Coincidence of Lyapunov exponents for random walks in weak

35594_600_AOP368.pdf

The Annals of Probability

2008, Vol. 36, No. 4, 1528-1583

DOI:10.1214/00-AOP368

©Institute of Mathematical Statistics, 2008

COINCIDENCE OF LYAPUNOV EXPONENTS FOR RANDOM

WALKS IN WEAK RANDOM POTENTIALS

1

BYMARKUSFLURY

University of Zürich

We investigate the free energy of nearest-neighbor random walks onZd , endowed with a drift along the rst axis and evolving in a nonnegative ran- dom potential given by i.i.d. random variables. Our main result concerns the ballistic regime in dimensionsd4, at which we show that quenched and an- nealed Lyapunov exponents are equal as soon as the strength of the potential is small enough.

1. Introduction and main results.

1.1.Random walk in random potential.LetS=(S(n))nN

0 be a nearest- neighbor random walk on the latticeZ d , with start at the origin and drifthin the direction of the rst axis. We supposeSto be dened on a probability space (,F,P h )and we denote byE h the associated expectation. Such a random process is characterized by the distributions of its nite-step subpaths

S[n]def

=(S(0),...,S(n)), nN. In the nondrifting caseh=0, these distributions are uniform on the nearest- neighbor paths inZ d .Thatis,fornNandx

0the origin, we have

P 0 ?

S[n]=(x

0 ,...,x n )  =  1 2d  n for allx 1 ,...,x n Zd such thatx i Šx iŠ1 =1fori=1,...,n, the probabil- ity being zero elsewhere. The case of a nonvanishing drift is related back to the nondrifting case by means of the density function dP h S[n]

Š1dP

0 S[n] Š1 =exp(h·S 1 (n)) E 0 [exp(h·S 1 (n))],nN,(1.1) withS1 denoting the rst component ofS.

Received October 2006; revised September 2007.

1 Supported by the Swiss National Fund under contract nos. 20-55648.98, 20-63798.00 and 20-

100536/1.

AMS 2000 subject classifications.Primary 60K37; secondary 34D08, 60K35. Key words and phrases.Random walk, random potential, Lyapunov exponents, interacting path potential. 1528

COINCIDENCE OF LYAPUNOV EXPONENTS1529

The random walkSis aMarkov chainwithindependent growths, which means the following. Supposem,n?N 0 and set

S[m,n]

def =(S(m),...,S(n)). Whenx 0 ,...,x n ?Z d fulfillP h [S[n]=(x 0 ,...,x m )]>0, we have P h ?

S[m,n]=(x

m ,...,x n )|S[m]=(x 0 ,...,x m ) ? =P h ?

S[n-m]=(x

m -x m ,...,x n -x m ) ? . We will constantly make use of this property and will simply refer to it as the Markov property(committing a slight abuse of standard terminology). In addition to the influence of the drift, we wantSto underlie the influence of a random potential on the lattice. To this end, letV=(V x ) x?Z dbeafamilyof independent, identically distributed random variables, independent of the random walk itself, with essinfV x =0andEV dx <∞. To avoid trivialities, we also assume P[V x =0]<1. Using the random potentialV, we are now able to introduce path measures for the random walk. Thereby, we distinguish between the so-calledquenchedsetting, where the path measure depends on the concrete realization of the potentialV, and theannealedsetting, where the measure depends on averaged values of the potential only.

Thequenched path potentialis given by

? qu

V,β

(N) def =β N ? n=1 V S(n) ,N?N, where the so-calledinverted temperatureβ≥0 is a parameter for the strength of the potential. Thequenched path measureis defined by means of the density function dQ qu

V,h,β,N

dP hdef =exp(-? qu

V,β

(N)) Z qu

V,h,β,N

,N?N, where the normalization Z qu

V,h,β,Ndef

=E h [exp(-? qu

V,β

(N))],N?N, is called thequenched partition function. The quenched setting defines a discrete- time model for a particle moving in a random medium. Here, the path measure is itself random, the randomness coming from the random environmentV. Under a concrete realization of the path measure, the walker jumps from site to site, thereby trying to stay in regions where the potential takes on small values. The drift, however, implies a restriction in the search for such an “optimal strategy" by imposing a particular direction on the walk.

1530M. FLURY

Theannealed path measureis defined by means of the density function dQ an h,β,N dP hdef =Eexp(-? qu

V,β

(N)) EZ qu

V,h,β,N

,N?N, andEZ qu

V,h,β,N

is called theannealed partition function.While our main interest lies in the quenched setting, the annealed model no longer depends on the realiza- tions of the environment and is thus easier to handle. A walker under the annealed measure finds himself in a similar situation as in the quenched setting. To see this, observe that the quenched potential can be expressed by ? qu

V,β

(N)=β ? x?Z d ? x (N)V x , where ? x (N) def = N ? n=1 1 {S(n)=x} denotes the number of visits to the sitex?Z d by theN-step random walkS[1,N].

Anannealed path potentialis given by

? an β (N) def = ? x?Z d ? anβ (? x (N)), N?N, where ? anβ (t) def =-logEexp(-tβV x ), t?R + ,(1.2) is a nonnegative function which is concave increasing by the Hölder inequality. Now, by the independence assumption onV, it is easily seen that dQ an h,β,N dP h =exp(-? an β (N)) Z an h,β,N , where the normalizing constant Z an h,β,Ndef =E h [exp(-? an β (N))] equals the annealed partition functionEZ qu

V,h,β,N

. By the concavity of? anβ ,the more often the random walk intersects its own path, the smaller the potential? an β . Therefore, on the one hand, it is convenient for the walker to return to places he has already visited, while on the other hand, he is urged to proceed in the direction of the drift. In a similar model in a continuous setting, namely Brownian motion in a Pois- sonian potential, the contrary influence of drift and potential on the long-time be- havior of the walk was first studied by A. S. Sznitman. By means of the pow- erful method of enlargement of obstacles, he established a precise picture in both

COINCIDENCE OF LYAPUNOV EXPONENTS1531

quenchedandannealedsettings(seeChapter5ofhisbook[12]).Amonghisresults there is an accurate description of a phase transition fromlocalizationfor large? (or smallh)todelocalizationfor small?(or largeh). In the delocalized phase, the random walk isballistic, that is, the displacement ofS(N)from the origin grows of orderO(N), while in the localized phase, the walk behavessub-ballistic,that is, the displacement is of ordero(N). The analogous results for the discrete setting have been established by Zerner in [15] and Flury in [7].

1.2.Lyapunov exponents. The above results on the transition from sub-

ballistic to ballistic behavior are based on large deviation principles for the random walk under the path measures and on phase transitions for thequenchedandan- nealed free energies logZ qu

V,h,?,N

and logZ an h,?,N . The free energies are important values for the study of the path measures. The mo- ments of the path potential under these measures, for instance, may be evaluated by differentiating the free energies with respect to the inverted temperature. For a more direct motivation in the context of random branching processes, and a thor- ough study of the one-dimensional case, we refer to [8] by Greven and den Hol- lander. The main subject of the present article is the long-time behavior of the free energies. We first deal with the phase transitions in this behavior, as established in [7]. The associated phase diagrams coincide with the ones for the random walk itself. They are characterized by values from the so-calledpoint-to-hyperplane setting. Forh>0,?0andLN,weset Z qu

V,h,?,Ldef

=   n=1 E h [exp(Š? qu V,? (n));{S 1 (n)=L}], Z an h,?,Ldef =   n=1 E h [exp(Š? an ? (n));{S 1 (n)=L}]. T HEOREMA.For any?0,there are continuous,nonnegative functions m qu (·,?)andm an (·,?)onR + such that m qu (h,?)=Šlim L 1 Llog Z qu

V,h,?,L

, m an (h,?)=Šlim L 1 Llog Z an h,?,L for allh>0,as well as continuous,nonnegative functionsm qu (·,?)andm an (·,?) onR + ,the so-calledquenchedandannealed Lyapunovexponents,such that m qu (h,?)=Šlim N 1 NlogZ qu

V,h,?,N

,

1532M. FLURY

m an (h,β)=-lim

N→∞

1 NlogZ an h,β,N for allh≥0,where the convergence in the quenched setting isP-almost surely and inL 1 (P),and where the limits no longer depend on the realizations ofV.

