Scaling limit of the complex mobility matrix for the random conductance model on \mathbbT^d_N

2025/12/17 by Faggionato, Alessandra, Salvi, Michele
#60K37 #74Q10 #82D30 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2512.15506

Abstract

We consider a continuous-time random walk on the d-dimensional torus \mathbbT^d_N=ℤ^d/N 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 a deterministic limiting matrix σ(ω) and provide different characterizations of σ(ω).

Citations

• A martingale approach to time-dependent and time-periodic linear response in Markov jump processes
• Scaling limit of the directional conductivity of random resistor networks on simple point processes
• Stochastic homogenization of random walks on point processes
• Einstein relation and steady states for the random conductance model
• Einstein relation for reversible diffusions in random environment

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