Characterization of random walks on space of unordered trees using efficient metric simulation

Farah Ben-Naoum, Christophe Godin, Romain Azaïs

Abstract: The simple random walk on Zp<math><msup is="true"><mrow is="true"><mi mathvariant="double-struck" is="true">Z</mi></mrow><mrow is="true"><mi is="true">p</mi></mrow></msup></math> shows two drastically different behaviors depending on the value of p<math><mi is="true">p</mi></math>: it is recurrent when p∈{1,2}<math><mrow is="true"><mi is="true">p</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">∈</mo><mrow is="true"><mo is="true">{</mo><mn is="true">1</mn><mo is="true">,</mo><mn is="true">2</mn><mo is="true">}</mo></mrow></mrow></math> while it escapes (with a rate increasing with p<math><mi is="true">p</mi></math>) as soon as p≥3<math><mrow is="true"><mi is="true">p</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">≥</mo><mn is="true">3</mn></mrow></math>. This classical example illustrates that the asymptotic properties of a random walk provides some information on the structure of its state space. This paper aims to explore analogous questions on space made up of combinatorial objects with no algebraic structure. We take as a model for this problem the space of unordered unlabeled rooted trees endowed with Zhang edit distance. To this end, it defines the canonical unbiased random walk on the space of trees and provides an efficient algorithm to evaluate its escape rate. Compared to Zhang algorithm, it is incremental and computes the edit distance along the random walk approximately 100 times faster on trees of size 500 on average. The escape rate of the random walk on trees is precisely estimated using intensive numerical simulations, out of reasonable reach without the incremental algorithm.