Moreover,for allh,β≥0,we have

m qu (h,β)=  λ h ,ifm qu (0,β)≥h, λ h -λ

¯hqu

(h,β) ,ifm qu (0,β)¯han (h,β) ,ifm an (0,β)0and¯h an (h,β) >0are determined by m qu (¯h qu (h,β),β)=h-¯h qu (h,β), m an (¯h an (h,β),β)=h-¯h an (h,β) and whereλ hdef =logE 0 [exp(h·S 1 (1))]. TheoremAisprovedin[7] for drifts in arbitrary directions and for a more general annealed potential which we introduce at the beginning of Section2. With regard to the difference in notation, observe that, withe 1 being the first unit vector, m qu (h,β)=λ h -lim n→∞ 1 nlogZ h·e 1 n,ω ,P-a.s., m an (h,β)=λ h -lim n→∞ 1 nlogZ h·e 1 n and that by Corollary C of [7], m qu (¯h,β)=1 α ?λ

¯h·e1

(e 1 )-¯h, m an (¯h,β)=1 β ?λ

¯h·e1

(e 1 )-¯h, where the notation on the right-hand sides is from [7] (with potentialV β ={βV x } and?=? V β ). For the latter equalities, observe also that the fact that the random walk is already stopped at its first entrance into the hyperplane has no effect on the limits in Corollary C of [7] (as will become clear in Section2.1). In accordance with the long-time behavior of the random walk itself, in Theo- remA, we have the following picture for the behavior of the free energies: in the sub-ballistic regime, the walker remains near the origin in the annealed case and in regions with small potential in the quenched case. Therefore, since lim t→∞ ? anβ (t) t=essinfV x =0,

COINCIDENCE OF LYAPUNOV EXPONENTS1533

the contribution from the potential then vanishes whenNbecomes large. What remains is the probability of staying in an only slow-growing region, contributing the value λ h =lim n→∞ 1 nlogE 0 ? exp  h·S 1 (n)  . In the ballistic regime, on the other hand, the walk obeys the drift and dislocates with a nonvanishing velocity. As a consequence, the path potential and the “spa- tial part" of the density for the drift must not be neglected, as they contribute the subtraction termλ

¯hqu

(h,β) , respectivelyλ

¯han

(h,β) , to the corresponding Lyapunov exponent (see [7] for a rigorous interpretation of this last point).

1.3.Main results and preliminaries. Our first new result concerns the simpler

annealed setting. Forh>0, the critical parameterβ anc (h)for the phase transition is given by m an (0,β anc (h))=h, where existence and uniqueness ofβ anc (h)will be explained in Remark2.10of

Section2.1.

T

HEOREMB. (a)For anyh,β≥0,we have

Z an h,β,N ≤exp(-m an (h,β)N), N?N, andm an is continuous onR + ×R + .Moreover,for anyh>0andβ 0 <β anc (h), there existsK h,β 0 <∞such that forβ≤β 0 ,we have Z an h,β,N ≥exp(-m an (h,β)N) K h,β 0 ,N?N. (b)m an is analytic on the open set{(h,β)?(0,∞) 2 :β?=β anc (h)}. Part (a) of TheoremBis established in Section3.1, essentially by subadditivity arguments. The sub-ballistic part in (b) is a straightforward consequence of The- oremA. The more complicated ballistic part is proved in Section3.2by renewal techniques. The next theorem is the main result of this article. It concerns dimensionsd≥4 and nonvanishing driftsh>0. It states that quenched and annealed Lyapunov ex- ponents coincide once the strength of the potential is chosen to be small enough. T HEOREMC.Supposed≥4andh>0.There then existsβ 0 >0such that m qu (h,β)=m an (h,β) for allβ≤β 0 .Moreover,whenV x is essentially bounded,there existsK fr.e. <∞ such that

E|logZ

qu

V,h,β,N

-logZ an h,β,N |≤K fr.e. 

1+β⎷N

for allN?Nandβ≤β 0 .

1534M. FLURY

Coincidence of Lyapunov exponents has been conjectured by Sznitman in [12]. It emerged from the fact that a similar result is true for the much simpler case of directed polymers in random potentials. There, the random walk(S(n)) n?N is re- placed by((ξ(n),n)) n?N ,where(ξ(n)) n?N is a standardd-dimensional walk. The coincidence of quenched and annealed Lyapunov exponents ford≥3andsmall disorder was first proven by Imbrie and Spencer in [10], using cluster expansion techniques, and then by Bolthausen in [2] and Albeverio and Zhou in [1], using martingale techniques. Martingale arguments have also been used in the more re- cent work on directed polymers in [5]and[3]. The situation considered here is much more delicate and, unfortunately, it does not seem possible to implement martingale techniques. We therefore take recourse to different methods, mainly renewal techniques and arguments from Ornstein-

Zernike theory.

The crucial result for the proof of TheoremCis an estimate on the second moment of the quenched partition function. T

HEOREMD.Supposed≥4andh>0.There are thenβ

0 >0and K s.m. <∞such that E(Z qu

V,h,β,N

) 2 ≤K s.m. (Z an h,β,N ) 2 for allN?Nandβ≤β 0 . In order to achieve a heuristic understanding of TheoremD, we consider two in- dependent copiesS 1 =(S 1 (n)) n?N 0 andS 2 =(S 2 (n)) n?N 0 of the random walkS.

Forx?Z

d ,weset ? 1x (N) def = N  n=1 1 {S 1 (n)=x} and? 2 x (N) def = N  n=1 1 {S 2 (n)=x} .

Recall that we haveZ

an h,β,N =EZ qu

V,h,β,N

, by the independence assumption on the potential. In a similar way, and by the independence ofS 1 andS 2 , we obtain E(Z qu

V,h,β,N

) 2 =E h exp  -  x?Z d ? anβ  ? 1 x (N)+? 2 x (N)  , (EZ qu

V,h,β,N

) 2 =E h exp  -  x?Z d ? anβ (? 1 x (N))+? anβ (? 2 x (N))  , whereE h denotes the expectation with respect to the product measureP h ?P h .

With the further notation

?

β,Ndef

=  x?Z d ? anβ (? 1 x (N))+? anβ (? 2 x (N))-? anβ  ? 1 x (N)+? 2 x (N) (1.3)

COINCIDENCE OF LYAPUNOV EXPONENTS1535

and withE an h,β,N the expectation with respect to the annealed product path measure Q an h,β,N ?Q an h,β,N , we thus have E(Z qu

V,h,β,N

) 2 (EZ qu

V,h,β,N

) 2 =E an h,β,N [exp(?

β,N

)].(1.4) Observe further that the only nonvanishing summands in (1.3) are the ones associ- ated to thosex?Z d that are visited by both random walks up to timeN.Fromthe concavity of? anβ , we therefore obtain ?

β,N

≤? anβ (1)  x?R 1 (N)∩R 2 (N) ? 1x (N)+? 2 x (N),(1.5) whereR j (N) def ={S j (n):n=1,...,N}forj=1,2. This finally gives us the fol- lowing picture of the situation: in the ballistic regime, the random walksS 1 andS 2 under the annealed path measure obey the drift and evolve in the direction of the first axis. While they do that, one expects them to move away from each other in the(d-1)-dimensional “vertical" direction as soon as the dimension of the lattice is large enough. The conditiond≥4 appears to be the right one since the “vertical distance" is then transient. As a consequence, the paths ofS 1 and S 2 are supposed to intersect only finitely many times. Forβsmall enough, the right-hand side of (1.4) should then stay bounded asN→∞because of (1.5)and lim

β↓0

? anβ (1)=0. The equality of the Lyapunov exponents, knowing that the quenched free en- ergy is deterministic, is obtained from TheoremDby rather elementary meth- ods. On the other hand, the estimate of the speed of convergence for the free energy has a more complicated derivation, requiring a concentration inequal- ity for the quenched free energy. In this particular model, the usual concen- tration estimate is not sharp enough. For this reason, we replaceZ qu

V,h,β,N

by a modified partition function. Forh>0,β≥0,k?R + andN?N,wede- fine Z qu

V,h,β,k,Ndef

=E h e -β  x?Zd ? x (N)V x (ω) ;  x?Z d ? x (N) 2 ≤kN . The justification for such a replacement is given in the following lemma, which is proved in Section3.3. L

EMMAE.Supposeh>0andβ

0 <β anc (h),and letK h,β 0 be chosen accord- ingtoTheoremB.For anyε<1/K h,β 0 ,there then existsk ε <∞such that EZ qu

V,h,β,k

ε ,N ≥εZ an h,β,N

1536M. FLURY

for allNNand?? 0 . By TheoremA, TheoremDand LemmaE, we now have the means to prove the coincidence of the Lyapunov exponents and to estimate the speed of convergence for the free energy. P ROOF OFTHEOREMC. By the quenched part of TheoremA,wehave lim N 1

NElogZ

qu

V,h,?,N

=Šm qu (h,?) and thusm qu (h,?)?m an (h,?), by Jensen"s inequality. In order to obtain the in- verted estimate, observe that P  lim N 1 NlogZ qu

V,h,?,N

?Šm an (h,?)  ?liminf N P  Z qu

V,h,?,N

?1 2EZ qu

V,h,?,N

 and that the left-hand side is either one or zero since the limit is deterministic (again by TheoremA). The Schwarz inequality now implies EZ qu

V,h,?,N

 1 2 EZ qu

V,h,?,N

+(E(Z qu

V,h,?,N

) 2 ) 1/2 P  Z qu

V,h,?,N

? 1 2 EZ qu

V,h,?,N

 1/2 , which leads to the Paley-Zigmund inequality P  Z qu

V,h,?,N

?1 2EZ qu

V,h,?,N

 ?1 4(EZ qu

V,h,?,N

) 2 E(Z qu

V,h,?,N

) 2 .(1.6)

The lower estimate form

qu (h,?)thus follows from TheoremD. We proceed to the estimate for the speed of convergence. Assume that V def =esssupV x <. We first investigate the modified partition functionZ qu

V,h,?,k,N

.ForNN,let M qu h,?,k,N beamedianoflogZ qu

V,h,?,k,N

, that is, a real numberM qu h,?,k,N with

P[logZ

qu

V,h,?,k,N

M qu h,?,k,N ]? 1 2 andP[logZ qu

V,h,?,k,N

?M qu h,?,k,N ]? 1 2 .

