Localization transition for a polymer near an interface

LOCALIZATION TRANSITION FOR A POLYMER

NEAR AN INTERFACE1

BY ERWINBOLTHAUSEN ANDFRANK DENHOLLANDER

University of Zurich and University of Nijmegen

_¨

Ž .

Consider the directed process i, S_{i iG 0} where the second component

Ž .

is simple random walk on_{⺪ S s 0 . Define a transformed path measure0}

w Ž . Ž .x

by weighting each n-step path with a factor exp ␭Ý_{1F iF n} ␻ q h sign S ._{i i}

Ž .

Here, ␻i iG 1is an i.i.d. sequence of random variables taking values"1

Ž . w .

with probability 1r2 acting as a random medium , while␭ g 0, ⬁ and

w .

hg 0, 1 are parameters. The weight factor has a tendency to pull the

path towards the horizontal, because it favors the combinations S_i) 0, ␻ s q1 and S - 0, ␻ s y1. The transformed path measure describes ai i i

heteropolymer, consisting of hydrophylic and hydrophobic monomers, near an oil᎐water interface.

We study the free energy of this model as nª ⬁ and show that there Ž .

is a critical curve ␭ ª h ␭ where a phase transition occurs betweenc

Ž .

localized and delocalized behavior in the vertical direction . We derive several properties of this curve, in particular, its behavior for ␭x0. To obtain this behavior, we prove that as _{␭, hx0 the free energy scales to its} Brownian motion analogue.

0. Introduction and main results. In this paper we solve a problem

Ž .

that was posed by Garel, Huse, Leibler and Orland 1989 and studied by

Ž .

Sinai 1993 . It involves a two-dimensional directed random polymer interact-ing with two solvents separated by an interface. Dependinteract-ing on the

interac-Ž .

tion, the polymer either stays near the interface localization or wanders

Ž .

away from it delocalization . The main problem is to determine the phase transition curve.

0.1. A random walk model. To define the model we need two ingredients;

Ž .

1. Ss S_{i iG 0}: a simple random walk on⺪ starting at the origin, where P, E denote its probability law and expectation.

Ž .

2. ␻ s ␻i iG1: an i.i.d. sequence of random variables taking values"1 with probability 1r2, where ⺠, ⺕ denote its probability law and expectation.

Received March 1996; revised June 1996.

1

Partially supported by Swiss National Science Foundation Contract 20-41925.94.

AMS 1991 subject classifications. 60F10, 60J15, 82B26.

Key words and phrases. Random walk, Brownian motion, random medium, large deviations,

phase transition.

w . w .

Fix ␭ g 0, ⬁ and h g 0, 1 . Given ␻, define a transformed probability law

Qn␭, h, ␻ on n-step paths by setting

␭, h, ␻ n dQn n 1 0.1 S s exp ␭ ␻ q h ⌬ ,

Ž

.

_dP

_Ž

Ž

i

.

is0

_.

_Z␭, h, ␻

_Ý

Ž

i

.

i n is1 where sign S ,

Ž

i

.

if Si/ 0, 0.2 ⌬ s

Ž

.

i

½

sign S

Ž

iy1

.

, if Sis 0 ␭, h, ␻ _{w Ž .}

and Zn is the normalizing constant or partition sum. In 0.2 we could x

put ⌬ s 0 if S s 0. This would be a site rather than a bond model.i i

␭, h, ␻ _{Ž .}n

We view Qn as modelling the following situation. Think of i, Si is0 as a directed polymer on⺪2_{, consisting of n monomers represented by the bonds}

in the path. The lower half plane is ‘‘water,’’ the upper half plane is ‘‘oil.’’ The monomers are of two different types, occurring in a random order indexed by ␻. Namely, ␻ s y1 means that monomer i ‘‘prefers water,’’ ␻ s q1 meansi i that it ‘‘prefers oil.’’ Since ⌬ s y1 when monomer i lies in the water andi

Ž .

⌬ s q1 when it lies in the oil, we see that the weight factor in 0.1i ‘‘encourages matches and discourages mismatches.’’ For hs 0 both types of monomers interact equally strongly with the water and with the oil, being

Ž .

attracted by one and repelled by the other. However, for hg 0, 1 the monomers preferring oil have a stronger interaction with both the solvents than the monomers preferring water. The parameter␭ is the overall interac-tion strength and plays the role of inverse temperature.

Ž .

REMARK. In 0.1 we could put the h-dependence in the probability law of

Ž . Ž .

␻, for instance, by picking ⺠ ␻ s "1 s 1 " h r2 and writing ␭Ý ␻ ⌬ ini i i i the exponent. This would describe a polymer where the two types of monomers occur with different densities but interact equally strongly with the solvents.

Ž

Alternatively we could make a mix of the two types of h-dependence or even .

allow for more general ␻-sequences with exponential moments . For the proofs in this paper it is a slight advantage that h enters into the exponent. Nevertheless, all results carry over with only minor changes in the proofs.

The way in which the polymer behaves near the interface is the result of a Ž

competition between energy and entropy. The energy is minimal i.e., the .

weight is maximal when all the monomers are placed in their preferred solvent, but this strategy has low entropy. On the other hand, the entropy is maximal when the polymer makes large excursions away from the interface,

Ž .

but this strategy typically has high energy i.e., the weight is small . What do we expect will happen under Q␭, h, ␻_n as nª ⬁?

2. ␭ ) 0, h s 0. The polymer will want to stay close to the interface, so that it can place as many monomers as possible in their preferred solvent and produce low energy. Indeed, wandering away from the interface would result in a misplacing of about half the monomers. The polymer can reduce this fraction by crossing the interface at a positive frequency. This lowers the entropy, but only by a small amount if the crossing frequency is small.

Ž .

The estimates in Sinai 1993 show that for this strategy the gain exceeds the loss; that is, we have localization.

3. ␭ ) 0, h­1. Now wandering away is again the winning strategy, simply because the monomers preferring water barely interact with either the water or the oil. By moving away in the upward direction the polymer can match all the monomers that prefer oil, thereby producing almost the minimal energy and almost the maximal entropy; that is, we have delocal-ization.

The above intuitive picture seems to suggest that there is a critical curve

Ž .

in the ␭, h -plane separating the localized from the delocalized phase. It is the goal of the present paper to prove the existence of this critical curve and to derive some of its properties.

In order to give a precise definition of the two phases, we need the

Ž .

following preliminary result proved in Section 1 .

w . w .

THEOREM 1. For every ␭ g 0, ⬁ and h g 0, 1 ,

1 _{␭, h, ␻}

0.3 lim log Z s␾ ␭, h

Ž

.

n

Ž

.

n

nª⬁

exists⺠-a.s. and is nonrandom.

