On the Probability Generating Functional for Point Processes.

Daley, D. J.; Vere-Jones, D.

doi:10.21236/ada147692

Cited by 7 publications

(23 citation statements)

References 3 publications

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“…Due to (164) and the symmetry p ω (t, x, y) = p ω (t, y, x), in order to conclude the proof of Lemma 17. Hence, similarly to [22], we get (using the symmetry of p ω (s, follows from stationarity and it is standard [7]. We prove Item (iii).…”

Section: Proof Of Theoremsupporting

confidence: 53%

“…Thanks to (36) and Lemma 16.1 (proved below) applied to the function ψ(r) := 1/(1 + r d+1 ), the limits (37) and (38) follow by the same arguments used in the proofs of Corollary 2.5 and Lemma 6.1 in [11,Sections 6,7]. We give the proof for (37), since (38) easily follows from (37) as in the proof of Corollary 2.5 in [11] due to some estimates presented below.…”

Section: Proof Of Theoremmentioning

confidence: 92%

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Stochastic homogenization in amorphous media and applications to exclusion processes

Faggionato

2019

Preprint

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We consider random walks on marked simple point processes with symmetric jump rates and unbounded jump range. We prove homogenization properties of the associated Markov generators. As an application, we derive the hydrodynamic limit of the simple exclusion process given by multiple random walks as above, with hard-core interaction, on a marked Poisson point process. The above results cover Mott variable range hopping, which is a fundamental mechanism of phonon-induced electron conduction in amorphous solids as doped semiconductors. Our techniques, based on an extension of two-scale convergence, can be adapted to other models, as e.g. the random conductance model.

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Section: Proof Of Theoremsupporting

confidence: 53%

Section: Proof Of Theoremmentioning

confidence: 92%

Stochastic homogenization in amorphous media and applications to exclusion processes

Faggionato

2019

Preprint

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“…On N one defines a special metric d (cf. [10,App. A2.6]) such that a sequence (ξ n ) converges to ξ in N if lim n→∞ dξ n (x)f (x) = dξ(x)f (x) for any bounded continuous function f : R d → R vanishing outside a bounded set.…”

Section: Models and Main Resultsmentioning

confidence: 99%

“…[3,27,30,32]). With our notation, it is obtained by taking as Ω the space of configurations of marked simple point processes [10].…”

Section: 1mentioning

confidence: 99%

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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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Regularity of biased 1D random walks in random environment

Faggionato¹,

Salvi²

2019

ALEA

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We study the asymptotic properties of nearest-neighbor random walks in 1d random environment under the influence of an external field of intensity λ ∈ R. For ergodic shift-invariant environments, we show that the limiting velocity v(λ) is always increasing and that it is everywhere analytic except at most in two points λ− and λ+. When λ− and λ+ are distinct, v(λ) might fail to be continuous. We refine the assumptions in [34] for having a recentered CLT with diffusivity σ 2 (λ) and give explicit conditions for σ 2 (λ) to be analytic. For the random conductance model we show that, in contrast with the deterministic case, σ 2 (λ) is not monotone on the positive (resp. negative) half-line and that it is not differentiable at λ = 0. For this model we also prove the Einstein Relation, both in discrete and continuous time, extending the result of [25].

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On the Probability Generating Functional for Point Processes.

Cited by 7 publications

References 3 publications

Stochastic homogenization in amorphous media and applications to exclusion processes

Stochastic homogenization in amorphous media and applications to exclusion processes

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

Regularity of biased 1D random walks in random environment

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