Left–right crossings in the Miller–Abrahams random resistor network and in generalized Boolean models

Faggionato, Alessandra; Mimun, Hlafo Alfie

doi:10.1016/j.spa.2021.03.001

Cited by 3 publications

(8 citation statements)

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“…In addition, the application of Theorem 1 to the Miller-Abrahams resistor network is a main tool in the proof of Mott's law in [16]. For other rigorous results on the Miller-Abrahams resistor network we refer to [17,18].…”

Section: 1mentioning

confidence: 99%

Scaling limit of the conductivity of random resistor networks on simple point processes

Faggionato¹

2021

Preprint

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We consider random resistor networks with nodes given by a simple point process on R d and with random conductances. The length range of the electrical filaments can be unbounded. We assume that the randomness is stationary and ergodic w.r.t. the action of the group G, given by R d or Z d . This action is covariant w.r.t. the action of G by translations on the Euclidean space. Under very basic and minimal assumptions we prove that a.s. the suitably rescaled directional conductivity of the resistor network along the principal directions of the effective homogenized matrix D converges to the corresponding eigenvalue of D times the intensity of the simple point process. The above result covers plenty of models including e.g. the standard conductance model on Z d (for which we improve the existing results), the Miller-Abrahams resistor network for conduction in amorphous solids (to which we can now extend the bounds in agreement with Mott's law previously obtained in [8,19,20] for Mott's random walk), resistor networks on the supercritical cluster in lattice and continuous percolations (solving an open problem for Bernoulli supercritical percolation on Z d ), resistor networks on crystal lattices and on Delaunay triangulations.

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Section: 1mentioning

confidence: 99%

Scaling limit of the conductivity of random resistor networks on simple point processes

Faggionato¹

2021

Preprint

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“…Due to (23), (38) implies (37). We will prove (38) by Rayleigh monotonicity law for resistor networks and suitable previous results on LR crossings obtained in [15]. Theorems 7.6 and 7.9 have the following immediate consequence: Corollary 7.10 (Mott's law).…”

Section: Main Results For Ppp[ρ ν] D ≥mentioning

confidence: 74%

“…We point out that (36) is an immediate consequence of (35) since we have (23). The strategy to prove (35) will be to use the variational formula (15) for D(β) for clever choices of the test function f (partially inspired by the proof of [27, Theorem 3.12] and several scaling considerations discussed in the next sections).…”

Section: Main Results For Ppp[ρ ν] D ≥mentioning

confidence: 99%

“…The following fact is obtained by applying [15, Thm. 1] using arguments similar to the ones in [15,Section 8] to check the hypotheses. We detail its derivation in Appendix C. We mention that [15]…”

Section: Thenmentioning

confidence: 99%

“…Our proof of the above Mott's law is based on previous results in percolation [14,15] and homogenization [12]. The analysis of left-right (LR) crossings in the supercritical phase provided in [15] uses the FKG inequality which is satisfied when the energy marks have a fixed sign (case ν ∈ V + ) but is violated otherwise (case ν ∈ V), as discussed in Appendix D. Once the results of [15] are extended to the MA resistor network with energy marks of arbitrary signs, then our analysis here provides Mott's law (2) also for the case ν ∈ V, which is the realistic one. Hence, in Corollary 7.10 we obtain Mott's law for ν ∈ V under a suitable Ansatz on the LR crossings.…”

Section: Introductionmentioning

confidence: 98%

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Mott's law for the Miller-Abrahams random resistor network and for Mott's random walk

Faggionato¹

2023

Preprint

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Mott's variable range hopping (v.r.h.) is the phonon-induced hopping of electrons in disordered solids (as doped semiconductors) within the regime of strong Anderson localization. It was introduced by N. Mott to explain the anomalous low temperature conductivity decay corresponding now to the so called Mott's law. We provide a rigorous derivation of this Physics law for two effective models of Mott v.r.h.: the Miller-Abrahams random resistor network and Mott's random walk. We also determine the constant multiplying the power of the inverse temperature in the exponent in Mott's law, which was an open problem also at heuristic level. Our proof uses also our previous results of homogenization and percolation for the above models, which are exhaustive when the energy marks have a fixed sign. For energy marks with different signs the expected percolation property of left-right crossings in the supercritical phase (on which we are currently working) is assumed as an Ansatz.

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Phase transition of disordered random networks on quasi-transitive graphs

Liu,

Xiang

2024

Electron. J. Probab.

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Left–right crossings in the Miller–Abrahams random resistor network and in generalized Boolean models

Cited by 3 publications

References 8 publications

Scaling limit of the conductivity of random resistor networks on simple point processes

Scaling limit of the conductivity of random resistor networks on simple point processes

Mott's law for the Miller-Abrahams random resistor network and for Mott's random walk

Phase transition of disordered random networks on quasi-transitive graphs

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