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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

o

6 (2000), p. 737-748

<http://www.numdam.org/item?id=AIHPB_2000__36_6_737_0>

© Gauthier-Villars, 2000, tous droits réservés.

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(2)

Velocity and Lyapounov exponents of

some

random 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 environment

satisfying

Kalikow’s condition in terms of the

Lyapounov exponents

which have

previously

been used in the context of

large

deviations. © 2000

Editions scientifiques

et médicales Elsevier SAS Key words: Random walk in random environment, Law of large numbers, Lyapounov exponents

AMS classification: 60K37, 60F 15

RESUME. - Nous

exprimons

la vitesse

asymptotique

des marches aleatoires en environnement aleatoire

qui

verifient la condition de Kalikow en function des

exposants

de

Lyapounov qui

ont

dej a

ete utilises dans le contexte des

grandes

deviations. @ 2000

Editions scientifiques

et

médicales Elsevier SAS

1 Current address: Department of Electrical Engineering, Technion, Haifa 32000, Israel.

E-mail: zemer@ee.technion.ac.il.

(3)

738 M.P W. ZERNER / Ann. Inst. Henri Poincare 36 (2000) 737-748

0.

INTRODUCTION

AND RESULT

There was

recently

some progress in

understanding

the

asymptotic

behavior of multidimensional random walks in random environment

(RWRE),

see

[9-11 ]

and

[13].

On the one

hand,

in

[ 11 ]

certain renewal times are

constructed,

which enable one to prove a law of

large

numbers

(see [ 11 ] )

and a central limit theorem for the walk

(see [10]).

This

approach

leads to an

expression

for the

asymptotic velocity

of the random walk in terms of these renewal times. On the other

hand,

one uses in

[ 13]

certain

Lyapounov exponents

to derive the rate function for some

large

deviation

principle

for the walk. The purpose of the

present

note is to link these two different

concepts by giving

an alternative

expression

for

the

asymptotic velocity

of the walk in terms of the

Lyapounov exponents.

Let us

present

the

precise

model and recall some of the results mentioned above. We

assign

to the lattice sites z E

tld (d &#x3E; 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 are

uniformly

bounded away from 0. That

is,

we assume that there is some fixed K &#x3E; 0 such that IL is

supported

on the set

PK

of 2d -vectors with

K and

~e pee)

= 1. The random variables

w(z,

z +

e)

can then

be realized as the canonical

projections

on the

product

space Q :=

~

endowed with the canonical

product cr -algebra

and the

product

measure

P :_ Given such an environment c~, the values z +

e)

serve

as transition

probabilities

for the Markov chain called random walk in random environment. This walk moves on

Zd

and is for fixed

starting point

x E

Zd

defined on the

sample

space

(Zd)N

endowed with the so-called

quenched

measure which satisfies

for all e E

Zd with lei == 1

and

all n &#x3E;

0. The so-called annealed measures

Px,

x E

Zd,

are then defined as the semi-direct

products Px :

= P x

Px,w

on Q x The

corresponding expectations

are denoted

by Ex,w

and

Ex, respectively.

While the one-dimensional case is very well understood

(see

e.g.

[7, 4,3]

and the references

therein)

there are still many open

questions

for

higher

dimensions.

For d &#x3E;

2 this model has been introduced

by

Kalikow

[5]

and has

subsequently

been studied in

[6]

and

[ 1 ] and,

as mentioned

above, recently

in

[9-11 ]

and

[ 13].

(4)

For the

study

of

large

deviations of

X n / n

under

typical

measures

Po,úJ

in

general dimension,

one introduces in

[13] 0,

y E

Rd

and

c~ E

~2,

the transform

of the first passage time

H(y)

:= 0:

Xn

=

[y]}

of the walk

through

the lattice

point [y]

E

Zd,

which is closest to y. It turns out that

e~, (0,

ny,

c~) decays exponentially

as n - oo with a deterministic rate the so-called

Lyapounov exponent,

which

depends

on the

direction. The next theorem shows that even more holds.

THEOREM A

(Point-to-point Lyapounov exponents

and

shape

theo-

rem

[13,

Theorem A and

Proposition 3]. -

There exists a continuous

function a:[O,oo) [0,(0), (h, x)

H which is concave

increasing in h

and

homogeneous

and convex in x, and a set

SZl

ç; S2

of full

P-measure such that

the following

holds: For all

03BB

0 and all

sequences with ym E

Rd

and

|ym |

- oo,

Furthermore,

The law of

large

numbers derived in

[ 11 ]

does not make any use of these

Lyapounov exponents

but of a condition which has

already

been

introduced

by

Kalikow in

[5].

