A NNALES DE L ’I. H. P., SECTION B
M ARTIN P. W. Z ERNER
Velocity and Lyapounov exponents of some random walks in random environment
Annales de l’I. H. P., section B, tome 36, n
o6 (2000), p. 737-748
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Velocity and Lyapounov exponents of
somerandom walks in random environment
Martin P.W.
ZERNER1
Departement Mathematik, ETH, CH-8092 Zurich, Switzerland
Manuscript received in 30 August 1999, revised 22 March 2000
© 2000 Editions scientifiques et medicales Elsevier SAS. All rights reserved
ABSTRACT. - We express the
asymptotic velocity
of random walks in random environmentsatisfying
Kalikow’s condition in terms of theLyapounov exponents
which havepreviously
been used in the context oflarge
deviations. © 2000Editions scientifiques
et médicales Elsevier SAS Key words: Random walk in random environment, Law of large numbers, Lyapounov exponentsAMS classification: 60K37, 60F 15
RESUME. - Nous
exprimons
la vitesseasymptotique
des marches aleatoires en environnement aleatoirequi
verifient la condition de Kalikow en function desexposants
deLyapounov qui
ontdej a
ete utilises dans le contexte desgrandes
deviations. @ 2000Editions scientifiques
etmédicales Elsevier SAS
1 Current address: Department of Electrical Engineering, Technion, Haifa 32000, Israel.
E-mail: zemer@ee.technion.ac.il.
738 M.P W. ZERNER / Ann. Inst. Henri Poincare 36 (2000) 737-748
0.
INTRODUCTION
AND RESULTThere was
recently
some progress inunderstanding
theasymptotic
behavior of multidimensional random walks in random environment
(RWRE),
see[9-11 ]
and[13].
On the onehand,
in[ 11 ]
certain renewal times areconstructed,
which enable one to prove a law oflarge
numbers(see [ 11 ] )
and a central limit theorem for the walk(see [10]).
Thisapproach
leads to anexpression
for theasymptotic velocity
of the random walk in terms of these renewal times. On the otherhand,
one uses in[ 13]
certain
Lyapounov exponents
to derive the rate function for somelarge
deviation
principle
for the walk. The purpose of thepresent
note is to link these two differentconcepts by giving
an alternativeexpression
forthe
asymptotic velocity
of the walk in terms of theLyapounov exponents.
Let us
present
theprecise
model and recall some of the results mentioned above. Weassign
to the lattice sites z Etld (d > 1)
i.i.d. 2d- dimensional vectors(03C9(z,
z + with a common distribution~c and with
positive components
which add up to one and areuniformly
bounded away from 0. That
is,
we assume that there is some fixed K > 0 such that IL issupported
on the setPK
of 2d -vectors withK and
~e pee)
= 1. The random variablesw(z,
z +e)
can thenbe realized as the canonical
projections
on theproduct
space Q :=~
endowed with the canonical
product cr -algebra
and theproduct
measureP :_ Given such an environment c~, the values z +
e)
serveas transition
probabilities
for the Markov chain called random walk in random environment. This walk moves onZd
and is for fixedstarting point
x EZd
defined on thesample
space(Zd)N
endowed with the so-calledquenched
measure which satisfiesfor all e E
Zd with lei == 1
andall n >
0. The so-called annealed measuresPx,
x EZd,
are then defined as the semi-directproducts Px :
= P xPx,w
on Q x The
corresponding expectations
are denotedby Ex,w
andEx, respectively.
While the one-dimensional case is very well understood
(see
e.g.[7, 4,3]
and the referencestherein)
there are still many openquestions
for
higher
dimensions.For d >
2 this model has been introducedby
Kalikow
[5]
and hassubsequently
been studied in[6]
and[ 1 ] and,
as mentionedabove, recently
in[9-11 ]
and[ 13].
