Some Criteria for the Transience of Simple Random Walks on Trees | ||
| The Egyptian Statistical Journal | ||
| Article 13, Volume 35, Issue 1, June 1991, Pages 175-190 | ||
| Document Type: Original Article | ||
| DOI: 10.21608/esju.1991.315004 | ||
| Author | ||
| H. Konsowa Mokhtar | ||
| Abstract | ||
| The main object of this paper is to use the connection between random walks and electrical networks to obtain much simpler proofs for the results on transience and volumes of trees, given in [7], than Woess' proofs. Woess in his paper [7] gives criteria for transience and recurrence of simple random walks (SRW) on a tree T using a geometric quantity called volume which is defined below. It turns out that the volume of T corresponding to Woess' harmonic weight function is the effective resistance of T considered as an electrical network in which every edge is one-ohm resistor, while his ordinary volume is the energy dissipation of the unit flow from the root r of T out to infinity that divides equally at every branching vertex of T. The recurrence-transience concept is more directly connected to the ideas of resistance and energy dissipation than to the abstract notion of volume. | ||
| Keywords | ||
| Simple Random Walks; Tree; Volume; Transience; Recurrence; Electrical Networks; Effective Resistance; Energy Dissipation | ||
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