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Once-excited random walks on general trees

Duy-Bao Le, Tuan-Minh Nguyen

Abstract

We study once-excited random walks on general trees, modeled by placing a single "cookie" at each vertex. Each cookie acts as a metaphorical reward that is consumed upon the first visit to the vertex where the cookie is placed. On that initial visit, the walk is in an excited state and behaves like a biased random walk. Once the cookie is consumed, the process reverts to a symmetric random walk on all subsequent visits. We consider a random environment in which the bias parameters are independent random variables. We prove that the process exhibits a sharp phase transition between transience and recurrence on general trees with polynomial growth, where the critical threshold is determined by the branching-ruin number of the tree.

Once-excited random walks on general trees

Abstract

We study once-excited random walks on general trees, modeled by placing a single "cookie" at each vertex. Each cookie acts as a metaphorical reward that is consumed upon the first visit to the vertex where the cookie is placed. On that initial visit, the walk is in an excited state and behaves like a biased random walk. Once the cookie is consumed, the process reverts to a symmetric random walk on all subsequent visits. We consider a random environment in which the bias parameters are independent random variables. We prove that the process exhibits a sharp phase transition between transience and recurrence on general trees with polynomial growth, where the critical threshold is determined by the branching-ruin number of the tree.
Paper Structure (12 sections, 10 theorems, 156 equations, 1 figure)

This paper contains 12 sections, 10 theorems, 156 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Strong construction of GOERW
  3. Rubin's construction of $\mathbf{X }$
  4. Restrictions and Extensions
  5. Quasi-independent percolation
  6. Preliminary notations and results
  7. Ruin percolation
  8. Criteria of transience and recurrence for GOERW
  9. Once-excited random walks in random environment
  10. Appendix
  11. Gambler's ruin
  12. Hoeffding's inequality

Key Result

theorem 1

Assume that where $(\alpha_v)_{v\in V\setminus\{\rho\}}$ are i.i.d. non-negative random variables with $m:=\mathbf{E}[1/(\alpha_v+1)]$. Then, under the annealed law, the process ${\rm OERW}(\boldsymbol{\lambda})$ is a.s. transient when ${\rm br}_r(\mathcal{T})> 2-m$ and it is a.s. recurrent when ${\rm br}_r(\ma

Figures (1)

  • Figure 1: The two shortest paths connecting $\rho$ with $e_1=\{e_1^-, e_1^+\}$ and $e_2=\{e_2^-, e_2^+\}$ with the last common edge $e=\{e^-,e^+\}$.

Theorems & Definitions (19)

  • theorem 1
  • theorem 2
  • Remark 3
  • proposition 1: Theorem 5.19 in LP2016
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof
  • ...and 9 more