The function␾ is the specific free energy of the polymer. It is immediate from Ž0.1 and 0.3 that. Ž . ␾ ␭, h is continuous, nondecreasing and convex in bothŽ .

w

variables. Note that our model makes perfect sense for ␭, h g ⺢. Obviously,

Ž .

in this larger parameter space,␾ ␭, h is everywhere finite, is symmetric and convex in both variables and hence is also continuous and unimodal in both

x

variables. Moreover, it is easy to show that

0.4 ␾ ␭, h G ␭h.

Ž

.

Ž

.

Ž . 1r2Ž .

Indeed, since P ⌬ s q1 for 1 F i F n ; Crn_i nª ⬁ , it follows that

n ␭, h, ␻ Zn s E exp

ž

␭

_Ý

Ž

␻ q h ⌬i

.

i

/

is1 n G exp ␭

_Ý

Ž

␻ q h q O log ni

.

Ž

.

is1 0.5

Ž

.

s exp ␭hn q o n , ⺠-a.s.,

Ž

.

where in the last step we use the strong law of large numbers for␻. Thus we

Ž .

polymer wanders away in the upward direction. This leads us to the following definition.

DEFINITION 1. We say that the polymer is:

Ž .a localized if ␾ ␭, h ) ␭h,Ž . Ž .b delocalized if ␾ ␭, h s ␭h.Ž .

Ž .

In case a the polymer is able to beat on an exponential scale the trivial strategy of moving upward. It is intuitively clear that this is only possible by crossing the interface at a positive frequency, which means that the path

Ž .

measure localizes near the interface in a strong sense. In case b , on the other hand, the polymer is not able to beat the trivial strategy on an exponential scale. In principle it could still do better on a smaller scale, but

w

we do not expect this at least not in the interior of the region described by Ž .xb . We shall not derive any properties of the path measure, but just stick to

Ž .

the above definition. See Section 0.4 for a further discussion. Our first main theorem reads as follows.

Ž . Ž . Ž .

THEOREM 2. For every ␭ g 0, ⬁ there exists h ␭ g 0, 1 such that the_c

polymer is: localized if 0F h - hc

Ž .

␭ , 0.6

Ž

.

delocalized if hG hc

Ž .

␭ . Moreover, Ž . w .

␭ ª h ␭ is continuous and nondecreasing on 0, ⬁ ,c 0.7

Ž

.

Ž . Ž .

lim_␭ª⬁ h_c ␭ s 1, lim_{␭x 0} h_c ␭ s 0.

The proof of Theorem 2 is given in Section 2. It will also provide upper and Ž .

lower bounds on h_c ␭ , namely: 1 i lim sup h ␭ F 1,

Ž .

c

Ž .

␭ ␭x0 1 0.8

Ž

.

Ž .

ii lim inf hc

Ž .

␭ ) 0, ␭ ␭x0 1 3

iii lim␭ 1 y h ␭ g log 2, log 2 .

Ž

.

Ž

c

Ž .

.

2 2

␭ª⬁

The two ingredients of the continuous model are two standard Brownian motions on⺢, denoted by:

Ž .1 Bs BŽ _{t t}._{G 0}, Ž .2 ␤ s ␤Ž _{t t}. _{G 0},

˜ ˜

˜ ˜

both starting at the origin. We write P, E, respectively,⺠, ⺕, to denote their

Ž . Ž .

probability law and expectation. Similarly as in 0.1 and 0.2 , the

trans-˜

␭, h, ␤

formed probability law Q_t on paths of length t, given ␤, is defined by

␭, h, ␤

˜

dQt 1 t 0.9 B s exp ␭ ⌬ d␤ q h ds .

Ž

.

_˜

Ž

Ž

s

.

0FsFt

.

_˜

_{␭, h, ␤}

_H

s

Ž

s

.

dP Z_t 0 Here, sign B ,

Ž

s

.

if Bs/ 0, 0.10 ⌬ s

Ž

.

s

½

_{0, if B s 0,} s w .

the first integral is an Ito integral, and the parameters

_ˆ

␭, h are both in 0, ⬁ .

Ž .

The analogue of Theorem 1 proved in Section 3 reads as follows.

w .

THEOREM 3. For every ␭, h g 0, ⬁ ,

1 _{␭, h, ␤}

˜

˜

0.11 lim log Z s␾ ␭, h

Ž

.

t

Ž

.

t tª⬁

˜

exists⺠-a.s. and is nonrandom.

˜

Ž .

The function ␾ has the same qualitative properties as ␾ in 0.3 , including

Ž .

the lower bound in 0.4 . Therefore we can maintain the same distinction between phases as in Definition 1.

The Brownian scaling property tells us that

0.12 B ,␤ s aB 2, a␤ 2 for all a) 0,

Ž

.

Ž

s s

.

sG0 D

Ž

sr a sr a

.

sG0

where s means equality in distribution. This implies that, for fixed_D ␭, h and as a random variable in ␤,

˜

␭, h , ␤

_˜

a␭, ah, ␤₂

0.13 Z s Z for all tG 0 and a ) 0.

Ž

.

t D tr a

Hence

1

˜

˜

0.14 ␾ ␭, h s ␾ a␭, ah for all a) 0.

Ž

.

Ž

.

₂

Ž

.

a

˜

Ž .

It immediately follows from 0.14 that ␾ has the following scaling form:

˜

2

Ž . Ž . w . Ž .

␾ ␭, K␭ s SS K ␭ for K g 0, ⬁ , with K ª SS K con-0.15

Ž

.

Ž . tinuous, nondecreasing and convex, satisfying SS K G K.

Ž .

Ž x

THEOREM 4. There exists Kcg 0, 1 such that

SS K

Ž

.

s K if K G K ,c SS K

Ž

.

) K if 0 F K - K .c 0.16

Ž

.

˜

˜

Ž . Ž . Ž .

By 0.15 , Theorem 4 implies that␾ ␭, h s ␭h for h G K ␭ and ␾ ␭, h ) ␭hc for h- Kc␭; that is, the phase separation curve is the straight line ␭ ª K ␭.c Although the picture here looks fairly simple, the complexity of the model is hidden in the constant K , which seems to be a very ungainly and complexc object. We have rough bounds on K , but nothing like a sequence of boundsc that could be expected to converge to K .c

0.3. Weak interaction limit. We are now ready to formulate our main results concerning the weak interaction limit of the random walk model and its relation to the Brownian motion model.

w .

THEOREM 5. For every ␭, h g 0, ⬁ ,

1

˜

0.17 lim ␾ a␭, ah s ␾ ␭, h .

Ž

.

_a2

Ž

.

Ž

.

ax0 Ž .

