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A cohesive account on the ergodic behaviours and scaling limits of Random Walks in Cooling Random Environments

Luca Avena, Conrado da Costa

Abstract

Transport in disordered media is a central theme in probability and statistical physics, where randomness in the underlying medium produces phenomena such as localization, anomalous scaling, and slow relaxation. A paradigmatic model for transport in disordered media is that of Random Walks in Random Environments (RWRE), which has been extensively studied since the 1970's and is by now well understood in one dimension. More recently, several works have explored perturbations of models of transport in disordered media aimed at interpolating between static disorder and fully homogenized dynamics. Random walks in cooling random environments (RWCRE), introduced in this context, constitute a key example: the environment is dynamically resampled at prescribed times and kept fixed in between, giving rise to a delicate ``quasi-ergodic'' structure in time allowing to interpolate between homogeneous random walks and classical RWRE. The purpose of this paper is twofold. A first goal is to offer an original survey on the main results for RWCRE in 1d: recurrence criteria, law of large numbers, large deviations, and fluctuation phenomena across different resampling regimes. Those results have been derived in a series of recent works and are complemented here with a number of new statements aiming at presenting a unified phenomenological picture. As a second goal, we try to extract a coherent conceptual picture that highlights structural mechanisms -- such as ergodic limits, persistence under perturbations and replacement principles -- that extend beyond the specific setting of RWCRE and are relevant for a broader class of disordered systems that can be perturbed by introducing independent resetting in the same fashion.

A cohesive account on the ergodic behaviours and scaling limits of Random Walks in Cooling Random Environments

Abstract

Transport in disordered media is a central theme in probability and statistical physics, where randomness in the underlying medium produces phenomena such as localization, anomalous scaling, and slow relaxation. A paradigmatic model for transport in disordered media is that of Random Walks in Random Environments (RWRE), which has been extensively studied since the 1970's and is by now well understood in one dimension. More recently, several works have explored perturbations of models of transport in disordered media aimed at interpolating between static disorder and fully homogenized dynamics. Random walks in cooling random environments (RWCRE), introduced in this context, constitute a key example: the environment is dynamically resampled at prescribed times and kept fixed in between, giving rise to a delicate ``quasi-ergodic'' structure in time allowing to interpolate between homogeneous random walks and classical RWRE. The purpose of this paper is twofold. A first goal is to offer an original survey on the main results for RWCRE in 1d: recurrence criteria, law of large numbers, large deviations, and fluctuation phenomena across different resampling regimes. Those results have been derived in a series of recent works and are complemented here with a number of new statements aiming at presenting a unified phenomenological picture. As a second goal, we try to extract a coherent conceptual picture that highlights structural mechanisms -- such as ergodic limits, persistence under perturbations and replacement principles -- that extend beyond the specific setting of RWCRE and are relevant for a broader class of disordered systems that can be perturbed by introducing independent resetting in the same fashion.
Paper Structure (29 sections, 8 theorems, 127 equations, 1 figure, 1 table)

This paper contains 29 sections, 8 theorems, 127 equations, 1 figure, 1 table.

Table of Contents

  1. RWRE, cooling, and perturbation via independent resetting
  2. Outline of the paper
  3. Models, stucture, questions & methods
  4. General framework
  5. RWRE and RWCRE
  6. Static environments
  7. Cooling dynamic environments
  8. Relation to the patchwork construction
  9. Types of resampling maps
  10. Asymptotic behaviours at glance: persistence, perturbation, and cross-over driven by resampling
  11. Methods: block decomposition, mass profiles, and replacement principles
  12. Homogenization for non-cooling maps
  13. LLN from LDP
  14. CLT from the LDP
  15. Recurrence and transience properties in 1d
  16. ...and 14 more sections

Key Result

Lemma 4.1

Given a pair $(\alpha,\tau)$, let be the tail sigma algebra generated by the associated RWCRE $X$. If $A \in {\mathcal{T}}$ then ${\mathbb {P}}(A) \in \{0,1\}$.

Figures (1)

  • Figure 1: The decomposition of RWCRE in pieces of RWRE as presented in \ref{['eq:X_dec']}.

Theorems & Definitions (20)

  • Definition 2.1: RWRE: quenched and annealed laws
  • Definition 2.2: RWCRE: kernel formulation
  • Remark 3.1: On the CLT: Lindeberg--Feller vs. Bryc-type criteria
  • Lemma 4.1: Tail triviality for RWCRE
  • Theorem 4.2: Stability of recurrence/transience, AveChidCosdHol22
  • Theorem 5.1: Annealed rate $n$ LDP for general resampling maps
  • Corollary 5.2: Persistence of annealed LDP for cooling maps
  • proof
  • Theorem 5.3: Weak LLN, AvedCos24game
  • proof
  • ...and 10 more