For the

precise

definition of this condition consider strict subsets U of

Zd

and denote the exterior

boundary

of U

by

On U U a U one defines an

auxiliary

Markov chain with transition

probability

(5)

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

that

where U runs over all connected strict subsets of

tld containing

0. For

sufficient conditions which

imply

Kalikow’s condition and are easier to check see

[5,

pp.

759-760]

and

[11, Proposition 2.4].

The limit theorems

of

[ 11 ]

and

[10]

assume Kalikow’s condition and

rely

on the

following

renewal structure. For fixed f E one constructs successive times

1,

which have the

property

that

Po-a. s.

For details we

again

refer to

[ 11 ] . Although

these times are not

stopping

times with

respect

to the canonical filtration of

they

are very useful as is shown

by

the

following

theorem. Here D :=

inf{n ~

0:

Xn .

.~

Xo . .~ {

is the first time the walk reaches a level

strictly

below

its

starting

level.

THEOREM B

(Renewal

structure

[11, Proposition 1.2, Corollary 1.5,

Theorem

2.3], [10,

Lemma

1.2]). -

Assume Kalikow’s condition relative

to some .~ E

II~d B { 0 { .

Then

Po -a. s.,

0 = : to tl t2 ... tk ... and under

P0, (X03C41, 03C41), (X t2 -

t2 -

03C41),

..., tk+1 - ...

are

independent

variables.

Furthermore, Po[D

=

00]

&#x3E; 0 and

(X t2 -

t2 -

tl ),

..., tk+1 -

tk)

are distributed under

Po

as

tl )

under

Po [ - ~ D

=

00].

Moreover

under

Po[ . D

=

00],

t{ has

finite expectation

and

has 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

of

large

numbers

[ 1 l,

Theorem

2.3], [ 10, Propo-

sition

1.6]). -

Assume Kalikow’s condition relative to some .~ E

II~d B {0{.

Then

Po -a. s.,

Moreover, ao(x)

= 0

if and only if x

= tv

for

some t

~

0.

(6)

In the

following

we shall

only

need the

if part

of the last statement of Theorem

C,

which is the easier

part.

It follows from the fact that in the transient case

Po-a. s.

- 0 as n - oo which is a

consequence of

( 1 )

and the Borel-Cantelli lemma.

However,

if one uses the full

equivalence given

in the last statement of Theorem C then under Kalikow’s condition the Oth

Lyapounov exponent

ao characterizes at least the direction

I

of the

velocity

v . If one

additionally

assumes the so-called

nestling property,

which is a

centering

condition for the

environment,

see

[13],

then it follows from

[13,

Section

5,

Remark

3]

and

[10, Proposition 5.10]

that v is characterized

by

where is the

right-hand

side derivative of

aÀ(v)

with

respect

to À at ~ = 0.

In the

present

note we combine the two

approaches

of

[ 11 ]

and

[ 13]

without

assuming

the

nestling property.

We assume Kalikow’s condition

only

and derive a different formula for the

asymptotic velocity v

in terms

of the

Lyapounov exponents.

To this

end,

we first use the

point-to-point exponents

aA to define some new

exponents

which could be called

point-to-hyperplane exponents.

This name will be

justified

in Lemma 2.

DEFINITION. - For

~, &#x3E;

0 and f E set

Our main result is the

following.

THEOREM 1. - Assume that the set

of l

E relative to which

Kalikow’s condition holds is not empty. Then on this set the

right

hand

side derivative

exists,

is smooth in

.~,

and

satisfies

the

identity

where L’ is the

asymptotic velocity defined in

Theorem C.

(7)

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 and

justify

its

name

point-to-hyperplane exponent by showing

that

YÀ (f)

is

essentially

the

Laplace

transform of the

hyperplane-hitting

time

LEMMA 2

(Point-to-hyperplane exponents). -

For all .~ E the

function y. (.~) : [0, oo)

---+

[0,(0),

~, r-+ y~,

(.~)

is continuous and concave

increasing. Furthermore,

where

II

.

112

is the Euclidean norm.

Moreover,

on a set

Q2

ç; S2

of full

P-measure

for

all 03BB &#x3E; 0 and all l E

Proof -

The map h r-+ is concave

increasing

since it is the infimum of the concave

increasing

functions h r-+

c~(~),

see Theorem A.