Also, letB

Ndef ={xZ d :x 1 N}and letf:[Š1,1] B N Rbe given by f k (v) def =logE h e

Š?V

 xBN ∞ x (N)v x ;  xB N ∞ x (N) 2 kN forv=(v x ) xB N [Š1,1] B N . We then obviously have logZ qu

V,h,?,k,N

=f k (V x /V) xB N .

COINCIDENCE OF LYAPUNOV EXPONENTS1537

Since the functionfis convex by the Hölder inequality, the setsf -1 ((-∞,a]) fora?Rare also convex. In addition, for anyv,w?[-1,1] B N ,wehave |f k (v)-f k (w)| ≤sup  βV  x?B N ? x |v x -w x |:??N B N 0 with  x?B N ? 2x ≤kN  ≤βV⎷kN  x?B N |v x -w x | 2  1/2 by the Cauchy-Schwarz inequality for sums. This means thatf k is Lipschitz con- tinuous with Lipschitz constant at mostβV⎷ kN. We can thus apply Theorem 6.6 of [13] to obtain the concentration inequality

P[|logZ

qu

V,h,β,k,N

-M qu h,β,k,N |≥t]≤4exp  -t 2

16β

2 V 2 kN  ,t?R + .(1.7)

Next, we will find an estimate for

E|logZ

qu

V,h,β,k,N

-logEZ qu

V,h,β,k,N

| by adding and subtractingM qu h,β,k,N . To this end, observe that (1.7) implies that

E|logZ

qu

V,h,β,k,N

-M qu h,β,k,N |=  ∞ 0

P[|logZ

qu

V,h,β,k,N

-M qu h,β,k,N |>t]dt ≤4  ∞ 0 exp  -t 2

16β

2 kV 2 N  dt(1.8) =8Vβ⎷ kπN.

It remains to find an estimate for|logEZ

qu

V,h,β,k,N

-M qu h,β,k,N |. By the definition of the median and an application of the Markov inequality, we have 1 2 ≤P[logZ qu

V,h,β,k,N

≥M qu h,β,k,N ]≤EZ qu

V,h,β,k,N

e -M qu h,β,k,N and therefore M qu h,β,k,N -logEZ qu

V,h,β,k,N

≤log2.(1.9)

Now, lett

k,Ndef =log( 1 2 EZ qu

V,h,β,k,N

)-M qu h,β,k,N . Since we are looking for an upper bound, we can supposet k,N ≥0. Again from (1.7), we obtain P  Z qu

V,h,β,k,N

≥1 2EZ qu

V,h,β,k,N

 =P[logZ qu

V,h,β,k,N

-M qu h,β,k,N ≥t k,N ] (1.10) ≤4exp  -t 2k,N

16β

2 V 2 kN  .

1538M. FLURY

Moreover, analogously to (1.6), we know that

P ? Z qu

V,h,,k,N

1 2EZ qu

V,h,,k,N

? 1 4(EZ qu

V,h,,k,N

) 2 E(Z qu

V,h,,k,N

) 2 .(1.11) For 0 < c (h)chosen according to TheoremD,and>0andk  according to

TheoremE, we further have

(EZ qu

V,h,,k

 ,N ) 2 E(Z qu

V,h,,k

 ,N ) 2  2 (EZ qu

V,h,,N

) 2 E(Z qu

V,h,,N

) 2  2 K s.m. (1.12) for all 0 andNN. A combination of (1.10), (1.11)and(1.12) then implies that t k  ,N K 1 N for all 0 andNN, where the constantK 1 is given by K 1def =4V ? k  log16K s.m.  2 .

By the definition oft

k  ,N , we therefore have logEZ qu

V,h,,k

 ,N ŠM qu h,,k  ,N K 1

N+log2(1.13)

and thus, as consequence of (1.8), (1.9)and(1.13),

E|logEZ

qu

V,h,,k

 ,N

ŠlogZ

qu

V,h,,k

 ,N |K 2

N+log2(1.14)

for all 0 andNN,whereK 2def =8V k  +K 1 . It remains to transfer (1.14) to the unmodified partition functions. By the trian- gle inequality,|logEZ qu

V,h,,N

ŠlogZ

qu

V,h,,N

|is bounded by logEZ qu

V,h,,N

ŠlogEZ

qu

V,h,,k

 ,N +|logEZ qu

V,h,,k

 ,N

ŠlogZ

qu

V,h,,k

 ,N |

ŠlogZ

qu

V,h,,k

 ,N +logEZ qu

V,h,,k

 ,N

ŠlogEZ

qu

V,h,,k

 ,N +logZ qu

V,h,,N

. In order to handle the last summand in the above formula, recall also that ElogZ qu

V,h,,N

logEZ qu

V,h,,N

, by Jensen"s inequality. From (1.14) and LemmaE, we thus obtain

E|logEZ

qu

V,h,,N

ŠlogZ

qu

V,h,,N

| 2log ? EZ qu

V,h,,N

EZ qu

V,h,,k

 ,N ∞ +2E|logEZ qu

V,h,,k

 ,N

ŠlogZ

qu

V,h,,k

 ,N | Š2log+2K 2

N+2log2

COINCIDENCE OF LYAPUNOV EXPONENTS1539

for allβ≤β 0 andN?N. By settingK fr.e.def =2max{K 2 ,log2-logε}, the proof of TheoremCis completed. For the proof of TheoremB, TheoremDand LemmaE, it remains to consider the annealed setting only. In Section2, we deal with the “point-to-hyperplane" set- ting. That is, under the annealed path measure, we analyze finite pathsS[n]with S 1 (n)=Lfor fixedL?N. In the first two subsections, such paths are approxi- mated by more specific paths, the so-called bridges, and a renewal formalism is found by introducing irreducibility for bridges. In Section2.3, we then prove the existence of a gap between the exponential decay rates of arbitrary and irreducible bridges. In Section3.1, we introduce an analogous renewal formalism for the “fixed- number-of-steps" setting. In Section3.2, the exponential gap from Section2.3is transferred to that setting, implying the crucial part of TheoremB, namely analyt- icity of the Lyapunov exponent in the ballistic regime. With the renewal formalism and the exponential gap, we then also have the means to prove LemmaE,whichis done in Section3.3. Finally, Section4is devoted to the proof of the second-moment estimate in TheoremD. It is based on a local decay estimate for sums of independent random variables and again on the gap between the exponential behaviors of arbitrary and irreducible bridges.

2. Endpoint in given hyperplane.As in the previous section, we consider a

nearest-neighbor random walkS=(S(n)) n?N 0 onZ d , with start at the origin and drifthin the direction of the first axis, defined on a probability space(?,F,P h ). In this section, we investigate the behavior of finite random paths with start at the origin and endpoint in a hyperplane H Ldef ={(ξ 1 ,...,ξ d )?Z d :ξ 1 =L}, exponentially weighted by a nonrandom path potential? β . We therefore allow a more general setting for? β than for the annealed path potential? anβ from Sec- tion1. More precisely, we assume that?:R + →R + is a nonconstant, concave increasing function with lim t→0 ?(t)=?(0)=0(2.1) and lim t→∞ ?(t)=∞,(2.2) lim t→∞ ?(t) t=0.(2.3)

1540M. FLURY

Forβ≥0, we define?

β :R + →R + by ? β (t) def =?(tβ), t?R + .

The function?

β plays the role of the annealed potential? anβ from Section1, with coincidence in the case ?(t)=-logEexp(-tV x ).(2.4) Assumption (2.3) is needed for TheoremAonly. Under (2.4), it corresponds to the assumption essinfV x =0 from the quenched setting, and (2.2) is equivalent to P h [V x =0]<1. The path potential is now defined as in the annealed setting from the previous section: forβ≥0andN,M?N 0 withM≤N,set ? β (N) def =  x?Z d ? β (? x (N)), ? β (M,N) def =  x?Z d ? β (? x (M,N)), where ? x (N) def = N  n=1 1 {S(n)=x} ,? x (M,N) def = N  n=M+1 1 {S(n)=x} denote the number of visits to the sitex?Z d by the random walkS[1,N], respec- tivelyS[M+1,N]. We derive some elementary properties of the path potential? β arising from the assumptions on? β .ForN,M?N 0 withM≤N,let R(N) def ={x?Z d :? x (N)≥1},

R(M,N)

def ={x?Z d :? x (M,N)≥1} denote the sets of sites visited byS[1,N], respectivelyS[M+1,N]. L

EMMA2.1. (a)For anyβ≥0andN?N

0 ,we have ? β (1)?R(N)≤? β (N)≤? β (1)N. (b)For anyβ≥0andM,N?N 0 withM≤N,we have ? β (N)≤? β (M)+? β (M,N).

Moreover,ifω?{R(M)∩R(M,N)=?},we have

? β (N,ω)=? β (M,ω)+? β (M,N,ω).

COINCIDENCE OF LYAPUNOV EXPONENTS1541

(c)For any??0andM 1 ,M 2 ,NN 0 withM 1 M 2 N,we have ? ? (N)?? ? (M 1 ,M 2 ).

