Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on transience of generalized many-dimensional excited random walks

Rodrigo B. Alves, Giulio Iacobelli, Glauco Valle

Abstract

We consider a variation of the Generalized Excited Random Walk (GERW) in dimension $d\ge 2$ where the lower bound on the drift for excited jumps is time-dependent and decays to zero. We show that if the lower bound decays slower that $n^{-β}$ ($n$ is time), for $β$ depending on the transitions of the process, the GERW is transient in the direction of the drift.

A note on transience of generalized many-dimensional excited random walks

Abstract

We consider a variation of the Generalized Excited Random Walk (GERW) in dimension where the lower bound on the drift for excited jumps is time-dependent and decays to zero. We show that if the lower bound decays slower that ( is time), for depending on the transitions of the process, the GERW is transient in the direction of the drift.
Paper Structure (8 sections, 5 theorems, 80 equations, 1 figure)

This paper contains 8 sections, 5 theorems, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definition of the model and main result
  3. Proof of Theorem \ref{['thm:main']}
  4. Acknowledgements:
  5. Conflict of interest:
  6. Data Availability:
  7. Proof of Proposition \ref{['prop41']}
  8. Proof of Proposition \ref{['prop42_pnn0']}

Key Result

Theorem 1.1

Let $X$ be a $\lambda_n$- GERW in direction $\ell$ with excitation set $A \supset \mathbb{M}_{\ell}$. There exists $\beta_0 < 1/6$ such that if for some $n_0 \in \mathbb{N}$, $\lambda >0$ and $\beta<\beta_0$, we have $\lambda_n \ge \lambda (n_0+n)^{-\beta}$ for every $n\ge 1$, then

Figures (1)

  • Figure 1: $H(a,b)$ is the strip of $\mathbb{R}^d$ corresponding to the points $x$ such that $a\le x\cdot \ell \le b$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 1.1
  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Remark 1.2
  • proof : Proof of Theorem \ref{['thm:main']}
  • Remark 1.3
  • proof : Proof of Proposition \ref{['prop43_pnn0-1']}
  • Lemma A.1
  • ...and 1 more