Although 0.17 is intuitively plausible, the estimates needed for its proof are quite delicate. The reason is that our paths carry exponential weight factors, which are very sensitive to fluctuations. One should keep in mind that, at least in the localized region, the path exhibits a behavior that has an exponentially small probability under the free path measure. It is therefore clear that the result cannot be proved by a routine application of invariance principles.

We shall not prove Theorem 5 separately, as it is a consequence of the more powerful but more technical Theorem 6 below. A proof of Theorem 5

Ž .

would be simpler and more transparent than that of Theorem 6 given in Section 4. However, the unfortunate fact is that Theorem 5 alone does not lead to a determination of the tangent at ␭ s 0 of the phase separation curve in the discrete model. In fact, it only yields

1 0.18 lim inf h ␭ G K .

Ž

.

c

Ž .

c ␭ ␭x0 Ž . Ž .

Indeed, pick K- K . Then, by 0.15 ᎐ 0.17 ,_c 1

˜

0.19 lim ␾ a, aK s ␾ 1, K ) K.

Ž

.

_a2

Ž

.

Ž

.

ax0 Ž . 2 Ž .

This implies ␾ a, aK ) Ka and hence h a ) aK for small enough a,_c

Ž .

which proves 0.18 after letting ax0 followed by K ­ K . It is clear that ac

Ž .

statement like 0.17 does not yield 1

0.20 sup h ␭ F K ,

Ž

.

c

Ž .

c

˜

_{Ž . Ž .} 2

simply because␾ 1, K s K for K G K does not imply that ␾ a, aK s a Kc for small enough a.

In order to remedy this situation, we introduce the ‘excess’ free energies ␺ ␭, h s ␾ ␭, h y ␭h,

Ž

.

Ž

.

0.21

Ž

.

˜

˜

␺ ␭, h s ␾ ␭, h y ␭h,

Ž

.

Ž

.

˜

so that the delocalized region is characterized by␺ s 0, respectively, ␺ s 0. Our main result for the weak interaction limit is the following.

X _{Ž .} X

THEOREM 6. Fix␭ ) 0. Let h ) 0, h G 0 and ␳ ) 0 satisfy 1 q ␳ h - h.

Then 1 _X

˜

␺ a␭, ah F 1 q ␳ ␺ ␭, h ,

Ž

.

Ž

.

Ž

.

2 a 0.22

Ž

.

1 _X

˜

␺ ␭, h F 1 q ␳

Ž

.

Ž

.

_a2␺ a␭, ah

Ž

.

for small enough a.

˜

Theorem 6 and the continuity of ␾ and ␾ obviously imply Theorem 5. Theorem 6 is also sufficiently strong to give us the following corollary.

COROLLARY1. 1 0.23 lim h ␭ s K .

Ž

.

c

Ž .

c ␭ ␭x0 Ž . Ž . X

To get 0.20 from the first line in 0.22 , pick hs K ,c ␳ ) 0 and ␭ s 1,

˜

Ž . Ž . Ž Ž . .

hs 1 q 2␳ K . Since ␺ 1, K s 0, it follows that ␺ a, a 1 q 2 ␳ K s 0_{c c c}

Ž . Ž .

and hence h a_c F a 1 q 2␳ K for small enough a. Now let ax0 and ␳ x0._c The idea behind Theorem 6 is that by slightly varying h we can dominate the errors that arise in the approximation of the random walk by the Brownian motion.

REMARK. Theorem 6 can be shown to carry over to the version of the

model where the h-dependence sits in the probability law of ␻. For the Brownian motion model there is no distinction between the two versions. Apparently, the weak interaction limit is largely independent of the details of the model. This is essentially a stability result. Stability is crucial for our understanding of the localization problem, and typically hard to prove for path measures with exponential weight factors.

0.4. Open problems. Our distinction between the localized and the delo-calized phase, as given in Definition 1, is in terms of the specific free energy rather than the path measure itself. We would like to show that in the

Ž .

␭, h, ␻_{Ž .}

1. For fixed i, does Qn Sig ⭈ converge to a nondegenerate limit law as

nª ⬁?

Ž . ␭, h, ␻Ž< 4<

2. Is there a ds d ␭, h ) 0 such that lim_nª⬁Q_n 1F i F n: S s 0 rn_i

w x.

g d y␧, d q ␧ s 1 for all ␧ ) 0?

No doubt the answer is ‘‘yes’’ in the localized phase and ‘‘no’’ in the delocal-ized phase, but this remains to be proven. Other interesting questions are: How does the free energy behave close to the critical curve? How large are the excursions of the path away from the horizontal?

Ž .

Sinai 1993 proved that if ␭ ) 0, h s 0, then the path localizes in the Ž .

following sense: there exist numbers ␭ ) 0, ␦ ␭ ) 0 and random variables

Ž . Ž . n₀ ␻ , k ␻ such that₀ ␭, 0, ␻ _{< <} y␦ Ž␭.k sup Qn

Ž

Si ) k F e

.

␥ ␥ log nFiFnylog n 0.24

Ž

.

for kG k0

Ž

␻ , n G n ␻ , ⺠-a.s.

.

0

Ž

.

We expect that Sinai’s arguments can be extended to cover the whole localized region.

Ž . Ž .

One could hope to make some progress on problems 1 and 2 above by looking at the times when the path intersects the interface. In the localized

Ž .

region these times admit a Gibbsian description in the limit as nª ⬁ . However, this leads to a Gibbs measure with a random long-range potential having both signs, which is a notoriously difficult object. Nevertheless, we expect that a limiting measure exists and that it has exponentially decaying correlations.

Even the delocalized region is not trivial. It seems intuitively clear that, at

w Ž .x

least in the interior of this region i.e., for h) hc ␭ , the path just behaves as simple random walk conditioned to stay positive, which is well known to have Brownian scaling with the so-called Brownian meander as limiting measure wsee Bolthausen 1976 . However, it appears to be difficult to exclude theŽ .x possibility of rare returns to the interface.

Ž .

Grosberg, Izrailev and Nechaev 1994 obtain localization for the case where ␻ is periodic instead of random.

Ž . ␭, 0, ␻

Albeverio and Zhou 1996 prove that if ␭ ) 0, h s 0, then log Z_n

Ž .

satisfies a LLN and a CLT as a random variable in ␻ . However, there is no description of the mean and the variance. They further show that

␭, 0, ␻ _{Ž .} H Qn ⺠ d␻ -a.s. both max



jy i: S s S s 0, S / 0 for i - k - j ,i j k

4

0Fi-jFn 0.25

Ž

.

< < max S_i 0FiFn

are of order log n as nª ⬁, which is typical for a localized path.

Ž . Ž .

However, the quenched version described in the present paper is qualita-tively very different and considerably more complex.

1. Proof of Theorem 1. The proof consists of two parts. In Lemma 1 we prove that the claim holds when the random walk is constrained to return to the origin at time 2 n. In Lemma 2 we show how to remove this constraint.

