Scaling limit of the complex mobility matrix for the random conductance model on $\mathbb{T}^d_N$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915682018394112 |
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| author | Faggionato, Alessandra Salvi, Michele |
| author_facet | Faggionato, Alessandra Salvi, Michele |
| contents | 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 $σ(ω)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15506 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scaling limit of the complex mobility matrix for the random conductance model on $\mathbb{T}^d_N$ Faggionato, Alessandra Salvi, Michele Probability Mathematical Physics 60K37, 82D30, 74Q10 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 $σ(ω)$. |
| title | Scaling limit of the complex mobility matrix for the random conductance model on $\mathbb{T}^d_N$ |
| topic | Probability Mathematical Physics 60K37, 82D30, 74Q10 |
| url | https://arxiv.org/abs/2512.15506 |