Moreover,if≤{R(M

1 )R(M 2 ,N)=},we have ? ? (N,≤)?? ? (M 1 ,≤)+? ? (M 2 ,N,≤). P ROOF. (a) For the lower bound, we use the monotonicity of→ ? to obtain ? ? (N)=  xR(N) → ? (∞ x (N))?  xR(N) → ? (1)=→ ? (1)◦R(N). For the upper estimate, we inductively apply the concavity of→ ? to obtain ? ? (N)=  xZ d → ? (∞ x (N))  xZ d → ? (1)∞ x (N)=→ ? (1)N. (b) By the concavity of→ ? and (2.1), we have ? ? (N)=  xZ d → ?  ∞ x (M)+∞ x (M,N)   xZ d → ? (∞ x (M))+→ ? (∞ x (M,N)) =? ? (M)+? ? (M,N).

For≤{R(M)R(M,N)=},wehave

? ? (N,≤)=  xR(M,≤) → ? (∞ x (N,≤))+  xR(M,N,≤) → ? (∞ x (N,≤)) =  xR(M,≤) → ? (∞ x (M,≤))+  xR(M,N,≤) → ? (∞ x (M,N,≤)) =? ? (M,≤)+? ? (M,N,≤). (c) The monotony of→ ? implies that ? ? (N)=  xZ d → ? (∞ x (N))?  xZ d → ? (∞ x (M 1 ,M 2 ))=? ? (M 1 ,M 2 ).

For≤{R(M

1 )R(M 2 ,N)=},wehave ? ? (N,≤)?  xR(M 1 ,≤) → ? (∞ x (N,≤))+  xR(M 2 ,N,≤) → ? (∞ x (N,≤)) ?  xR(M 1 ,≤) → ? (∞ x (M 1 ,≤))+  xR(M 2 ,N,≤) → ? (∞ x (M 2 ,N,≤)) =? ? (M 1 ,≤)+? ? (M 2 ,N,≤).∅

1542M. FLURY

2.1.Masses for paths and bridges. We start with a few comments on the

process of the first components ofS,thatis, S 1def =(S 1 (n)) nN 0 .

The processS

1 is itself a random walk onZ, again with independent increments and drift in the positive direction. It can be expressed by S 1 (n)= n  m=1  S 1 (m)ŠS 1 (mŠ1) fornN, where the random variables(S 1 (m)ŠS 1 (mŠ1)) mN are independent and identically distributed. SinceE h S 1 (n) >0forh>0, we then have P h [S 1 (n)asn]=1, by the strong law of large numbers. This convergence property, as we show next, implies transience to the processS 1 , which is here equivalent to the fact that the probability (h) def =P h [S 1 (n) >0forallnN](2.5) is strictly greater than zero: forh>0, and withmN 0 denoting the last time the random walkS 1 is inLN 0 ,wehave 1=   m=0 P h [S 1 (m)=L,S 1 (m+n) > Lfor allnN] =   m=0 P h [S 1 (n)=L]P h [S 1 (n) >0forallnN] and therefore   m=0 P h [S 1 (m)=L]=1 (h)<.(2.6) Observe, in particular, that the left-hand side of (2.6) does not depend onLN 0 . R

EMARK2.2. Foreveryh>0andLN,wehave

P h [H ŠL <] =e

Š2Lh

,(2.7) where the stopping time H

ŠLdef

=inf{nN:S 1 (n)=ŠL} denotes the time of the random walk"s first visit to the hyperplaneH ŠL .

COINCIDENCE OF LYAPUNOV EXPONENTS1543

PROOF. The Markov property implies that

P h [H -L <∞] =P h [H -1 <∞] L . We can thus restrict our attention to the caseL=1. Also by the Markov property, we have P h [H -1 <∞] =P h [S 1 (1)=-1]+P h [S 1 (1)=0]P h [H -1 <∞] +P h [S 1 (1)=1]P h [H -1 <∞] 2 , which is a quadratic equation in the variableP h [H -1 <∞], with solutions 1 and P h [S 1 (1)=-1]/P[S 1 (1)=1]. To find the correct one among these two solutions, observe that

α(h)=P

h [S 1 (1)=1]P h [H -1 =∞], again by the Markov property. We thus have P h [H -1 <∞] =1-α(h) P h [S 1 (1)=1]<1 sinceα(h) >0by(2.6), and therefore P h [H -1 <∞] =P h [S 1 (1)=-1] P[S 1 (1)=1]=e -2h , where, at the second step, the concrete definition ofP h in Section1is used.?

We now return to random walks onZ

d . In the present setting, for anyh>0,

β≥0andL?N, the counterpart of

Z an h,β,L from the point-to-hyperplane setting of Section1is given by G h,β (L) def = ∞  N=1 E h [exp(-? β (N));{S 1 (N)=L}].(2.8) R EMARK2.3. The drifth>0 ensures that (2.8) is finite. More exactly, for anyh>0,β≥0andL?N,wehave P h [S(1)=1] L e -? β (1)L ≤G h,β (L)≤1

α(h)e

-? β (1)L . P

ROOF. It is plain that

G h,β (L)≥E h [exp(-? β (L));{S 1 (L)=L}], which implies the lower estimate. Forω?{S 1 (N)=L}, we obviously haveL≤ ?R(N,ω). From Lemma2.1(a), we thus obtain E h [exp(-? β (L));{S 1 (N)=L}] ≤e -? β (1)L P h [S 1 (N)=L]

1544M. FLURY

for allL?N, from which the lower estimate now follows by (2.6).

We are interested in the exponential behavior of

G h,β (L)asL→∞.Thisbe- havior is easier to study when the expectations in (2.8) are restricted to so-called bridges. D

EFINITION2.4. Supposeω??andN,M?N

0 withM≤N. The finite pathS[M,N](ω)is called abridgeif S 1 (M,ω) < S 1 (n,ω)≤S 1 (N,ω) is valid forn=M+1,...,N. In that case, thespanof the bridgeS[M,N](ω)is given byS 1 (N,ω)-S 1 (M,ω).

ForL?NandM,N?N

0 withM≤N,wedefine br(L;N) def ={S[N]is a bridge of spanL}, br(L;M,N) def ={S[M,N]is a bridge of spanL}.

Forh,β≥0andL?N, we further define

b h,β (L;N) def =E h [exp(-? β (N));br(L;N)], B h,β (L) def = ∞  N=1 b h,β (L). R

EMARK2.5. Foranyh,β≥0andL?N,wehave

P h [S(1)=1] L e -? β (1)L ≤B h,β (L)≤e -? β (1)L . P ROOF. The lower estimate is proved as in Remark2.3. For the upper esti- mate, observe that b h,β (L;N)≤b h,0 (L;N)e -? β (1)L forL≤Nby Lemma2.1(a). By the Markov property, we furthermore have b h,0 (L;N)=P h [0≤S 1 (n) < S 1 (N)=Lfor 0≤nCOINCIDENCE OF LYAPUNOV EXPONENTS1545 The well-known subadditive limit lemma (see, e.g., page 9 of [11]) states the following. Let(a n ) n?N be a sequence of real numbers with thesubadditivity prop- erty a n+m ≤a n +a m for allm,n?N.Wethenhave lim n→∞ a n n=inf ł a n n:n?N - . We want to apply the subadditive limit lemma to(-log B h,β ) L?N . The subad- ditivity property is a consequence of the following lemma. L

EMMA2.6.Forh,β≥0andL

1 ,L 2 ?N,we have B h,β (L 1 +L 2 )≥B h,β (L 1 )B h,β (L 2 ).

Moreover,for nonvanishing drifth>0,we have

B h,β (L 1 +L 2 )≤1

α(h)

B h,β (L 1 )B h,β (L 2 ). P ROOF. In order to obtain the lower estimate, observe that br(L 1 +L 2 ;N)? N-1 M=1 br(L 1 ;M)∩br(L 2 ;M,N), where the right-hand side is a union of disjoint sets and whereMdenotes the time of the unique visit ofS[N]to the hyperplaneH L 1 .Forω?br(L 1 ;M)∩ br(L 2 ;M,N), we further have ? β (N,ω)=? β (M,ω)+? β (M,N,ω) by Lemma2.1(b). By splitting over all possible values ofMand using the Markov property to renew the random walkSat that time, we obtain B h,β (L 1 +L 2 )≥ ∞ 

N=1N-1

 M=1 E h  e -? β (M) 1 br(L 1 ;M) e -? β (M,N) 1 br(L 2 ;M,N)  = ∞ 

N=1N-1

 M=1 b h,β (L 1 ;M)b h,β (L 2 ;N-M) = B h,β (L 1 )B h,β (L 2 ), which proves the lower estimate of the lemma.