Fix ␭ and h. Define

2 n U ␻ ,

_

₄

1.1 Z s E exp ␭ ␻ q h ⌬ 1 S s 0 ,

Ž

.

2 n

ž

_Ý

Ž

i

.

i 2 n

/

is1

where we recall the notation introduced in Section 0.1.

Ž . ␻,U

LEMMA 1. The limit limnª⬁1r2n log Z2 n exists and is constant ⺠-a.s.

PROOF. We need the following three properties.

␻,U ␻,U T2 m_␻,U

Ž .

I. Z2 n G Z2 m Z2 ny2 m for all 0F m F n, with T the left-shift T␻ si ␻iq1.

Ž . Ž ␻,U.

II. nª 1r2n ⺕ log Z2 n is bounded from above.

Ž . Ž .

III. ⺠ T␻ g ⭈ s ⺠ ␻ g ⭈ .

Ž .  4

Property I follows from 1.1 by inserting an extra indicator 1 S2 ms 0 and

using the Markov property of S at time 2 m. Property II holds because ⺕ log Z␻ ,U _{F log ⺕ Z}␻ ,U

Ž

2 n

.

Ž

2 n

.

2 n

2 n



4

s log E cosh

ž

Ž

␭

.

exp ␭h

_Ý

⌬ 1 S s 0i 2 n

/

is1

1.2

Ž

.

F 2n log cosh

Ž

␭ q ␭h .

.

Ž ␻,U.

Property III is trivial. Thus, ␻ ª log Z2 n nG 0 is a superadditive process. It

w Ž .

therefore follows from the superadditive ergodic theorem Kingman 1973 ,

x Ž . ␻,U

Theorem 1 that limnª⬁1r2n log Z2 n converges⺠-a.s. and in mean, and is measurable w.r.t. the tail ␴-field of ␻. Since the latter is trivial, the limit is constant⺠-a.s. I

Our original partition sum was 2 n ␻ 1.3 Z s E exp ␭ ␻ q h ⌬ ,

Ž

.

2 n

ž

Ý

Ž

i

.

i

/

is1 Ž .

which is 1.1 but without the indicator. Thus, in order to prove Theorem 1 we < Ž ␻ ␻ .< must show that this indicator is harmless as nª ⬁. Since log Z rZ_{2 n 2 nq1} F

Ž .

␭ 1 q h , it will suffice to consider n even.

PROOF. The lower bound is obvious. The upper bound is proved as follows. By conditioning on the last hitting time of 0 prior to time 2 n, we may write

n 2 n U U ␻ ␻ , ␻ , Z2 ns Z2 n q

_Ý

Z2 ny2 kE exp

ž

␭

_Ý

Ž

␻ q h ⌬i

.

i ks1 is2 ny2 kq1 q y =1 A



n , kj An , k

4

S2 ny2 ks 0

/

1.4

Ž

.

n _a 2 n k U U ␻ , ␻ , s Z2 n q

_Ý

Z2 ny2 k E exp

ž

␭

_Ý

Ž

␻ q h ⌬i

.

i bk ks1 is2 ny2 kq1 q y =1 B



n , kj Bn , k

4

S2 ny2 ks 0 .

/

Here we abbreviate the events

q

_

₄

A_{n , k}s S ) 0 for 2n y 2k q 1 F i F 2n ,_i q

_

₄

B_{n , k}s S ) 0 for 2n y 2k q 1 F i - 2n, S s 0_{i 2 n} 1.5

Ž

.

and similarly for Ayn, k, Byn, k, and their probabilities

q _< y _< aks P A

Ž

n , k S2 ny2 ks 0 s P A

.

Ž

n , k S2 ny2 ks 0 ,

.

1.6

Ž

.

_q < y _< bks P B

Ž

n , k S2 ny2 ks 0 s P B

.

Ž

n , k S2 ny2 ks 0

.

Žboth independent of n . The reason for the second equality in 1.4 is that. Ž . ⌬ s q1 for all 2n y 2k q 1 F i F 2n on the events Aq _{, B}q _{and ⌬ s y1}

i n, k n, k i

y y _{Ž .}

for all 2 ny 2k q 1 F i F 2n on the events A , B_{n, k n, k} ␻ is fixed .

Next, there exist C , C1 2) 0 such that a F C rkk 1 1r2 and bkG C rk2 3r2 for

Ž .

all kG 1. Moreover, without the factor a rb the last sum in 1.4 is preciselyk k

Z2 n␻,U. Hence C_{1 U} ␻ ␻ , 1.7 Z F 1 q n Z . I

Ž

.

2 n

ž

/

2 n C₂

Lemmas 1 and 2 complete the proof of Theorem 1.

2. Proof of Theorem 2. The proof proceeds in a sequence of five steps,

w Ž .x

organized as Sections 2.1 and 2.2. Define recall 0.21

2.1 ␺ ␭, h s ␾ ␭, h y ␭h.

Ž

.

Ž

.

Ž

.

Let 2.2 DDs



␭, h : ␺ ␭, h s 0

4

Ž

.

Ž

.

Ž

.

Ž .

be the region of delocalization see Definition 1 . Ž . 2.1. Existence, continuity and monotonicity of hc ␭ .

Ž . Ž .

STEP 1. If ␭, h g DD, then ␭ q ␦, h q ␧ g DD for all ␦, ␧ G 0 satisfying

Ž .

n _{Ž . Ž .}

PROOF. Since ␭Ýis1 ␻ q h s ␭hn q o n ⺠-a.s., we have the followingi

w Ž .x

equivalence recall that ␺ G 0 by 0.4 : ␺ ␭, h s 0

Ž

.

n

1

m lim log E exp

ž

␭

_Ý

Ž

␻ q h ⌬ y 1i

. Ž

i

.

/

F 0, ⺠-a.s. n

nª⬁ _is1

2.3

Ž

.

Ž .

Thus, to prove the claim we must show that if the r.h.s. of 2.3 holds for Ž␭, h , then it also holds for ␭ q ␦, h q ␧ . To see this, write. Ž .

n ␭ q ␦ ␻ q h q ␧ ⌬ y 1

Ž

.

_Ý

Ž

i

. Ž

i

.

is1 n s␭

_Ý

Ž

␻ q h ⌬ y 1i

. Ž

i

.

is1 2.4

Ž

.

n q

_Ý

␦ ␻ q h q ␧␭ q ␦␧ ⌬ y 1 .

Ž

i

.

Ž

i

.

is1

Since ⌬ F 1 and_i ␻ G y1, the last sum is less than or equal to 0 when_i

Ž . ␦ y1 q h q ␧␭ G 0. I w . For ␭ g 0, ⬁ define

w

x

2.5 h ␭ s inf h g 0, 1 : ␭, h g D



D

4

.