For the
study
oflarge
deviations ofX n / n
undertypical
measuresPo,úJ
in
general dimension,
one introduces in[13] 0,
y ERd
andc~ E
~2,
the transformof the first passage time
H(y)
:= 0:Xn
=[y]}
of the walkthrough
the latticepoint [y]
EZd,
which is closest to y. It turns out thate~, (0,
ny,c~) decays exponentially
as n - oo with a deterministic rate the so-calledLyapounov exponent,
whichdepends
on thedirection. The next theorem shows that even more holds.
THEOREM A
(Point-to-point Lyapounov exponents
andshape
theo-rem
[13,
Theorem A andProposition 3]. -
There exists a continuousfunction a:[O,oo) [0,(0), (h, x)
H which is concaveincreasing in h
andhomogeneous
and convex in x, and a setSZl
ç; S2of full
P-measure such thatthe following
holds: For all03BB
0 and allsequences with ym E
Rd
and|ym |
- oo,Furthermore,
The law of
large
numbers derived in[ 11 ]
does not make any use of theseLyapounov exponents
but of a condition which hasalready
beenintroduced
by
Kalikow in[5].
For theprecise
definition of this condition consider strict subsets U ofZd
and denote the exteriorboundary
of Uby
On U U a U one defines an
auxiliary
Markov chain with transitionprobability
740 M.P.W. ZERNER / Ann. Inst. Henri Poincare 36 (2000) 737-748
where is the exit time from U. For f E Kalikow’s condition relative to f is the
requirement
thatwhere U runs over all connected strict subsets of
tld containing
0. Forsufficient conditions which
imply
Kalikow’s condition and are easier to check see[5,
pp.759-760]
and[11, Proposition 2.4].
The limit theoremsof
[ 11 ]
and[10]
assume Kalikow’s condition andrely
on thefollowing
renewal structure. For fixed f E one constructs successive times
1,
which have theproperty
thatPo-a. s.
For details we
again
refer to[ 11 ] . Although
these times are notstopping
times with
respect
to the canonical filtration ofthey
are very useful as is shownby
thefollowing
theorem. Here D :=inf{n ~
0:Xn .
.~Xo . .~ {
is the first time the walk reaches a levelstrictly
belowits
starting
level.THEOREM B
(Renewal
structure[11, Proposition 1.2, Corollary 1.5,
Theorem
2.3], [10,
Lemma1.2]). -
Assume Kalikow’s condition relativeto some .~ E
II~d B { 0 { .
ThenPo -a. s.,
0 = : to tl t2 ... tk ... and underP0, (X03C41, 03C41), (X t2 -
t2 -03C41),
..., tk+1 - ...are
independent
variables.Furthermore, Po[D
=00]
> 0 and(X t2 -
t2 -
tl ),
..., tk+1 -tk)
are distributed underPo
astl )
underPo [ - ~ D
=00].
Moreover
underPo[ . D
=00],
t{ hasfinite expectation
andhas some ,
finite exponential
moment.It follows from this result that under Kalikow’s condition relative to .~
the walk has a
non-vanishing velocity
as stated in the next theorem.THEOREM C
(Law
oflarge
numbers[ 1 l,
Theorem2.3], [ 10, Propo-
sition
1.6]). -
Assume Kalikow’s condition relative to some .~ EII~d B {0{.
Then
Po -a. s.,
Moreover, ao(x)
= 0if and only if x
= tvfor
some t~
0.In the
following
we shallonly
need theif part
of the last statement of TheoremC,
which is the easierpart.
It follows from the fact that in the transient casePo-a. s.
- 0 as n - oo which is aconsequence of
( 1 )
and the Borel-Cantelli lemma.However,
if one uses the fullequivalence given
in the last statement of Theorem C then under Kalikow’s condition the OthLyapounov exponent
ao characterizes at least the direction
I
of thevelocity
v . If oneadditionally
assumes the so-callednestling property,
which is acentering
condition for the
environment,
see[13],
then it follows from[13,
Section
5,
Remark3]
and[10, Proposition 5.10]
that v is characterizedby
where is the
right-hand
side derivative ofaÀ(v)
withrespect
to À at ~ = 0.In the
present
note we combine the twoapproaches
of[ 11 ]
and[ 13]
without
assuming
thenestling property.
