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

Marek Biskup*, Xin Chen, Takashi Kumagai, Jian Wang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)847-889
Number of pages43
JournalProbability Theory and Related Fields
Volume180
Issue number3-4
DOIs
Publication statusPublished - 2021 Aug
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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