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Aging and sub-aging for one-dimensional random walks amongst random conductances

David A. Croydon, Daniel Kious, Carlo Scali

Abstract

We consider random walks amongst random conductances in the cases where the conductances can be arbitrarily small, with a heavy-tailed distribution at 0, and where the conductances may or may not have a heavy-tailed distribution at infinity. We study the long time behaviour of these processes and prove aging statements. When the heavy tail is only at 0, we prove that aging can be observed for the maximum of the process, i.e. the same maximal value is attained repeatedly over long time-scales. When there are also heavy tails at infinity, we prove a classical aging result for the position of the walker, as well as a sub-aging result that occurs on a shorter time-scale.

Aging and sub-aging for one-dimensional random walks amongst random conductances

Abstract

We consider random walks amongst random conductances in the cases where the conductances can be arbitrarily small, with a heavy-tailed distribution at 0, and where the conductances may or may not have a heavy-tailed distribution at infinity. We study the long time behaviour of these processes and prove aging statements. When the heavy tail is only at 0, we prove that aging can be observed for the maximum of the process, i.e. the same maximal value is attained repeatedly over long time-scales. When there are also heavy tails at infinity, we prove a classical aging result for the position of the walker, as well as a sub-aging result that occurs on a shorter time-scale.
Paper Structure (21 sections, 42 theorems, 308 equations, 1 figure)

This paper contains 21 sections, 42 theorems, 308 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Model(s)
  4. Main results
  5. Outline of the proof
  6. Limit processes and refined statements of main results
  7. Limit processes and limit environments
  8. Restatement of the main results
  9. Coupling and convergence of the environment
  10. Convergence of point processes
  11. Coupling and convergence of the discrete environment
  12. Random walk estimates
  13. Couplings
  14. Aging estimates under assumption \ref{['RW']}
  15. Sub-aging estimates under assumption \ref{['RWT']}
  16. ...and 6 more sections

Key Result

Theorem 1.3

Under RW and Assumption AssumptionWeakBias, for all $0<\alpha_0<1$, the following aging statement holds. There exists an explicit function $\theta:(1,\infty)\to(0,1)$ such that, for all $h>1$,

Figures (1)

  • Figure 1: Visualisation the interval around $\{j_{\ell}(n), j_{\ell}(n) + 1\}$. The two points are visualised as collapsed for simplicity.

Theorems & Definitions (84)

  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Proposition 3.4
  • ...and 74 more