Gaussian, stable, tempered stable and mixed limit laws for random walks in cooling random environments

Avena, Luca; Da Costa, Conrado; Peterson, Jonathon

Abstract

Random Walks in Cooling Random Environments (RWCRE) is a model of random walks in dynamic random environments where the entire environment is resampled along a fixed sequence of times, called the "cooling sequence", and is kept fixed in between those times. This model interpolates between that of a homogeneous random walk, where the environment is reset at every step, and Random Walks in (static) Random Environments (RWRE), where the environment is never resampled. In this work we focus on the limiting distributions of one-dimensional RWCRE in the regime where the fluctuations of the corresponding (static) RWRE is given by a s-stable random variable with s ∈ (1, 2). In this regime, due to the two extreme cases (resampling every step and never resampling, respectively), a crossover from Gaussian to stable limits for sufficiently regular cooling sequence was previously conjectured. Our first result answers affirmatively this conjecture by making clear critical exponent, norming sequences and limiting laws associated with the crossover which demonstrates a change from Gaussian to s-stable limits, passing at criticality through a certain generalized tempered stable distribution which have not appeared as limits of random walks in dynamic random environments previously. We then explore the resulting RWCRE scaling limits for general cooling sequences. On the one hand, we offer sets of operative sufficient conditions that guarantee asymptotic emergence of either Gaussian, s-stable or generalized tempered distributions from a certain class. On the other hand, we give explicit examples and describe how to construct irregular cooling sequences for which the corresponding limit law is characterized by mixtures of the three above mentioned laws. To obtain these results, we need and derive a number of refined asymptotic results for the static RWRE with s ∈ (1, 2) which are of independent interest. Résumé. ???. MSC2020 subject classifications: Primary 60K37, 60E07; secondary 60G50, 60K50

Citation

Avena, L., Da Costa, C., & Peterson, J. (in press). Gaussian, stable, tempered stable and mixed limit laws for random walks in cooling random environments. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques,

• Journal Article Type: Article
• Acceptance Date: Nov 5, 2023
• Deposit Date: Jan 28, 2024
• Journal: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
• Print ISSN: 0246-0203
• Publisher: Institute Henri Poincaré
• Peer Reviewed: Peer Reviewed
• Keywords: Random walk; dynamic random environment; resetting dynamics; stable laws; anomalous diffusion
• Public URL: https://durham-repository.worktribe.com/output/1903061
• Publisher URL: https://imstat.org/journals-and-publications/annales-de-linstitut-henri-poincare/annales-de-linstitut-henri-poincare-accepted-papers/