Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scaling limit of the complex mobility matrix for the random conductance model on $\mathbb{T}^d_N$

Alessandra Faggionato, Michele Salvi

TL;DR

The paper establishes a quenched, infinite-volume limit for the complex mobility matrix σ_N^ξ(ω) of a random walk in random conductances on the torus, under finite second-moment conductances. Using the two-scale homogenization framework, it derives deterministic σ(ω) and presents multiple equivalent representations, including relations to an auxiliary field θ and the generator L. The analysis extends to finite-range but long-range neighbourhoods and handles the challenges posed by zero spectral gap without relying on uniform ellipticity. This work provides rigorous fluctuation-dissipation-type identities for disordered media and contributes a foundational limit theory for AC transport in random environments.

Abstract

We consider a continuous-time random walk on the $d$-dimensional torus $\mathbb{T}^d_{N}=\mathbb{Z}^d/N \mathbb{Z}^d$, possibly with long-range, but finite, jumps. The law of the jumps is regulated by a random environment $ξ$ yielding a stationary and ergodic field of random conductances. The complex mobility matrix $σ_N^ξ(ω)$ measures the linear response of the random walk to a $\cos(ωt)$-type oscillating external field. By investigating the homogenization properties of the medium, and assuming in addition that the conductances have finite second moment, we show that, for almost every realization of the environment $ξ$, the complex mobility matrix $σ_N^ξ(ω)$ converges as $N\to+\infty$ to a deterministic limiting matrix $σ(ω)$ and provide different characterizations of $σ(ω)$.

Scaling limit of the complex mobility matrix for the random conductance model on $\mathbb{T}^d_N$

TL;DR

The paper establishes a quenched, infinite-volume limit for the complex mobility matrix σ_N^ξ(ω) of a random walk in random conductances on the torus, under finite second-moment conductances. Using the two-scale homogenization framework, it derives deterministic σ(ω) and presents multiple equivalent representations, including relations to an auxiliary field θ and the generator L. The analysis extends to finite-range but long-range neighbourhoods and handles the challenges posed by zero spectral gap without relying on uniform ellipticity. This work provides rigorous fluctuation-dissipation-type identities for disordered media and contributes a foundational limit theory for AC transport in random environments.

Abstract

We consider a continuous-time random walk on the -dimensional torus , possibly with long-range, but finite, jumps. The law of the jumps is regulated by a random environment yielding a stationary and ergodic field of random conductances. The complex mobility matrix measures the linear response of the random walk to a -type oscillating external field. By investigating the homogenization properties of the medium, and assuming in addition that the conductances have finite second moment, we show that, for almost every realization of the environment , the complex mobility matrix converges as to a deterministic limiting matrix and provide different characterizations of .

Paper Structure

This paper contains 13 sections, 24 theorems, 149 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model and main result
  3. Extension to the long-range model
  4. Some algebraic identities satisfied by $\sigma(\omega)$
  5. Preliminary results
  6. Set $\Omega_{\rm typ}$ of typical environments
  7. Measures $\mu_\varepsilon$ and $\nu_\varepsilon^\xi$
  8. 2-scale convergence in $L^2( \mu_\varepsilon)$
  9. 2-scale convergence in $L^2( \nu _\varepsilon^{\zeta})$
  10. $\varepsilon$--gradient
  11. Proof of Theorem \ref{['teo1bis']} (I): 2-scale convergence
  12. Proof of Theorem \ref{['teo1bis']} (II): Conclusion
  13. Proof of Proposition \ref{['zap']}

Key Result

Proposition 2.2

Suppose that the probability measure ${\mathbb P}$ on $\Omega$ is stationary and ergodic w.r.t. the action of the group ${\mathbb Z} ^d$ and that the covariant relation covariante is satisfied. In addition suppose that ${\mathbb E} [c_{0,z}(\xi)]<+\infty$ for all $z\in\mathcal{N}$. Consider the bili with domain $\mathcal{D}(\mathcal{E}):=\{ f\in L^2_R({\mathbb P} )\,:\, \sum_{z\in\mathcal{N}}{\mat

Figures (3)

  • Figure 1: Periodization procedure in 1d for nearest-neighbour conductances. On the first line: conductances $(c_{x,x+1})_{x\in\mathbb Z}$ (e.g. $c_{0,1}=D$). On the second line: periodized conductances $(c^{(5)}_{x,x+1})_{x\in\mathbb Z}$.
  • Figure 2: Portion of a periodized environment in ${\mathbb Z} ^2$ with $N=3$. The grey region corresponds to $[0,N)^d$. Different bond thicknesses represent different conductances.
  • Figure 3: Periodization procedure in 1d for nearest-neighbours and second-nearest-neighbours conductances ($\mathcal{N}=\{\pm e_1, \pm 2 e_1\}$ and $\mathcal{N}_*=\{e_1, 2e_1\}$). On the first line: conductances $(c_{x,x+w})_{x\in\mathbb Z,\,w\in \mathcal{N}_*}$ (e.g. $c_{0,1}=D$ and $c_{0,2}=d$). On the second line: periodized conductances $(c^{(5)}_{x,x+w})_{x\in\mathbb Z,\,w\in\mathcal{N}_*}$.

Theorems & Definitions (52)

  • Remark 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 2.5: Good neighbourhood
  • Definition 2.6: Periodized rates
  • Theorem 2.7
  • Proposition 2.8
  • proof
  • Corollary 2.9
  • ...and 42 more