TY - JOUR

T1 - Quenched invariance principle for a class of random conductance models with long-range jumps

AU - Biskup, Marek

AU - Chen, Xin

AU - Kumagai, Takashi

AU - Wang, Jian

N1 - Funding Information:
We thank anonymous referees for their helpful comments and corrections. This research has been supported by National Science Foundation (US) Awards DMS-1712632 and DMS-1954343, the National Natural Science Foundation of China (No.?11871338), JSPS KAKENHI Grant Number JP17H01093, the Alexander von Humboldt Foundation, the National Natural Science Foundation of China (Nos.?11831014 and?12071076), the Program for Probability and Statistics: Theory and Application (No. IRTL1704) and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ). The non-Kyoto based authors would like to thank RIMS at Kyoto University for hospitality that made this project possible. A version of this paper by two of the present authors was previously posted on arXiv [25] and was subsequently withdrawn due to an error in one of the key arguments. The present paper subsumes the parts of [25] that are worth saving. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funding Information:
We thank anonymous referees for their helpful comments and corrections. This research has been supported by National Science Foundation (US) Awards DMS-1712632 and DMS-1954343, the National Natural Science Foundation of China (No. 11871338), JSPS KAKENHI Grant Number JP17H01093, the Alexander von Humboldt Foundation, the National Natural Science Foundation of China (Nos. 11831014 and 12071076), the Program for Probability and Statistics: Theory and Application (No. IRTL1704) and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ). The non-Kyoto based authors would like to thank RIMS at Kyoto University for hospitality that made this project possible. A version of this paper by two of the present authors was previously posted on arXiv [] and was subsequently withdrawn due to an error in one of the key arguments. The present paper subsumes the parts of [] that are worth saving.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2021/8

Y1 - 2021/8

N2 - We study random walks on Zd (with d≥ 2) among stationary ergodic random conductances { Cx,y: x, y∈ Zd} 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∈ZdC0,x|x|2 and q-th moment of 1 / C,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 2d 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 2d, 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.

AB - We study random walks on Zd (with d≥ 2) among stationary ergodic random conductances { Cx,y: x, y∈ Zd} 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∈ZdC0,x|x|2 and q-th moment of 1 / C,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 2d 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 2d, 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.

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U2 - 10.1007/s00440-021-01059-z

DO - 10.1007/s00440-021-01059-z

M3 - Article

AN - SCOPUS:85110450939

VL - 180

SP - 847

EP - 889

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 3-4

ER -