Recurrence/Transience criteria for excited random walks with\n finite-drift cookie stacks

2021/03/09 by Zachary Letterhos, Letterhos, Zachary
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #60J85 (Secondary) #60K35 (Primary) #Diffusion and Search Dynamics #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2103.05570

openalex publication_date 2021/03/09 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We consider excited random walk (ERW) on \ℤ in environments with\nidentical stacks of infinitely many cookies at each site, subject to the\nconstraint that the total drift per site \δ = \∑ (2p_j - 1) is finite.\nBuilding on the methods of Kozma, Orenshtein, and Shinkar (arXiv:1311.7439), we\nshow that ERW in finite-drift environments is recurrent when |\δ|<1 and\ntransient when |\δ|>1. In the case |\δ|=1 we prove that ERW is\nrecurrent under mild assumptions on the environment. In addition, we show that\nERW may be transient when |\δ|=1, an interesting new behavior that was\nnot present in previously studied models.\n

Related

• Functional limit laws for recurrent excited random walks with periodic cookie stacks
• Excited Random Walk in a Markovian Environment
• Excited random walks with Markovian cookie stacks
• Excursions and occupation times of critical excited random walks
• Positively and negatively excited random walks on integers, with branching processes