journal = {Mathematical Proceedings of the Cambridge Philosophical Society}, keywords = {electrical_networks random_walk random_walks_on_graphs resistance_distance}, month = {4}, number = 02, numpages = {14}, pages = {181--194}, timestamp = {2014-08-25T18:12:11.000+0200}, title = {Random walk and electric currents in networks}, url = {http://journals.cambridge.org/article_S0305004100033879}, volume = 55, year = 1959 }