Ž

.

c

Ž .

Ž

.

Ž Ž ..

By continuity of ␺ , we have ␭, h ␭ g Dc D. It therefore follows from Step 1

Ž . Ž .

that ␭, h g DD for all h) h_c ␭ , so that the localized and the delocalized

Ž . phase are separated by a single critical curve: ␭ ª h ␭ ._c

Ž . Ž . w .

STEP 2. i ␭ ª h ␭ is continuous and nondecreasing on 0, ⬁ ._c

Ž .ii ␭ ª ␭ 1 y h ␭ is continuous and nondecreasing on 0, ⬁ .Ž _cŽ .. w .

Ž . Ž .

PROOF. i We know that ␺ ␭, h G 0 is convex in ␭ with boundary value

Ž . Ž . Ž .

␺ 0, h s 0. Therefore, if ␭, h f DD then also ␭ q ␦, h f DD for all ␦ ) 0.

Ž .

Hence ␭ ª h ␭ is nondecreasing. Step 1 shows that its slope at the point ␭_c

Ž Ž ..

is bounded from above by 1y hc ␭ r␭. Since this is finite for ␭ ) 0, we get

Ž . Ž .

continuity on 0,⬁ . Continuity at ␭ s 0 follows from Step 3 i . Ž .ii This is easily deduced from Step 1.I

Ž . 2.2. Bounds on hc ␭ .

Ž . Ž . Ž .

STEP 3. hc ␭ F 1r2␭ log cosh 2␭ . Consequently: Ž .i lim sup_{␭x 0}Ž1r␭ h ␭ F 1,. _cŽ .

1

Ž .ii lim inf_␭ª⬁ ␭ 1 y h ␭ G log 2.Ž _cŽ .. ₂

Ž .

PROOF. The claim will follow once we prove that ␭, h g DD for all hG

Ž .

Estimate ␺ ␭, h from above as follows:

n

1

␺ ␭, h s lim

Ž

.

⺕ log E exp

ž

ž

␭

_Ý

Ž

␻ q h ⌬ y 1i

. Ž

i

.

/

/

n

nª⬁ _is1

n

1

F lim inf log E ⺕ exp

ž

ž

␭

_Ý

Ž

␻ q h ⌬ y 1i

. Ž

i

.

/

/

n nª⬁ _is1 2.6

Ž

.

n 1 1 1 y2␭Ž1qh. y2␭Žy1qh.

s lim inf log E

_Ł

1_{⌬ sy14} e q e .

i

ž

/

n 2 2

nª⬁ is1

Ž

The first equality is a direct consequence of the superadditivity see Section .

1 . The r.h.s. is less than or equal to 0 as soon as the term between square brackets is less than or equal to 1.I

Ž . Ž .

STEP 4. lim inf_{␭x 0} 1r␭ h ␭ ) 0._c

PROOF. The idea is to find a strategy of the polymer for which the

Ž .

contribution to the free energy exceeds ␭h see Definition 1 . The computa-tions below are easy but a bit lengthy, due to a necessary fine-tuning of constants. The proof comes in three parts.

Ž .i As was shown in Section 1,

1 ␻ 2.7 ␾ ␭, h s lim ⺕ log Z

Ž

.

Ž

.

Ž

n

.

n nª⬁ ␻ _{w Ž .x} ␻

with Zn our partition sum see 1.3 . We begin by rewriting Zn in terms of the excursions of S away from the origin. To that end, define

␩ s 0,0 ␩jq1s inf i )



␩ : S s 0 ,j i

4

jG 0, ␶ s max j G 0: ␩ F nn



j

4

2.8

Ž

.

and 2.9 ␰ x s log cosh x.

Ž

.

Ž

.

Let ␶n 2.10 H S,␻ s ␰ ␭ ␻ q h q ␰ ␭ ␻ q h .

Ž

.

n

Ž

.

_Ý

ž

_Ý

Ž

i

.

/

ž

_Ý

Ž

i

.

/

js1 igŽ␩jy1,␩jx igŽ␩ , n␶n x

Then, using the up᎐down symmetry of S for each excursion, we can write

␻

2.11 Z s E exp H S,␻ .

Ž

.

n

Ž

n

Ž

.

.

Ž .ii The length of a typical free excursion has distribution f given by

l

_'

2

2.12 z f l s 1 y 1 y z ,

Ž

.

_Ý

Ž .

l

which is the generating function for the probability of first return to the

Ž .

be looking for a strategy of the path in which the excursions have distribution 1 l ␥

_'

2 2.13 f l s f l 1y␥ , lG 1.

Ž

.

Ž .

Ž .

_ž

_/

1y␥ Ž This corresponds to a random walk with drift ␥ towards the origin i.e.,

1

Ž w Ž .x. .

S_iq1y S s "1 with probability_i 2 1"␥ ysign S_i for iG 0 . Here 0 -␥ - 1 is a parameter we shall optimize over.

The following lemma is an intermezzo. Abbreviate ␻ s ÝI ig I␻ .i

w . w .

LEMMA 3. For all ␭ g 0, ⬁ and h g 0, 1

␥ _␥ ␾ ␭, h G sup

Ž

.

_1q␥

½

Ý

f

Ž .

l ⺕

Ž

␰ ␭␻

Ž

Ž0 , lxq␭hl

.

.

0-␥-1 l 2.14

Ž

.

_{1q␥ 1y␥} y log 1

Ž

q␥ y

.

log 1

Ž

y␥ .

.

5

2␥ 2␥

PROOF. Let P_n0 and P_n␥ denote the laws of simple random walk,

respec-Ž .

tively, random walk with drift␥, restricted to n-step paths. Then from 2.13 ,

␶ ␥ ␥ n ␥ _Ý f

_{Ž .}

l dPn n f l) ny␩␶n S s ␩ y ␩

Ž

.

Ž

.

Ž

i is0

.

Ł

j jy1 0 _f Ý f l dPn js1 l) ny␩

Ž .

2.15

Ž

.

␶n nr2 y␶ y1n 2 F 1 y

Ž

␥

.

Ž

1y␥

.

. Ž . Ž .

Using Jensen’s inequality, we get from 2.11 and 2.15 that

␥ dP_n

␻ ␥

log Zns log E exp H S,n

ž

n

Ž

␻ y log

.

_dP0

/

n dPn␥ ␥ ␥ G E H S,n

Ž

n

Ž

␻ y E log

.

.

n

ž

_dP0

/

n 2.16

Ž

.

n ␥ ␥ 2 G E H S,n

Ž

n

Ž

␻ q E ␶ q 1 log 1 y ␥ y log 1 y ␥ .

.

.

n

Ž

n

.

Ž

.

Ž

.

