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Quenched invariance principle for random walks in random environments admitting a cycle decomposition

Jean-Dominique Deuschel, Martin Slowik, Weile Weng

Abstract

We study a class of non-reversible, continuous-time random walks in random environments on $\mathbb{Z}^d$ that admit a cycle representation with finite cycle length. The law of the transition rates, taking values in $[0, \infty)$, is assumed to be stationary and ergodic with respect to space shifts. Moreover, the transition rate from $x$ to $y$, denoted by $c^ω(x,y)$, is a superposition of non-negative random weights on oriented cycles that contain the edge $(x,y)$. We prove a quenched invariance principle under moment conditions that are comparable to the well-known p-q moment condition of Andres, Deuschel, and Slowik [2] for the random conductance model. A key ingredient in proving the sublinearity is an energy estimate for the non-symmetric generator. Our result extends that of Deuschel and Kösters [12] beyond strong ellipticity and bounded cycle lengths.

Quenched invariance principle for random walks in random environments admitting a cycle decomposition

Abstract

We study a class of non-reversible, continuous-time random walks in random environments on that admit a cycle representation with finite cycle length. The law of the transition rates, taking values in , is assumed to be stationary and ergodic with respect to space shifts. Moreover, the transition rate from to , denoted by , is a superposition of non-negative random weights on oriented cycles that contain the edge . We prove a quenched invariance principle under moment conditions that are comparable to the well-known p-q moment condition of Andres, Deuschel, and Slowik [2] for the random conductance model. A key ingredient in proving the sublinearity is an energy estimate for the non-symmetric generator. Our result extends that of Deuschel and Kösters [12] beyond strong ellipticity and bounded cycle lengths.

Paper Structure

This paper contains 12 sections, 12 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Model and Results
  3. The model
  4. Results
  5. Quenched invariance principle
  6. Corrector construction and harmonic coordinates
  7. Properties of the corrector and local martingale
  8. Sublinearity of the corrector
  9. Local martingale QFCLT
  10. Vanishing corrector
  11. Proof of Theorem \ref{['thm:QFCLT']}
  12. Maximal inequality for harmonic functions

Key Result

Theorem 2.6

Suppose that $d \geq 2$ and Assumption ass:P-(i) and ass:p-q-moment hold. Then, the QFCLT holds for $X$ with a deterministic non-degenerate covariance matrix $\Sigma^{2}$.

Theorems & Definitions (41)

  • Remark 2.2
  • Definition 2.3: QFCLT
  • Remark 2.5
  • Theorem 2.6: QFCLT
  • Proposition 3.1
  • proof : Proof outline
  • Definition 3.2: Cycle condition
  • Definition 3.3: Cocycle property
  • Remark 3.4
  • Proposition 3.5
  • ...and 31 more