1546M. FLURY

The upper estimate is shown in a similar way. The event br(L 1 +L 2 ;N)is contained in the union N-1  M 1 =1N-1  M 2 =M 1 br(L 1 ;M 1 )∩{S 1 (M 2 )=L 1 }∩br(L 2 ;M 2 ,N), in whichM 1 ,forω?br(L 1 +L 2 ;N), may be chosen as the time of the first visit ofS[N](ω)to the hyperplaneH L 1 ,andwhereM 2 stands for the time of the last visit toH L 1 . By splitting over all possible values ofM 1 andM 2 , and using Lemma2.1(c) and the Markov property to renew the random walk at these times, we obtain B h,β (L 1 +L 2 ) ≤ ∞ 

N=1N-1

 M 1 =1N-1  M 2 =M 1 E h ? e -? β (M 1 ) 1 br(L 1 ;M 1 ) 1 {S 1 (M 2 )=L 1 } ×e ? β (M 2 ,N) 1 br(L 2 ;M 2 ,N)  = ∞ 

N=1N-1

 M 1 =1N-1  M 2 =M 1 b h,β (L 1 ;M 1 )P h [S 1 (M 2 -M 1 )=0]b h,β (L 2 ;N-M 2 ) = B h,β (L 1 )B h,β (L 2 ) ∞  m=0 P h [S 1 (m)=0], from which the upper estimate of the lemma follows by (2.6). P

ROPOSITION2.7.For anyh,β≥0,the mass

m B (h,β) def =lim

L→∞

-logB h,β (L) L of B h,β exists in[? β (1),∞),is continuous as function onR + ×R + and satises B h,β (L)≤e -m B (h,β)L (2.9) for allL?N.Moreover,for nonvanishingh>0,we have B h,β (L)≥α(h)e -m B (h,β)L (2.10) for allL?N. P

ROOF. The sequence(-logB

h,β (L)) L?N is subadditive by Lemma2.6.The subadditive limit lemma thus yields m B (h,β)=inf  -logB h,β (L) L:L?N  ?[-∞,∞),

COINCIDENCE OF LYAPUNOV EXPONENTS1547

which includes the existence of the limit and implies the estimate in (2.9). The lower bound? β (1)for the massm B (h,β)follows from Remark2.3. By the above expression, as an infimum of continuous functions, the mass m B is upper semicontinuous. In order to obtain lower semicontinuity, it is convenient to consider ? B

λ,β

(L) def =  → N=1 E 0 ≥ exp ≤ Š? β (N)ŠλN ? ;br(L;N) ? forλ?0andLN. By the definition ofP h in (1.1), we have ?m B (λ h ,β) def =lim L

Šlog

? B λ h ,β (L) L= m B (h,β)+h, whereλ h =logE 0 [exp(h·S 1 (1))]. It consequently suffices to show that?m B is lower semicontinuous. To see this, observe that for any fixedNN,themapβ? β (N)inherits the concavity of?. By the Hölder inequality, for any(λ,β),(λ  ,β  )R + ×R + and t[0,1], we thus have ? B tλ+(1Št)λ  ,tβ+(1Št)β (L)   → N=1 E 0 ≥ e

Št?

β (N)Š(1Št)? β (N)ŠtλŠ(1Št)λ  ;br(L;N) ?   → N=1 E 0 ≥ e Š? β (N)ŠλN ;br(L;N) ? t E 0 ≥ e Š? β (N)Šλ  N ;br(L;N) ? (1Št)  ? B

λ,β

(L) t ? B λ  ,β (L) (1Št) . Therefore, for any fixedLN, the negative logarithm of ? B

λ,β

(L)is concave as a function of(λ,β).Themass ?m B inherits this concavity and, as a consequence, is lower semicontinuous. It remains to prove (2.10). To this end, we consider ? B h,β (L) def =α(h) Š1 B h,β (L), LN.

The sequence(log

? B h,β (L)) LN is subadditive by Lemma2.6. As a consequence, we have

α(h)

Š1 B h,β (L)= ? B h,β (L)?e

Š?m

B (h,β)L for allLN, by the subadditivity limit lemma. Thereby,?m B (h,β)denotes the mass of ? B h,β and is given by ?m B (h,β) def =lim L

Šlog

? B h,β (L) L =lim L

ŠlogB

h,β (L)

L+logα(h)L=

m B (h,β).

1548M. FLURY

By means of the following lemma, the results onB

h,β in Proposition2.7can be transfered to G h,β . L

EMMA2.8.For anyh>0andβ?0,we have

α(h)

2 G h,β (L)B h,β (L) for allLN. P

ROOF.Theevent{S

1 (N)=L}is contained in the union

NŠ1

M 1 =0N M 2 =M 1 +1 {S 1 (M 1 )=0}br(L;M 1 ,M 2 ){S 1 (N)=L}, in whichM 1 andM 2 ,forω{S 1 (N)=L}, may be chosen as the time of the last return ofS[N](ω)to the hyperplaneH 0 ,respectively the firstvisitofS[M 1 ,N](ω) to the hyperplaneH L . By splitting over all possible values ofM 1 andM 2 ,and applying Lemma2.1(c) and the Markov property to renew the random walk at these times, we obtain that G h,β (L)is bounded by  

N=1NŠ1

 M 1 =0N  M 2 =M 1 +1 E h  1 {S 1 (M 1 )=0} e Š? β (M 1 ,M 2 ) 1 br(L;M 1 ,M 2 ) 1 {S 1 (N)=L}  =  

N=1NŠ1

 M 1 =0N  M 2 =M 1 +1 P h [S 1 (M 1 )=0]b h,β (L;M 2 ŠM 1 ) ×P h [S 1 (NŠM 2 )=0] = B h,β (L)    n=0 P h [S 1 (n)=0]  2 , from which the lemma follows, by (2.6).? C

OROLLARY2.9.For anyh>0andβ?0,we have

m G (h,β) def =lim L

ŠlogG

h,β (L) L= m B (h,β) and

α(h)e

Šm G (h,β)L G h,β (L)1

α(h)

2 e Šm G (h,β)L (2.11) for allLN. P ROOF. The corollary follows from Proposition2.7and Lemma2.8.?

COINCIDENCE OF LYAPUNOV EXPONENTS1549

REMARK2.10. For anyh>0, as anticipated in Section1.3, there exists a unique parameterβ c (h) >0, such that m G (0,β c (h))=h. P ROOF.Foranyh>0, by Corollary2.9and as shown in the proof of

Proposition2.7,themass

m G (h,β)is continuous and concave increasing in the variableβ?R + .Furthermore,wehavem G (h,0)=0by(2.6)andthe assumption?(0)=0,and lim

β→∞

m G (h,β)=∞byRemark2.3and theassump- tion lim t→∞ ?(t)=∞. This limiting behavior forβ→∞, in combination with the concavity of the mass shown above, moreover yields that the monotonicity of m G (h,β)inβ?R + is strict. This completes the proof of the remark.

2.2.Irreducible bridges and renewal results. For the rest of the this section,

we fix an integerp?Nand consider independent copies S j =(S j (n)) n?N 0 ,j=1,...,p, of the random walkS. We assume these copies to be defined on the product space (? p ,F ?p ),onwhichthep-fold product measure, in order to keep notation sim- ple, is again denoted byP h . We compose fromS 1 ,...,S p a random processS (p) with values in(Z d ) p by setting S (p) (n) def =(S 1 (n 1 ),...,S p (n p )), forn=(n 1 ,...,n p )?N p 0 ,and S (p)def =  S (p) (n) n?N p 0.

The processS

(p) inherits the Markov property fromS 1 ,...,S p in the following way: forM=(M 1 ,...,M p )andN=(N 1 ,...,N p )?N p 0 , we write

M≤N,if and only ifM

j ≤N j forj=1,...,p,

M j The origin inN

p 0 is denoted by 0. Suppose now thatM,N?N p 0 withM≤N,and x n =(x 1 n 1 ,...,x p n p)?(Z d ) p forn=(n 1 ,...,n p )?[0,...,N]. Then, if P h ? S (p) (m)=x m form?[0,...,M]  >0

1550M. FLURY

is valid, we have P h ≥ S (p) (n)=x n forn[M,...,N]|S (p) (m)=x m form[0,...,M] ? = p ? j=1 P h ≥ S j [M j ,N j ]=(x j M j ,...,x j N j )|S j [M j ]=(x j 0 ,...,x j M j ) ? = p ? j=1 P h ≥ S j [N j ŠM j ]=(x j M j Šx j M j ,...,x j N j Šx j M j ) ? =P h ≥ S (p) (n)=x M+n Šx M forn[0,...,NŠM] ? . This means that, similarly toSin the previous subsection, the processS (p) can be renewed at any timeM. In Definition2.4, the denomination bridge was introduced in the context of a single random walk. We want to generalize it to the present setting: forM,NN p 0 withMN, we write S (p) [N] def = ≤ S (p) (n) ? nN p 0 :nN , S (p) [M,N] def = ≤ S (p) (n) ? nN p 0 :MnN for finite subpaths ofS (p) in(Z d ) p . D

EFINITION2.11. Supposeω?

p andM,NN p 0 withMN. The finite pathS (p) [M,N](ω)is called abridgeif S 11 (M 1 ,ω)=S j 1 (M j ,ω)EMARK. By this definition, a finite pathS (p) [M,N]in(Z d ) p is a bridge if and only ifS 1 [M 1 ,N 1 ],...,S p [M p ,N p ]are bridges in the sense of Defini- tion2.4, with start and endpoint each in a common hyperplane. Also, observe that the definition includes the caseM=N,inwhichS (p) [M,N]is a bridge of span zero. At the beginning of this section, we introduced the path potential? β for the random walkS.Foranyj{1,...,p}, we denote the corresponding potential associated withS j by? j β .ForM,NN p 0 withMN, a path potential for the processS (p) is then given by ? (p) β (N) def = p → j=1 ? j β (N j ),

COINCIDENCE OF LYAPUNOV EXPONENTS1551

(2.12) ? (p) β (M,N) def = p  j=1 ? j β (M j ,N j ).