2  4

Now,␶ q 1 s min j G 0: ␩ ) n is a stopping time. Moreover, a straightfor-n j

␥_{Ž . Ž .}

ward calculation yields En ␩ s 1 q ␥ r␥. Therefore, the optional sampling1

theorem gives us 1 _␥ ␥ 2.17 lim E ␶ q 1 s .

Ž

.

n

Ž

n

.

n 1q␥ nª⬁ Ž . Ž . Ž ␻.

By stationarity of the ␻-sequence, the summands are functions of ␩ y ␩j jy1

only. Applying the optional sampling theorem again we get ␶n 1 ␥ lim En

_Ý

⺕

ž

␰ ␭

ž

_Ý

Ž

␻ q hi

.

/

/

n nª⬁

ž

_{js1 igŽ␩ ,␩x}

/

jy1 j 2.19

Ž

.

␩1 ␥ _␥ s En

ž

⺕

ž

␰ ␭

ž

_Ý

Ž

␻ q hi

.

/

/

/

. 1q␥ _js1

ŽTo handle the last excursion␩_{␶ q1}y␩ , note that ␰ is linearly bounded and_␶

n n

␥_.

that the excursion times have an exponential moment under P . Putting then estimates together we obtain the claim. I

Žiii The proof of Step 4 can now be complete as follows. Because. ␰ G 0 and ␰ is convex, we have ⺕

Ž

␰ ␭␻

Ž

Ž0 , lxq␭hl

.

.

G ⺠

Ž

␻Ž0 , lxG 0 ⺕

.

Ž

␰ ␭␻

Ž

Ž0 , 1xq␭hl ¬ ␻

.

Ž0 , lxG 0

.

2.20

Ž

.

1 G2␰ ␭⺕ ␻

Ž

Ž

Ž0 , lx¬␻Ž0 , lxG 0 q

.

␭hl .

.

Ž . 1r2

Next, note that there exists A) 0 such that ⺕ ␻_{Ž0, lx}¬␻_{Ž0, lx}G 0 G Al for

Ž . Ž .

all lG 1. Now pick h s␣␭ and ␥ s ␤␭ in Lemma 3, insert 2.13 and 2.20 ,

Ž . w Ž .lx 3r2Ž .

and use that f l ; 1 q y1 Brl lª ⬁ , to obtain

1 ␤ 2.21 lim inf ␾ ␭, ␣␭ G BI A,␣ , ␤ y ␤ ,

Ž

.

₂

Ž

.

Ž

.

2␣ ␣␭ ␭x0 where ⬁ dx 1 2

_'

2.22 I A,␣ , ␤ s exp y ␤ x ␰ A x q ␣ x .

Ž

.

Ž

.

H

_x3r2

ž

₂

/

Ž

.

0 Ž .

The constants ␣, ␤ can still be optimized. Pick M ) 2rBI A, 0, 0 and put

Ž .

␤ s M␣. Then, as ␣ x0, the r.h.s. of 2.21 converges to a number greater

Ž . 2

than 1. Therefore we have proved that ␾ ␭, ␣␭ ) ␣␭ for ␣, ␭ sufficiently

Ž .

small. This proves the claim in Step 4 recall Definition 1 .I

3

Ž Ž ..

STEP 5. lim_␭ª⬁ ␭ 1 y h ␭ F log 2.c 2

Ž .

PROOF. Recall Step 2 ii . The claim is proved as follows. As ␭ ª ⬁, the path will tend to make short excursions. Therefore we bound the partition sum from below by requiring all excursions to have length 2:

ŽUse the up᎐down symmetry of S for each excursion. It follows that recall. w Ž2.9 :.x 1 _␻ ␾ ␭, h s lim

Ž

.

⺕ log Z

Ž

2 n

.

2 n nª⬁ 1 1

w

x

G y log 2 q ⺕

Ž

␰ ␭ ␻ q ␻ q 2h

Ž

1 2

.

.

2 2 2.24

Ž

.

1 s y log 2 2 1 1 1 1 q

½

␰ 2␭ 1 q h q ␰ 2␭ 1 y h q ␰ 2␭h .

Ž

Ž

.

.

Ž

Ž

.

.

Ž

.

5

2 4 4 2 Ž . Ž y2 x. Ž .

Next, insert ␰ x s x y log 2 q O e xª ⬁ . Pick h s 1 y Mr␭ with M) 0 arbitrary. Then for␭ ª ⬁,

M M 1 3 1 ␾ ␭, 1 y

ž

/

G␭ 1 y

ž

/

q

½

My log 2 q ␰ 2 M

Ž

.

5

␭ ␭ 4 8 8 2.25

Ž

.

q O ey4␭ _.

Ž

.

3

As soon as MG log 2, the term between braces is greater than 0, implying2

Ž . w Ž . Ž .x

that ␭, 1 y Mr␭ f DD for ␭ sufficiently large cf. 2.2 and 2.3 . But then

Ž . w Ž .x Ž Ž ..

hc ␭ ) 1 y Mr␭ for ␭ sufficiently large cf. 2.6 , that is, ␭ 1 y h ␭ - M.c I

Ž . Ž . Ž .

Steps 2᎐5 prove Theorem 2 as well as Properties i ᎐ iii in 0.8 .

3. Proofs of Theorems 3 and 4. Essentially the same arguments as in

the proofs of Theorems 1 and 2 carry over to the continuous case. We only indicate which points need modification.

 4 Ž .

3.1. Proof of Theorem 3. We cannot insert 1 B_ts 0 , since P B s 0 s 0_t wcompare with 1.1 . However, this problem is easily handled through aŽ .x comparison argument. Recall the notation introduced in Section 0.2.

Define t U ␤ ,

˜

˜

< < 3.1 Z s inf E exp ␭ ⌬ d␤ q h ds 1 B F 1 B s x .



4

Ž

.

t

ž

_H

s

Ž

s

.

t 0

/

< <xF1 0 Then:

˜

␤,U

_˜

␤,U

_˜

Tu_{␤,U u u}

Ž .

I. Zt G Zu Ztyu for all 0F u F t, with T the left-shift T ␤ ss ␤uqsy␤ .u

˜

˜

␤,U

Ž . Ž .

II. tª 1rt ⺕ log Zt is bounded from above.

˜

_Ž u _.

_˜

_{Ž .}

III. ⺠ T ␤ g ⭈ s ⺠ ␤ g ⭈ for all u G 0.

Properties I and III are obvious. Property II holds because

t 1 2

˜

< < s log E exp

ž

2␭ t q ␭h ⌬ ds 1 B F 1 B s 0

_H

s



t

4

0

/

0 1 2 F t

Ž

2␭ q ␭h ,

.

where the equality follows from the martingale property

t ₁ t _{2 2}

˜

_w

3.3 ⺕ exp f s d␤ s exp f s ds , fg L 0, t .

