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...
Gespeichert in:
Veröffentlicht in: | Probability theory and related fields 2021-08, Vol.180 (3-4), p.847-889 |
---|---|
Hauptverfasser: | , , , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
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 |