Quenched invariance principle for a class of random conductance models with long-range jumps

We study random walks on  Z d (with d ≥ 2 ) among stationary ergodic random conductances { C x , y : x , y ∈ Z d } that permit jumps of arbitrary length. Our focus is on the quenched invariance principle (QIP) which we establish by a combination of corrector methods, functional inequalities and heat...

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Veröffentlicht in:Probability theory and related fields 2021-08, Vol.180 (3-4), p.847-889
Hauptverfasser: Biskup, Marek, Chen, Xin, Kumagai, Takashi, Wang, Jian
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Sprache:eng
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Zusammenfassung:We study random walks on  Z d (with d ≥ 2 ) among stationary ergodic random conductances { C x , y : x , y ∈ Z d } that permit jumps of arbitrary length. Our focus is on the quenched invariance principle (QIP) which we establish by a combination of corrector methods, functional inequalities and heat-kernel technology assuming that the p -th moment of ∑ x ∈ Z d C 0 , x | x | 2 and q -th moment of 1 / C 0 , x for  x neighboring the origin are finite for some p , q ≥ 1 with p - 1 + q - 1 < 2 / d . In particular, a QIP thus holds for random walks on long-range percolation graphs with connectivity exponents larger than 2 d in all d ≥ 2 , provided all the nearest-neighbor edges are present. Although still limited by moment conditions, our method of proof is novel in that it avoids proving everywhere-sublinearity of the corrector. This is relevant because we show that, for long-range percolation with exponents between d + 2 and 2 d , the corrector exists but fails to be sublinear everywhere. Similar examples are constructed also for nearest-neighbor, ergodic conductances in d ≥ 3 under the conditions complementary to those of the recent work of Bella and Schäffner (Ann Probab 48(1):296–316, 2020). These examples elucidate the limitations of elliptic-regularity techniques that underlie much of the recent progress on these problems.
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-021-01059-z