Ž

.

ž

_H

Ž .

s

/

2

_H

Ž .

Ž

.

.

0 0

˜

␤,U Ž .

Thus, ␤ ª log Zt tG 0 is a superadditive process.

In order to apply the superadditive ergodic theorem, we need an additional w regularity condition that is absent in the discrete time setting, namely see

Ž . x

Kingman 1973 , Theorem 4 the following property. IV.

˜

_<

˜

␤ ,U_<

3.4 ⺕ sup log Z - ⬁ for all T - ⬁,

Ž

.

ž

s , t

/

0Fs-tFT

˜

␤,U _{w .}

where Z_{s, t} is the partition sum over the time interval s, t ; that is,

t U ␤ ,

˜

˜

< < 3.5 Z s inf E exp ␭ ⌬ d␤ q h du 1 B F 1 B s x .



4

Ž

.

s , t

ž

_H

u

Ž

u

.

t s

/

< <xF1 s

To prove Property IV, we first note that, for all ␤,

˜

␤ ,U

_˜

_{< <}

3.6 inf Z G inf inf P B F 1 ¬ B s x ) 0 for all T - ⬁.

Ž

.

s , t

Ž

t s

.

0Fs-tFT 0Fs-tFT < <xF1

˜

ŽUse Jensen’s inequality together with E⌬ ' 0. Hence it suffices to prove_u . Ž3.4 without the absolute value signs. But this we may estimate as follows:.

˜

˜

␤ ,U

⺕

ž

sup log Zs , t

/

0Fs-tFT

t

˜ ˜

F log E ⺕

ž

ž

sup exp ␭ ⌬ d␤ q h du

_H

u

Ž

u

.

/

s 0Fs-tFT 3.7

Ž

.



4

=1 ¬ B ¬F 1 B s 0 .t s

/

Ž . Ž . t

The exponent in 3.7 is bounded from above by ␭h t y s q ␭ H ⌬ d␤ .s u u

˜

Moreover, we note that under the law ⺠ the last integral is just Brownian

2

_˜

motion, since⌬ s 1 almost everywhere P-a.s. Thus we obtain_u

U ␤ ,

˜

˜

˜

3.8 ⺕ sup log Z F␭hT q log ⺕ sup exp ␭ ␤ y ␤ .

Ž

.

ž

s , t

/

ž

Ž

t s

.

/

0Fs-tFT 0Fs-tFT

< <

But the last integral is finite, because 2␭ sup0F uF T ␤ has an exponentialu moment. This proves Property IV.

Properties I᎐IV guarantee that the superadditive ergodic theorem applies:

˜

␤,U

_˜

Ž .

lim_tª⬁1rt log Z_t converges ⺠-a.s. and in mean, and is 3.9

Ž

.

˜

Ž .

Thus we have the LLN for the quantity defined in 3.1 . In order to get it for < < 4

our original partition sum, it remains to remove 1 Bt F 1 and inf< x < F1 from Ž3.1 . This will be done in two pieces..

Define t U ␤ ,

˜

˜

< < Zt

Ž

x

.

s E exp

ž

␭ ⌬ d␤ q h ds 1 B F 1 B s x ,

_H

s

Ž

s

.



t

4

0

/

0 t ␤

˜

˜

Zt

Ž

x

.

s E exp

ž

␭ ⌬ d␤ q h ds B s x .

_H

s

Ž

s

.

0

/

0 3.10

Ž

.

˜

␤,U

_˜

␤,U Ž . Ž .

In 3.9 we have the LLN for Zt s inf< x < F1Zt x . The key estimates are

now

˜

␤ ,U

_˜

␤ ,U

_˜

␤ ,U

i Z F Z 0 F C ␤ Z for all t and ␤ ,

Ž .

t t

Ž .

Ž

.

t

3.11

Ž

.

˜

␤ ,U

_˜

␤

_˜

␤ ,U

ii Z 0 F Z 0 F CtZ 0 for all t and ␤.

Ž .

t

Ž .

t

Ž .

t

Ž .

Ž .

The lower bounds are trivial. The upper bound in ii is obtained from an Ž . almost literal transcription of the proof of Lemma 2. The upper bound in i follows from a coupling argument. Indeed, since two Brownian motions starting at 0, respectively, x, hit each other after a finite time a.s., we have

˜

␤,U

_˜

␤,U

_˜

< Ž . Ž .< Ž . Ž .

sup_{< x < F1} Zt 0rZt x F C ␤ with C ␤ - ⬁, ⺠-a.s.

˜

␤

_˜

␤

Ž . Ž .

The conclusion of 3.11 is that our original partition sum Zt s Z 0 hast

˜

˜

␤,U

Ž . Ž .

the same ⺠-a.s. constant growth rate as Zt in 3.1 .

Ž x w 3.2. Proof of Theorem 4. All we have to do is show that K_cg 0, 1 since

Ž .x Ž . Ž .

the rest follows from 0.15 . As this inclusion follows from 0.8 and 0.23 , strictly speaking there is no need to give a proof here. Still, we indicate a direct proof of the lower bound for Kc because it is instructive.

Fix ␭, h. In Section 3.1 we saw that 1 ␤

˜

˜

˜

3.12 ␾ ␭, h s lim ⺕ log Z .

Ž

.

Ž

.

_Ž

t

_.

t tª⬁

We begin by expressing our partition sum in terms of the excursions of B

 4 w .

away from the origin. Let NNs s G 0: B s 0 . Then 0, ⬁ R NN s D I is as j j w

countable union of disjoint open intervals having full measure Revuz and

Ž . x

Yor 1991 , Chapter XII . Let

 4

3.13 J s j: I ; 0, t j 0 ,

Ž

.

t



j

.

4

where we reserve the index 0 for the interval between t and the last hitting time of the origin prior to time t. Then, using the up᎐down symmetry of B for each excursion, we can write

␤

˜

˜

_{< <}

3.14 Z s E exp ␰ ␭␤ q ␭h I .

Ž

.

t

ž

_Ý

_Ž

Ij j

_.

/

Here, ␤ denotes the increment of ␤ over the set I, and ␰ was defined inI Ž2.9 . The representation in 3.14 is the continuous analogue of 2.10 and. Ž . Ž . Ž2.11 ..

˜

␥

_˜

␥

Fix␥ ) 0. Let P , E denote the probability law and expectation of Brown-ian motion with drift ␥ towards the origin. Then it follows from the

w Ž . x

Cameron᎐Martin formula Chung and Williams 1990 , Theorem 9.10 ,

re-w Ž . x

spectively, the Tanaka formula Revuz and Yor 1991 , Theorem VI.1.2 , that

˜

␥ dP B

Ž

.

Ž

s 0FsFt

.