Forh,β≥0,L?N

0 andM,N?N p 0 withM≤N,wenowdefine br p (L;N) def =  S (p) [N]is a bridge of spanL  , br p (L;M,N) def =  S (p) [M,N]is a bridge of spanL  andalsolet b p h,β (L;N) def =E h ? exp  -? (p) β (N) ;br p (L;N)  , B p h,β (L) def =  N?N p 0 b p h,β (L;N). R

EMARK2.12. By the independence ofS

1 ,...,S p ,wehave B p h,β (L)=(B h,β (L)) p (2.13) for allL?N. As a consequence, the mass m p B (h,β) def =lim

L→∞

-logB p h,β (L) L of B p h,β exists andm p B (h,β)=pm B (h,β). By Proposition2.7, we further have B p h,β (L)≤e -m p B (h,β)L (2.14) for allL?N. Moreover, for nonvanishingh>0, we have B p h,β (L)≥α(h) p e -m p B (h,β)L (2.15) for allL?N. Bridges allow a treatment using the tools of renewal theory. The decisive con- cepts for this treatment are the following. D

EFINITION2.13. Suppose thatL?Nandω?br

p (L;M,N)forM?N p 0 andN?N p withM1552M. FLURY

Moreover, the bridgeS

(p) [M,N](≤)is calledirreducibleifS 11 (N 1 ,≤)is its only breaking point.

Forh,??0,LN,MN

p 0 andNN p withMROPOSITION2.14.For allh,??0andLN,we have B p h,? (L)= L  k=1 ? p h,? (k)B p h,? (LŠk).(2.16) P

ROOF.For≤br

p (L;N),letk{1,...,L}denote the smallest breaking point forS (p) [N](≤). Then, there is a unique timenN p withnNsuch that S (p) [n](≤)is an irreducible bridge of spankandS (p) [n,N](≤)is a bridge of span

LŠk. We thus obtain

br p (L;N)= L  k=1  nN p :nN ir p (k;n)br p (LŠk;n,N), where the union is of disjoint sets. For≤ir p (k;n)br p (LŠk;n,N),wefurther have ? (p) ? (N,≤)=? (p) ? (n,≤)+? (p) ? (n,N,≤),

COINCIDENCE OF LYAPUNOV EXPONENTS1553

by Lemma2.1(b) applied to? 1β ,...,? p β . By using the Markov property to renew the processS (p) at timen, we thus haveB p h,β (L)equal to L  k=1  NN p  nN p :nN E h ? e Š? (p) β (n) 1 ir p (k;n) e Š? (p) β (n,N) 1 br p (LŠk;n,N)  = L  k=1  NN p  nN p :nN E h ? e Š? (p) β (n) 1 ir p (k;n)  E h ? e Š? (p) β (NŠn) 1 br p (LŠk;NŠn)  = L  k=1 ? p h,β (k)B p h,β (LŠk).∅ R EMARK. The subadditivity property of the sequence(ŠlogB(L)) LN , shown in Section2.1in a “straightforward" way, is also a consequence of the renewal equation forp=1.

Forh,β0,LN

0 andkN,weset a p h,β (L) def =B p h,β (L)e m p B (h,β)L andπ p h,β (k) def =? p h,β (k)e m p B (h,β)k . L

EMMA2.15.For anyh>0andβ0,we have

 kN π p h,β (k)=1and  kN kπ p h,β (k) <.(2.17) P

ROOF.ForLN

0 ,kNands[0,1],set A(s) def =   L=0 a p h,β (L)s L andP(s) def =   k=1 π p h,β (k)s k .

The renewal equation implies

A(s)=1+

  L=1 a p h,β (L)s L =1+   L=1L  k=1 π p h,β (k)s k a p h,β (LŠk)s

LŠk

=1+P(s)A(s).

By (2.14), we havea

p h,β (L)1forallLN 0 and thereforeA(s) <fors [0,1). As a consequence,

P(1)=lim

s1

P(s)=lim

s1

A(s)Š1

A(s)1.

1554M. FLURY

We have thus shown thatπ

p h,β (k)is a (nonperiodic) subprobability sequence with renewal sequencea p h,β (L). The first equation in (2.17) states thatπ p h,β (k)is re- current, which is equivalent toA(1)=∞.Ifπ p h,β (k)is now recurrent, then the renewal theorem (see, e.g., Theorem 4.2.2 in [11]) yields lim

L→∞

a p h,β (L)=1 ∞k=1 kπ p h,β (k).

The lemma thus follows from the estimate forB

p h,β (L)in (2.15), which states that a p h,β (L)≥α(h) p is true for allL?N.

By (2.14) and Lemma2.15, we know that

? p h,β (L)decays to zero faster than B p h,β (L)whenL→∞. In the next subsection, we will show that the difference in the decay velocity is even exponential. For this purpose, we will need the following result, stating that long intervals are unlikely to be free of breaking points. More precisely, we suppose that?,L?NwithL≥2?,M?N p 0 andN?N p with

M br ?p ? (L;M,N) def =  S (p) [M,N]is a bridge of spank,ofwhich S 1 (M 1 )+?,...,S 1 (N 1 )-?are no breaking points  and br ?p ? (L;N) def =br ?p ? (L;0,N).Forh>0andβ≥0, we further set b ?p ?,h,β (L;N) def =  N?N p E h ? exp  -? (p) β (N) ;br ?p ? (L;N)  , B ?p ?,h,β (L) def =  N?N p b ?p ?,h,β (L;N). L EMMA2.16.For anyh>0andβ≥0,there exists a decreasing function ε p h,β :R + →R + such thatlim

T→∞

ε p h,β (T)=0and B ?p ?,h,β (L)≤ε p h,β (L-2?)e -m p B (h,β)L for all?,L?NwithL≥2?. P ROOF. By the Markov property and Lemma2.1(b), with?denoting the largest breaking point smaller than?(or?=0 if there is no such point) andk denoting the smallest breaking point greater thanL-?,wehave B ?p ?,h,β (L)= ?-1  ?=0L  k=L-?+1 B p h,β (?)? p h,β (k-?)B p h,β (L-k) = ? 

˜?=1L-?



˜k=T+1

B p h,β (?-˜?)? p h,β (˜k+˜?)B p h,β (L-˜k-?)

COINCIDENCE OF LYAPUNOV EXPONENTS1555

withT=L-2?,˜k=k-?and˜?=?-?.Now,weset ε p h,β (T) def = ∞  j=T+2 jπ p h,β (j).(2.18)

From Lemma2.15, we know that lim

T→∞

ε p h,β (T)=0. Moreover, we have e m p B (h,β)L B ?p ?,h,β (L)≤ ∞ 

˜?=1∞



˜k=T+1

π p h,β (˜k+˜?) = ∞ 

˜?=1∞

 j=T+˜?+1 π p h,β (j) = ∞  j=T+2j-T-1 

˜?=1

π p h,β (j) = ∞  j=T+2 (j-T-1)π p h,β (j) < ε p h,β (T), which completes the proof.

2.3.Separation of the masses. In this subsection, we investigate the decay ve-

locity of weighted irreducible bridges. In particular, we will show that they decay exponentially faster than bridges without the irreducibility restriction. L

EMMA2.17.For allh,β≥0,the mass

m p ? (h,β) def =lim

L→∞

-log? p h,β (L) L of ? p h,β exists in[? β (1),∞),is continuous as function onR + ×R + and satisfies ? p h,β (L)≤1 pe 2(? β (1)+λ h ) e -m p ? (h,β)L (2.19) for allL?N,whereλ h =logE 0 [exp(h·S 1 (1))]. P

ROOF.Fori?{1,...,p},letE

idef =(δ i1 ,...,δ ip )?N p 0 . Then, for everyN? N p , the union p  i=1  N-2E i >M?N p ir p (L 1 ;M)∩{S i1 (M i +1)=L 1 +1} ∩{S i1 (M i +2)=L 1 }∩ir p (L 2 ;M+2E i ,N)

1556M. FLURY

consists of disjoint sets and is a subset of ir p (L 1 +L 2 ;N).Foranyωin that union, a double application of Lemma2.1(b) to the potentials? 1 ,...,? p yields ? (p) β (N,ω)≤? (p) β (M,ω)+2? β (1)+? (p) β (M+2E i ,N,ω) for the correspondingi?{1,...,p}andM?N p . By splitting over all possible values ofiandM, and renewing the processS (p) at the timesMandM+2E i , we therefore have ? p h,β (L 1 +L 2 ) ≥ p  i=1  N?N p  N-2E i >M?N p E h  e -? (p) β (M) 1 ir p (L 1 ;M) e -2? β (1) 1 {S i1 (M i +1)=L 1 +1} ×1 {S i1 (M i +2)=L 1 } e -? (p) β (M+2E i ,N) ×1 ir p (L 2 ;M+2E i ,N)  = p  i=1  N?N p  N-2E i >M?N p λ p h,β (L;M)e -2? β (1) P h [S 1 (1)=1] ×P h [S 1 (1)=-1]λ p h,β (L;N-M-2E i ) = p e 2(? β (1)+λ h ) ? p h,β (L 1 )? p h,β (L 2 ), where, at the last step, the concrete definition ofP h is used. The existence of m ? (h,β)in[-∞,∞), as well as the estimate in (2.19), now follows from the subadditive limit lemma applied to -log  p e 2(? β (1)+λ h ) ? p h,β (L)  ,L?N.