We assume Kalikow’s conditiononly
and derive a different formula for theasymptotic velocity v
in termsof the
Lyapounov exponents.
To thisend,
we first use thepoint-to-point exponents
aA to define some newexponents
which could be calledpoint-to-hyperplane exponents.
This name will bejustified
in Lemma 2.DEFINITION. - For
~, >
0 and f E setOur main result is the
following.
THEOREM 1. - Assume that the set
of l
E relative to whichKalikow’s condition holds is not empty. Then on this set the
right
handside derivative
exists,
is smooth in.~,
andsatisfies
theidentity
where L’ is the
asymptotic velocity defined in
Theorem C.742 M.P.W. ZERNER / Ann. Inst. Henri Poincare 36 (2000) 737-748 1. POINT-TO-HYPERPLANE EXPONENTS
In this section we list some basic
properties
of andjustify
itsname
point-to-hyperplane exponent by showing
thatYÀ (f)
isessentially
the
Laplace
transform of thehyperplane-hitting
timeLEMMA 2
(Point-to-hyperplane exponents). -
For all .~ E thefunction y. (.~) : [0, oo)
---+[0,(0),
~, r-+ y~,(.~)
is continuous and concaveincreasing. Furthermore,
where
II
.112
is the Euclidean norm.Moreover,
on a setQ2
ç; S2of full
P-measure
for
all 03BB > 0 and all l EProof -
The map h r-+ is concaveincreasing
since it is the infimum of the concaveincreasing
functions h r-+c~(~),
see Theorem A.Hence the
only possible
location ofdiscontinuity
ofYÀ (f)
could beÀ = 0.
However,
since is also continuous inÀ, y.(f)
is upper semicontinuous and thus continuous also in h = 0. The firstpart
of(7)
follows from
(2)
since for all À > 0 and x with;c - ~ ~ 1,
The second
part
is a consequence of thehomogeneity
of Theproof
of(8)
is similar to theproof
of[8, Corollary 1.9]
and of[12, Corollary 16]. Indeed,
observe that for any m and x EI~d
withjc’ ~ ~ 1,
and therefore
for m > Consequently,
for fixedc~ E
Q2
:=S21 (see
TheoremA),
forany h
> 0 andany £
E dueto (1),
which proves one
inequality
of(8).
For the reverse
inequality
setThe
complement
ofDm
contains theorigin
and is finite since h > 0.Denote
by Hm
the first time the walk visits the interiorboundary
of
Dm (see (3)).
Since the walk with start at theorigin
must pass theboundary
before it can hit thehalfspace {x: jc - ~ ~ m }
we findthat
for some
maximizing (see Fig. 1 ). Because #~(Dcm)
grows
only polynomially
in m weget
The term in
(9)
vanishes due to( 1 )
and the factthat
isbounded. The term in
(10)
islarger
thanIndeed,
since xm EDm
we are in one of the
following
cases:or due to
(2)
744 M.P.W. ZERNER / Ann. Inst. Henri Poincare 36 (2000) 737-748
Fig.
1. Sketch for theproof
of Lemma 2. The box is chosenlarge enough
toinclude the random contour of
~(0,
x,c~)
that touches thehyperplane x .f
= m.This proves
(8).
2. PROOF OF THEOREM 1
The
following
lemma holdsregardless
of Kalikow’s condition.LEMMA 3. - For all f E
Po-a.s.,
Proof -
Since03B3.(l)
is concave in 03BBby
Lemma 2 it follows from the firstpart
of(7)
thaty~+(2) >
0. Now fix some co EQ2 (see
Lemma2).