Hence the

only possible

location of

discontinuity

of

YÀ (f)

could be

À = 0.

However,

since is also continuous in

À, y.(f)

is upper semicontinuous and thus continuous also in h = 0. The first

part

of

(7)

follows from

(2)

since for all À &#x3E; 0 and x with

;c - ~ ~ 1,

The second

part

is a consequence of the

homogeneity

of The

proof

of

(8)

is similar to the

proof

of

[8, Corollary 1.9]

and of

[12, Corollary 16]. Indeed,

observe that for any m and x E

I~d

with

jc’ ~ ~ 1,

and therefore

for m &#x3E; Consequently,

for fixed

c~ E

Q2

:=

S21 (see

Theorem

A),

for

any h

&#x3E; 0 and

any £

E due

to (1),

(8)

which proves one

inequality

of

(8).

For the reverse

inequality

set

The

complement

of

Dm

contains the

origin

and is finite since h &#x3E; 0.

Denote

by Hm

the first time the walk visits the interior

boundary

of

Dm (see (3)).

Since the walk with start at the

origin

must pass the

boundary

before it can hit the

halfspace {x: jc - ~ ~ m }

we find

that

for some

maximizing (see Fig. 1 ). Because #~(Dcm)

grows

only polynomially

in m we

get

The term in

(9)

vanishes due to

( 1 )

and the fact

that

is

bounded. The term in

(10)

is

larger

than

Indeed,

since xm E

Dm

we are in one of the

following

cases:

or due to

(2)

(9)

744 M.P.W. ZERNER / Ann. Inst. Henri Poincare 36 (2000) 737-748

Fig.

1. Sketch for the

proof

of Lemma 2. The box is chosen

large enough

to

include the random contour of

~(0,

x,

c~)

that touches the

hyperplane x .f

= m.

This proves

(8).

2. PROOF OF THEOREM 1

The

following

lemma holds

regardless

of Kalikow’s condition.

LEMMA 3. - For all f E

Po-a.s.,

(10)

Proof -

Since

03B3.(l)

is concave in 03BB

by

Lemma 2 it follows from the first

part

of

(7)

that

y~+(2) &#x3E;

0. Now fix some co E

Q2 (see

Lemma

2).

We first show that

Pick some 0 y

Vo+(~)’

Then there are some

ho

&#x3E; 0 and 8 &#x3E; 0 with

where we used 0. Observe that for any

positive integer

m

Due to

(8)

this is for

large m

less than

Since this is summable in m due to

(13)

it follows from the Borel-Cantelli lemma that the left hand side of

(12)

is

Po,w-a.s.

at least y.

Letting

y

~’

proves

( 12).

For the

proof

of

( 11 )

we

distinguish

two cases.

If the sequence E

N,

is bounded from

above, ( 11 )

is obvious. If the sequence is unbounded from above then the sequence

tends to

infinity

as n - oo. Observe that

Tun (~) ~ ~

for all n. Hence the

left hand side of

( 11 )

is less than

due to

(12).

Here

LuJ

denotes the

largest integer

less than or

equal

to u. D

Proof of

Theorem 1. - Fix some l E relative to which Ka- likow’s condition holds. For the

proof

of

(5),

thanks to Lemma 3 we

only

need to derive the

inequality opposite

to the one in

( 11 ).

To this end we

first estimate yo, which is defined in terms of

quenched exponents,

from

(11)

746 M.P.W. ZERNER / Ann. Inst. Henri Poincare 36 (2000) 737-748

below

by

some annealed

exponent

as follows. We

proceed

as in the first

part

of the

proof

of

(8),

but this time we use

L 1 (I~)

convergence in

( 1 )

and Jensen’s

inequality

to

get

for all h &#x3E;

0,

Now fix some 8 &#x3E; 0 and set

Then for À &#x3E; 0

and m 0,

Recall from Theorem B that

E0[exp(cX03C41 . f)]

oo for some c &#x3E; 0.

Using

the

part

of Theorem B

concerning

the increments of

X03C4i

it follows

from a Cramer

type argument (see,

e.g.