The lower bound?

β (1)for the mass goes back to Remark2.3. Finally, the conti- nuity of m ? is derived by the same arguments that were used in Proposition2.7to show the continuity of m B .∅ The main result of this section is the derivation of a gap between the exponential decay rates of bridges and irreducible bridges. T

HEOREM2.18.For anyh>0andβ≥0,we have

m p ? (h,β) >m p B (h,β). The strategy for the proof of Theorem2.18was introduced in [4] (or see the more polished version of it in Chapter 4 of [11]) in the case of a single random

COINCIDENCE OF LYAPUNOV EXPONENTS1557

walk in absence of a potential. It was then extended in [14] to single random walks evolving under the “trap" potential ? trap β (N) def =β?R(N). Before we present the strategy for the proof, we introduce the essential concept of backtracks of bridges. D

EFINITION2.19. Suppose thatL?Nandω?br

p (L;M,N)forM?N p 0 andN?N p withMThe subpathS

j [m,n](ω)ofS j [M j ,N j ](ω)is called aj-backtrack(or simply backtrack) of the bridgeS (p) [M,N](ω)if (i)S j 1 (μ 1 ,ω)≤S j 1 (m,ω)forμ 1 =M j +1,...,m; (ii)S j 1 (n,ω)≤S j 1 (ν,ω) < S j 1 (m,ω)forν=m+1,...,n; (iii)S j 1 (n,ω) < S j 1 (μ 2 ,ω)forμ 2 =n+1,...,N j . If this is the case, then thespanof the backtrackS j [m,n](ω)is given by S j 1 (m,ω)-S j 1 (n,ω). A backtrackS j [m,n](ω)is said tocoveran integerkif S j 1 (n,ω)≤k1558M. FLURY small probability because of the drift. It is going to be important that the “total span"ofthesebacktracks remainslargeenough withrespecttothereduced number of points considered (i.e., the number of endpoints). More precisely, we proceed as follows. LetTand?be positive integers (to be specified) and setQ def =2?+T.ForlargeL?N,letk=k(L)be the greatest integer less than or equal to L Q -1andset A def ={Q,2Q,...,kQ}.

Now, letB={b

1 ,...,b τ }withb 1 <···τ+1def =L. We introduce two further items of notation for par- ticularN-step bridges of spanL:

•for?>0, let

ir p ?,B (L;N)?F ?p denote the set of allω?? p for whichS (p) [N](ω)is an irreducible bridge of spanLsuch that no point ofBis covered by a backtrack ofS (p) [N](ω)with a span larger than?;

•forσ?N

B , any pairwise disjoint decompositionB 1 ,...,B p ofB(possibly with some of theB j being empty) and eachj?{1,...,p},let br p,j B j ,σ| Bj (L;N j )?F ?p denote the set of allω?? p for whichS j [N j ](ω)is a bridge of spanLsuch that eachb?B j is covered by a backtrack ofS j [N j ](ω)with spanσ(b)which is not covering any othera?B j \{b},andlet br p B 1 ,...,B p ,σ (L;N) def = p ? j=1 br p,j B j ,σ| Bj (L;N j ). The following lemma realizes the aforementioned distinction on how the points ofA={Q,2Q,...,kQ}are covered by backtracks. L

EMMA2.20.The eventir

p (L;N)is contained in the union of ?

B?A:?B≥k/2

ir p ?,B (L;N) and ?

B?A:?B≥1

?

σ?N

B : ? b?B

σ(b)>k?/2

? B 1 ,...,B p ?B pairwise disjoint decomposition br p B 1 ,...,B p ,σ (L;N).

COINCIDENCE OF LYAPUNOV EXPONENTS1559

PROOF. Suppose thatωis in ir

p (L;N), but not in 

B?A,?B≥k/2

ir p ?,B (L;N).

There then exists a collection of backtracks ofS

(p) [N](ω), each of them of a span greater than?, which cover at leastk/2 of the points inA. Although some of them may cover several points inA, the sum of their spans is still greater than?k/2, since the distance between two points is at leastQ=T+2?.

We now inductively construct the setsB

1 ,...,B p .Forj?{1,...,p}, assume thatB 1 ,...,B j-1 are already constructed. For eachj-backtrack from our collec- tion, we then add toB j a single point from the complement ofB 1 ?···?B j-1 whichiscoveredbythisj-backtrack, but not covered by any otherj-backtrack (regardless of whether it is covered byi-backtracks fori?=j). If there is no such point, we remove this particularj-backtrack from the collection. The remaining backtracks still cover the same points inA, so the sum of their spans is still larger than?k/2.

By this construction, the setsB

1 ,...,B p are pairwise disjoint. Moreover, for anyj?{1,...,p},thesetB j has the property, that each of its points is covered by exactly one of the remainingj-backtracks. Consequently, if we set B def =B 1 ?···?B p , then there is aσ?N B with  b?B

σ(b)>?k/2andω?

br p B 1 ,...,B p ,σ (L;N).∅

ForB={b

1 ,...,b τ }?Awithb 1 <···EMMA2.21.For anyh>0andβ≥0,we have ? p h,β [ir p ?,B (L;N)]≤e -m p B (h,β)L (ε p h,β (T)α(h) -p )

τ+1

, whereα(h)is dened in(2.5)andε p h,β (T)is dened in(2.18). P

ROOF. Suppose thatω?ir

p ?,B (L;N)and recall thatb 0 =0andb

τ+1

=

L. For eachi?{1,...,τ+1},letm

i-1 =(m 1i-1 ,...,m p i-1 )?N p 0 andn i = (n 1i ,...,n p i )?N p be given by m j i-1def =min 

μ?{0,...,N

j }:S j 1 (μ ? ,ω)>b i-1 forμ<μ ? ≤N j  , n j idef =max 

ν?{1,...,N

j }:S j 1 (ν ? ,ω)≤b i for 0<ν ? ≤ν  forj=1,...,p. It is also convenient to choosem

τ+1def

=L.

1560M. FLURY

Sinceb

1 ,...,b τ are not covered by backtracks of a span greater than?and because the distance between these points is at leastQ=2?+T, it is clear that S j 1 (μ,ω)≤b i-1 +?τ+1  i=1 br ?p ? (b i -b i-1 ;m i-1 ,n i ), where br ?p ? was introduced at the end of Section2.2. An inductive application of

Lemma2.1(c) to the potentials?

1 ,...,? p further yields ? (p) β (N,ω)≥

τ+1

 i=1 ? (p) β (m i-1 ,n i ,ω). As a consequence, by the Markov property and Lemma2.16, an upper bound for ? p h,β [ir p ?,B (L;N)]is given by  N?N p  n 1 ,m 1 ,n 2 ,...,n τ ,m τ ?N p : 0=m 0 τ+1 =m

τ+1

=N E h

τ+1

 i=1 e -? (p) β (m i-1 ,n i ) 1 br ?p ? (b i -b i-1 ;m i-1 ,n i ) ×1 {S j 1 (m j i )=S j 1 (n j i )forj=1,...,p} =  N?N p  n 1 ,m 1 ,n 2 ,...,n τ ,m τ ?N p 0=m 0 τ+1 =m

τ+1

=Nτ+1  i=1 b ?p ?,h,β (b i -b i-1 ;n i -m i-1 ) ×P h [S j 1 (m j i -n j i )=0forj=1,...,p] = 

τ+1

 i=1 B ?p ?,h,β (b i -b i-1 )  m?N p p  j=1 P h [S j 1 (m j )=0]  τ ≤ 

τ+1

 i=1 ε p h,β (b i -b i-1 -2?)  e -m p B (h,β)L 1

α(h)

pτ , where, at the last step, (2.6) is used to identifyα(h). The lemma now follows from the facts thatα(h)≤1andb i -b i-1 ≥Q=T+2?fori=1,...,τ+1.∅

COINCIDENCE OF LYAPUNOV EXPONENTS1561

ForB={b

1 ,...,b τ }?Awithb 1 <···EMMA2.22.For anyh>0,β≥0andσ?N B ,we have  B 1 ,...,B p ?B pairwise disjoint decomposition ? p h,β [br p B 1 ,...,B p ,σ (L;N)]≤ψ p (h) τ e -m p B (h,β)L e -h  b?B

σ(b)

, whereα(h)is defined in(2.5)andψ p (h) def =p/α(h) 2 h. P

ROOF. We first restrict to the casep=1. ForL?N

0 andM,N?N 0 with

M≤N,wedefine

br ? (L;N) def ={-S[N]is a bridge of spanL}, br ? (L;M,N) def ={-S[M,N]is a bridge of spanL}. By the Markov property ofS, it is easily seen that P h [br ? (L;N)]=P h [-L=S 1 (N) < S 1 (n)≤0for0≤nBy Remark2.2, we thus have B ? h,0 (L) def = ∞  N=1 P h [br ? (L;N)]≤P h [H -L <∞] =e -2hL ,(2.20) whereH -L =inf{n?N:S 1 (n)=-L}.