We first show that
Pick some 0 y
Vo+(~)’
Then there are someho
> 0 and 8 > 0 withwhere we used 0. Observe that for any
positive integer
mDue to
(8)
this is forlarge m
less thanSince this is summable in m due to
(13)
it follows from the Borel-Cantelli lemma that the left hand side of(12)
isPo,w-a.s.
at least y.Letting
y
~’
proves( 12).
For theproof
of( 11 )
wedistinguish
two cases.If the sequence E
N,
is bounded fromabove, ( 11 )
is obvious. If the sequence is unbounded from above then the sequencetends to
infinity
as n - oo. Observe thatTun (~) ~ ~
for all n. Hence theleft hand side of
( 11 )
is less thandue to
(12).
HereLuJ
denotes thelargest integer
less than orequal
to u. D
Proof of
Theorem 1. - Fix some l E relative to which Ka- likow’s condition holds. For theproof
of(5),
thanks to Lemma 3 weonly
need to derive the
inequality opposite
to the one in( 11 ).
To this end wefirst estimate yo, which is defined in terms of
quenched exponents,
from746 M.P.W. ZERNER / Ann. Inst. Henri Poincare 36 (2000) 737-748
below
by
some annealedexponent
as follows. Weproceed
as in the firstpart
of theproof
of(8),
but this time we useL 1 (I~)
convergence in( 1 )
and Jensen’s
inequality
toget
for all h >0,
Now fix some 8 > 0 and set
Then for À > 0
and m 0,
Recall from Theorem B that
E0[exp(cX03C41 . f)]
oo for some c > 0.Using
thepart
of Theorem Bconcerning
the increments ofX03C4i
it followsfrom a Cramer
type argument (see,
e.g.[2, Chapter 2.2.1])
thatAm decays exponentially
as m ---+ oo with a ratedepending only
on 8 but not on /LOn the other
hand,
we can use thepart
of Theorem Bconcerning
theincrements
of il
to obtain for1,
Thus
decays exponentially
oo likeAm
but with a rate thattends to 0
if ~ B
0. Hence there are someho(s)
> 0 and some mo such that for all 003BB ho (s) and m
mo .Consequently,
weget by ( 14)
and( 15)
for all 0ho (s) ,
Since is concave
increasing
and continuous in hby
Lemma 2 wecan use the fact that
ao(v)
=0,
see TheoremC,
whichimplies
=0,
and
( 16)
toget
Note that
by
theCauchy-Schwarz inequality
the numerator in the lastexpression
is concavein À,
too, and vanishes for À = 0. Therefore the lastexpression equals
Now we let 8
B
0 and use Theorem C to arrive at the conclusion thatyo+ (.~ ) > 1/(~ ’ .~ )
asdesired,
which proves(5).
For the
proof
of(6),
that the set of directions relative to which Kalikow’s condition holds is open, cf.[10,
afterEq. (1.7)]. Indeed,
ifKalikow’s condition holds relative to f with 8f >
0,
then it is also fulfilled for f + h with h EJRd if h ~ ~.~
because ofThus
(5)
is also valid in aneighborhood
Thereforey«+(~)
=1/(~’ ~)
is smooth in f with
for all i =
1, ... , d
which proves(6).
Remark. - In one dimension Kalikow’s condition relative to f > 0 is
equivalent
to the existence of anon-vanishing velocity
of the walk intopositive direction,
cf.[ 11,
Remark2.5].
In order to evaluate formula(6)
for the
velocity
in this situation observe that it follows from theergodic
theorem that for f > 0 and
A ~ 0,
(cf. [12, (39)]).
Recall that we denoteby H (1)
thefirst-passage
timethrough
1. SinceYÀ(f)
= weget
748 M.P.W. ZERNER / Ann. Inst. Henri Poincare 36 (2000) 737-748
which is not very
explicit
but still the correctexpression
for thevelocity
v since
Eo,w[H(n)]/n
converges P-a.s. due toergodicity
toEo[77(l)].
ACKNOWLEDGEMENT
I thank Professor A.-S. Sznitman for
drawing
my attention to thequestion
answered in Theorem 1.REFERENCES
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