[2, Chapter 2.2.1])

that

Am decays exponentially

as m ---+ oo with a rate

depending only

on 8 but not on /L

On the other

hand,

we can use the

part

of Theorem B

concerning

the

increments

of il

to obtain for

1,

Thus

decays exponentially

oo like

Am

but with a rate that

tends to 0

if ~ B

0. Hence there are some

ho(s)

&#x3E; 0 and some mo such that for all 0

03BB ho (s) and m

mo .

Consequently,

we

get by ( 14)

and

( 15)

for all 0

ho (s) ,

Since is concave

increasing

and continuous in h

by

Lemma 2 we

can use the fact that

ao(v)

=

0,

see Theorem

C,

which

implies

=

0,

and

( 16)

to

get

(12)

Note that

by

the

Cauchy-Schwarz inequality

the numerator in the last

expression

is concave

in À,

too, and vanishes for À = 0. Therefore the last

expression equals

Now we let 8

B

0 and use Theorem C to arrive at the conclusion that

yo+ (.~ ) &#x3E; 1/(~ ’ .~ )

as

desired,

which proves

(5).

For the

proof

of

(6),

that the set of directions relative to which Kalikow’s condition holds is open, cf.

[10,

after

Eq. (1.7)]. Indeed,

if

Kalikow’s condition holds relative to f with 8f &#x3E;

0,

then it is also fulfilled for f + h with h E

JRd if h ~ ~.~

because of

Thus

(5)

is also valid in a

neighborhood

Therefore

y«+(~)

=

1/(~’ ~)

is smooth in f with

for all i =

1, ... , d

which proves

(6).

Remark. - In one dimension Kalikow’s condition relative to f &#x3E; 0 is

equivalent

to the existence of a

non-vanishing velocity

of the walk into

positive direction,

cf.

[ 11,

Remark

2.5].

In order to evaluate formula

(6)

for the

velocity

in this situation observe that it follows from the

ergodic

theorem that for f &#x3E; 0 and

A ~ 0,

(cf. [12, (39)]).

Recall that we denote

by H (1)

the

first-passage

time

through

1. Since

YÀ(f)

= we

get

(13)

748 M.P.W. ZERNER / Ann. Inst. Henri Poincare 36 (2000) 737-748

which is not very

explicit

but still the correct

expression

for the

velocity

v since

Eo,w[H(n)]/n

converges P-a.s. due to

ergodicity

to

Eo[77(l)].

ACKNOWLEDGEMENT

I thank Professor A.-S. Sznitman for

drawing

my attention to the

question

answered in Theorem 1.

REFERENCES

[1] Bricmont J., Kupiainen A., Random walks in asymmetric random environments,

Comm. Math. Phys. 142 (1991) 345-420.

[2] Dembo A., Zeitouni O., Large Deviations Techniques, Jones and Bartlett Publishers, Boston, 1993.

[3] Gantert N., Zeitouni O., Large deviations for one-dimensional random walk in a

random environment 2014 a survey, in: Révész P., Tóth B. (Eds.), Random Walks, Bolyai Society Mathematical Studies 9, Budapest, 1999, pp. 127-165.

[4] Hughes B.D., Random Walks and Random Environments, Vol. II, Clarendon Press, Oxford, 1996.

[5] Kalikow S.A., Generalized random walk in a random environment, Ann. Probab. 9 (1981) 753-768.

[6] Lawler G.F., Weak convergence of a random walk in a random environment, Comm.

Math. Phys. 87 (1982) 81-87.

[7] Solomon F., Random walks in random environment, Ann. Probab. 3 (1975) 1-31.

[8] Sznitman A.S., Shape theorem, Lyapounov exponents, and large deviations for Brownian motion in a Poissonian potential, Comm. Pure Appl. Math. 47 (1994) 1655-1688.

[9] Sznitman A.S., Slowdown and neutral pockets for a random walk in random environment, Probab. Theory Related Fields 115 (3) (1999) 287-323.

[10] Sznitman A.S., Slowdown estimates and central limit theorem for random walks in random environment, J. Europ. Math. Soc. (1999), to appear.

[11] Sznitman A.S., Zerner M.P.W., A law of large numbers for random walks in random environment, Ann. Probab. 27 (4) (1999) 1851-1869.

[12] Zerner M.P.W., Directional decay of the Green’s function for a random nonnegative potential on

Zd,

Ann. Appl. Probab. 8 (1) (1998) 246-280.

[13] Zerner M.P.W., Lyapounov exponents and quenched large deviations for multi- dimensional random walks in random environment, Ann. Probab. 26 (4) (1998) 1446-1476.

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