%0 Journal Article %1 MR3052399 %A Barlow, Martin T. %A Peres, Yuval %A Sousi, Perla %D 2012 %J Ann. Inst. Henri Poincaré Probab. Stat. %K Green_function colliding_random_walks random_walks_on_graphs recurrence %N 4 %P 922--946 %R 10.1214/12-AIHP481 %T Collisions of random walks %U http://dx.doi.org/10.1214/12-AIHP481 %V 48 %X A recurrent graph G has the infinite collision property if two independent random walks on G, started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton–Watson tree with finite variance conditioned to survive, the incipient infinite cluster in Zd with d ≥ 19 and the uniform spanning tree in Z2 all have the infinite collision property. For power-law combs and spherically symmetric trees, we determine precisely the phase boundary for the infinite collision property.

@article{MR3052399, abstract = {A recurrent graph G has the infinite collision property if two independent random walks on G, started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton–Watson tree with finite variance conditioned to survive, the incipient infinite cluster in Zd with d ≥ 19 and the uniform spanning tree in Z2 all have the infinite collision property. For power-law combs and spherically symmetric trees, we determine precisely the phase boundary for the infinite collision property. }, added-at = {2013-09-05T06:56:28.000+0200}, author = {Barlow, Martin T. and Peres, Yuval and Sousi, Perla}, biburl = {https://www.bibsonomy.org/bibtex/2ee0bed3655d01e41efa6328ed108ebbe/peter.ralph}, doi = {10.1214/12-AIHP481}, fjournal = {Annales de l'Institut Henri Poincar\'e Probabilit\'es et Statistiques}, interhash = {b2da50b266f8b2784615b935b828c34b}, intrahash = {ee0bed3655d01e41efa6328ed108ebbe}, issn = {0246-0203}, journal = {Ann. Inst. Henri Poincar\'e Probab. Stat.}, keywords = {Green_function colliding_random_walks random_walks_on_graphs recurrence}, mrclass = {60J10 (05C81 60J35 60J80)}, mrnumber = {3052399}, number = 4, pages = {922--946}, timestamp = {2013-09-05T06:56:28.000+0200}, title = {Collisions of random walks}, url = {http://dx.doi.org/10.1214/12-AIHP481}, volume = 48, year = 2012 }