Suppose now thatω?br

1B,σ

(L;N)and recall thatb 0 =0andb

τ+1

=L.For eachi?{1,...,τ}, by the definition of br

1B,σ

(L;N), there is at least one backtrack S[s i ,t i ](ω)ofS[N](ω)of spanσ(b i )satisfying b i-1 It is also convenient to chooses

τ+1def

=t

τ+1def

=Landσ

τ+1def

=0, in order to have P h [br ? (σ

τ+1

;s

τ+1

,t

τ+1

)]=0.

Fori?{1,...,τ+1},wenowdefine

m i-1def =min 

μ?{0,...,N}:S

1 (μ ? ,ω)>b i-1 forμ<μ ? ≤N  , n idef =max 

ν?{m

i-1 +1,...,N}:S 1 (ν ? ,ω)≤b i forμ<ν ? ≤ν  , such thatS[m i-1 ,n i ](ω)is a bridge of spanb i -b i-1 .By(2.21) and the maximal- ity conditions for backtracks, we havet i 1562M. FLURY fori=1,...,τ, by the definition ofn i+1 and (2.21) again. Moreover, since ? β (N,ω)≥

τ+1

 i=1 ? β (m i-1 ,n i ,ω) by Lemma2.1(c), the Markov property of the random walk yields ? 1 h,β [br

1B,σ

(L;N)] ≤  N?N  0=m 0 ···

τ+1

=s

τ+1

=t

τ+1

=m

τ+1

=N  q 1 ,...,q τ ?N,q

τ+1

=0,˜q 1 ,...,˜q τ ?N 0 : q 1 +˜q 1 =σ(b 1 ),...,q τ +˜q τ =σ(b τ ) E h

τ+1

 i=1 e -? β (m i-1 ,n i ) 1 br(b i -b i-1 ;m i-1 ,n i ) 1 {S 1 (s i )=S 1 (n i )+q i } ×1 br ? (σ(b i );s i ,t i ) 1 {S 1 (m i )=S 1 (t i )+˜q i } =  N?N  0=m 0 τ+1 =N 

τ+1

 i=1 b h,β (L;n i -m i-1 )  τ  i=1 P h ? br ? 

σ(b

i );t i -s i  ×  τ  i=1  q?N,˜q?N 0 :q+˜q=σ(b i ) P h [S 1 (s i -n i )=q i ]P h [S 1 (m i -t i )=˜q i ]  ≤ 

τ+1

 i=1 B h,β (L)  τ  i=1 B ? h,0 (L)  ×  τ  i=1  q?N,˜q?N 0 :q+˜q=σ(b i )∞  k=0 P h [S 1 (k)=q] ∞ 

˜k=1

P h [S 1 (˜k)=˜q]  =α(h) -2τ 

τ+1

 i=1 B h,β (b i -b i-1 )  τ  i=1 B ? h,0 (σ(b i ))  τ  i=1

σ(b

i )  , where, in the last step, equation (2.6) is used to identifyα(h). From Proposition2.7 and (2.20), we then obtain ? 1 h,β [br

1B,σ

(L;N)]≤α(h) -2τ e -m B (h,β)Lτ  i=1

σ(b

i )e -2hσ(b i ) ≤(α(h) 2 h) -τ e -m B (h,β)L e -h 

τi=1

σ(b

i ) ,

COINCIDENCE OF LYAPUNOV EXPONENTS1563

where, in the second estimate, the elementary inequalityxe -x ≤1forx≥0is used. We now proceed to the case of arbitraryp?N. We consider a pairwise disjoint decompositionB 1 ,...,B p ofB={b 1 ,...,b τ }. By the independence of the walks S 1 ,...,S p ,wethenhave ? p h,β [br p B 1 ,...,B p ,σ (L;N)]=  N?N p p ı j=1 E h [exp(-? j (N j ));br p,j B j ,σ| Bj (L;N)] = p ı j=1 ? 1 h,β [br 1 B j ,σ| Bj (L;N)] ≤(α(h) 2 h) τ e -pm B (h,β)L e -h  b?B

σ(b)

, where, in the last step, the estimate for a single random walk is used. Since there arep τ pairwise disjoint decompositionsB 1 ,...,B p ofB, the lemma now follows from m p B (h,β)=pm B (h,β)in Remark2.12.∅ P ROOF OFTHEOREM2.18. By means of Lemma2.20and Lemma2.22,we finally have the necessary tools to prove the mass gap for irreducible bridges. We first fixT?Nand??Nlarge enough such that we have

2(α(h)

-p ε p h,β (T)) 1/2 ≤1 2ande -?h/4 

1+ψ

p (h) 1-e -h/2  ≤1 2, and we setQ def =2?+T. Since there are 2 k subsets ofA={Q,...,kQ},wehave 

B?A:?B≥k/2

? p h,β [ir p ?,B (L;N)]≤e -m p B (h,β)L 2 k (α(h) -p ε p h,β (T)) 1+k/2 B?A:?B≥1 

σ?N

B :  b?B

σ(b)>k?/2

 B 1 ,...,B p ?B pairwise disjoint decomposition ? p h,β [br p B 1 ,...,B p ,σ (L;N)] ≤e -m p B (h,β)Lk 

τ=1

 k τ  ψ p (h) τ  σ 1 ,...,σ τ ?N: σ 1 +···+σ τ >k?/2 e -h(σ 1 +···+σ τ ) ≤e -m p B (h,β)L e -k?h/4k 

τ=1

 k τ  ψ p (h) τ  σ 1 ,...,σ τ ?N e (-h/2)(σ 1 +···+σ τ )

1564M. FLURY

ωe Šm p B (h,?)L e

Šk?h/4k

 ?=1  k ?  ? p (h)

1Še

Šh/2

 ? ωe Šm p B (h,?)L e

Šk?h/4

 1+? p (h)

1Še

Šh/2

 k ωe Šm p B (h,?)L 2 Šk .

Finally, we apply Lemma2.20and the estimatek?

L Q

Š2 to obtain

? p h,? (L)ω2e Šm p B (h,?)L 2 Šk

ω8e

Šm p B (h,?)L 2

ŠL/Q

and therefore m p ? (h,?)=lim L??

Šlog?

p h,? (L) L? m p B (h,?)+log2 Q> m p B (h,?), which proves the theorem.

3. Fixed number of steps.In this section, we consider finite random walks

S[N]for fixedN?N, again evolving under the influence of the path potential ? ? (N)=  x?Z d → ? (∞ x (N)), N?N. As introduced at the beginning of Section2, we assume→ ? to be given by → ? (t)=→(?t), t?R + , where→:R + ?R + is a concave increasing function satisfying lim t?0 →(t)=→(0)=0, as well as lim t?? →(t)=?and lim t?? →(t)/t=0.

3.1.Masses for paths and bridges. In the present setting, forh,??0, the

generalization of the annealed partition functionZ anh,? from Section1is given by G h,? (N) def =E h [exp(Š? ? (N))],N?N. R

EMARK3.1. Foranyh,??0andN?N,wehave

e

Š→

? (1)N ωG h,? (N)ω1, where the lower estimate follows from Lemma2.1(a). We are interested in the limiting exponential behavior ofG h,? (N).Theexis- tence of an associated mass was part of TheoremA(which, in the original pa- per [7], is shown in the present, more general setting). As we show next, it can also be obtained in a straightforward way by the subadditive limit lemma, which additionally delivers a bound for the speed of convergence.

COINCIDENCE OF LYAPUNOV EXPONENTS1565

PROPOSITION3.2.For anyh,??0,the mass

m G (h,?) def =lim N

ŠlogG

h,? (N) N ofG h,? exists in[0,→ ? (1)]and is continuous as a function onR + ×R + ,and G h,? (N)e Šm G (h,?)N is valid for allNN. P

ROOF. By Lemma2.1(b), for anyN

1 ,N 2 N,wehave G h,? (N 1 +N 2 )=E h  e Š? ? (N 1 +N 2 )  (3.1)?E h  e Š? ? (N 1 )Š? ? (N 1 ,N 1 +N 2 )  =G h,? (N 1 )G h,? (N 2 ), where, in the last step, the Markov property is used to renew the random walk at timeN 1 . Therefore, the existence of the massm G and the estimate in (3.2) are consequences of the subadditive limit lemma applied to the sequence (ŠlogG h,? (N)) NN and the bounds for the mass follow from Remark3.1.Fi- nally, the continuity ofm G is obtained by similar (but slightly simpler) arguments as used to prove the continuity of m B in Proposition2.7.∅ As in the point-to-hyperplane setting of Section2, it is convenient to introduce

N-step bridges. ForM,NN

0 withMN,wedefine Br(N) def ={S[N]is a bridge},

Br(M,N)

def ={S[M,N]is a bridge}.

Forh,??0andNN

0 ,wefurtherset B h,? (N) def =E h [exp(Š? ? (N));Br(N)].

Observe that we have

Br(N)=

LN 0 br(L;N)andB h,? (N)=  LN 0 b h,? (L;N), where the union is of disjoint sets and where br(L;N)={S[N]is a bridge of spanL}, b h,? (L;N)=E h [exp(Š? ? (N));br(L;N)] were introduced in Section2.1. R

EMARK3.3. For anyh,??0andNN,wehave

e

Š→

? (1)N B h,0 (N)B h,? (N)G h,? (N), where the lower estimate follows from Lemma2.1(a).

1566M. FLURY

Our interest lies with the ballistic regime{(h,?)?(0,?) 2 :?

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