0

˜

dP 1 t t 2

s exp

H



y␥ sign B

Ž

s

.

4

dBsy

H



y␥ sign B

Ž

s

.

4

ds

2 0 0 3.15

Ž

.

1 2 < < s exp ␥ L y B y ␥ t ,

Ž

t t

.

2 w .

where L is the local time at the origin in the time interval 0, t . Next,t

1 1

␥ ␥ ␥

˜

˜

Ž .

˜

Ž 

according to Tanaka’s formula under P , we have 2E L_t s2␥ q E B 1 B_{t t} 1

4. Ž . Ž . Ž .

) 0 s2␥ q O 1 . Therefore, substituting 3.15 into 3.14 and using Jensen’s inequality, we obtain

1 1 2 ␥

˜

˜ ˜

< < 3.16 ␾ ␭, h G y ␥ q lim sup ⺕ E ␰ ␭␤ q ␭h I .

Ž

.

Ž

.

ž

ž

_Ý

_Ž

I_j j

_.

/

/

2 tª⬁ t jgJt Ž .

It remains to compute the r.h.s. of 3.16 . This is essentially parallel to Ž2.18. Ž᎐ 2.22 . In order to be able to properly count excursions, one first has to. cut away the excursions that have length smaller than ␧ and then let ␧ x0. We leave this to the reader.

4. Proof of Theorem 6. Recall the notation introduced in Sections 0.1 and 0.2. Define for the random walk model,

␰ s 1_{i ⌬ sy14},

i

? @t

1

␺ ␭, h s ⺕ log E exp y2␭t

Ž

.

ž

ž

_Ý

␰ ␻ q hi

Ž

i

.

/

/

,

t _is1

4.1

Ž

.

␺ ␭, h s lim ␺ ␭, h

Ž

.

t

Ž

.

tª⬁

and for the Brownian motion model, ␰ s 1_{s ⌬ sy14},

s

1 t

˜

˜

˜

␺ ␭, h s ⺕ log E exp y2␭ ␰ d␤ q h dst

Ž

.

ž

ž

_H

s

Ž

s

.

/

/

,

t 0 4.2

Ž

.

˜

˜

␺ ␭, h s lim ␺ ␭, h .

Ž

.

t

Ž

.

tª⬁

By the law of large numbers for␻, respectively, ␤, ␾ ␭, h s ␺ ␭, h q ␭h,

Ž

.

Ž

.

4.3

Ž

.

˜

˜

␾ ␭, h s ␺ ␭, h q ␭h.

Ž

.

Ž

.

4.1. Outline of the proof of Theorem 6. Theorem 6 is proved by a series of approximation steps. Our approximations will depend on two auxiliary pa-rameters ␧ and ␦, where 0 - ␧ - ␦. Later on, we shall let t ª ⬁, ax0, ␧ x0,

Ž . 2

␦ x0 in this order . There will be no danger in assuming that tra , tr␧, ␧ra2_{, ␦r␧ are all integers, which we shall do in order to avoid a plethora of}

brackets.

Below we shall make a number of quite similar comparisons. In order to write these in a compact form, we introduce the following notation.

Ž . Ž .

DEFINITION 2. Let ft,␧ , ␦ a, h and g_X t,␧ , ␦ a, h be real-valued functions.

Ž . X

We write f$ g if for any 0 F h - h, ␳ ) 0 satisfying 1 q ␳ h - h the Ž . following is true: there exists ␦ such that for 0 - ␦ - ␦ there exists ␧ ␦_{0 0 0}

Ž .

such that for 0-␧ - ␧ there exists a ␧, ␦ such that_{0 0}

X 2 2 2 lim sup ft ,␧ , ␦

Ž

a, h

.

y 1q

Ž

␳ g

.

tŽ1q␳ . , ␧ Ž1q␳ . , ␦ Ž1q␳ .

Ž

a 1

Ž

q␳ , h

.

.

F 0 tª⬁ 4.4

Ž

.

for 0- a - a .₀ Here ␦ , ␧ , a may depend on h, h0 0 0 X, ␳. We write f , g if f $ g and g $ f. Note that $ is a transitive relation and therefore , is an equivalence relation.

The function for which we shall make such comparisons will be of the form 1

4.5 f a, h s ⺕ log E exp y2 aH a, h ,

Ž

.

t ,␧ , ␦

Ž

.

_t

Ž

Ž

t ,␧ , ␦

Ž

.

.

.

Ž .

where the Hamiltonian Ht,␧ , ␦ a, h is a random variable defined on the

Ž

product space of the random walk and the random medium having as .

probability measure the product of P and ⺠ . Similar functions will be considered for the Brownian motion and medium.

X X _{Ž .}

Now suppose that we want to prove f$ f , where f_{t,␧ , ␦} a, h has the

X _{Ž .}

Hamiltonian Ht,␧ , ␦ a, h . We can do this in the following way:

1. Split H into two parts

4.6 Hs HŽ I .q HŽ I I ._.

Ž

.

2. Apply Holder, Jensen and Fubini to get, for

¨

␳ ) 0,

Ž . X

3. The crucial point will be, for given 1q␳ h - h, to choose the splitting in such a way that

4.8 HŽ I .s HŽ I . _{a, h}X s H

2 2 2

X _{a 1q␳ , h}X

Ž

.

t ,␧ , ␦

Ž

.

tŽ1q␳ . , ␧ Ž1q␳ . , ␦ Ž1q␳ .

Ž

Ž

.

.

Ž I I . Ž I I . _Ž X_.

and that H s H_{t,␧ , ␦} a, h, h satisfies

1 _X

y1 Ž I I .

4.9 lim sup log E ⺕ exp y2 a 1 q␳ H a, h, h F 0

Ž

.

_t

Ž

Ž

Ž

.

t ,␧ , ␦

Ž

.

.

.

tª⬁

Ž .

with ␦, ␧, a chosen appropriately in the sense of Definition 2 .

Ž . Ž . X

Clearly, 4.6 ᎐ 4.9 imply f $ f .

Before we proceed, let us agree on some conventions about constants: A,

B, C are generic positive constants, not necessarily the same at different

occurrences. They may depend on h, hX, ␳, but not on the running parameters

t, a, ␧, ␦. Ž . Return to 4.1᎐4.3 . Let 1 2 ⌿t ,␧ , ␦

Ž

a, h

.

s _a2␺tr a

Ž

a, ah ,

.

4.10

Ž

.

˜

˜

⌿t ,␧ , ␦

Ž

a, h

.

s␺ 1, ht

Ž

.

Žwhich in fact do not depend on ␦, ␧, respectively, ␦, ␧, a . What we finally.

˜

want to prove is ⌿ , ⌿, since by Definition 2 this implies Theorem 6. In order to achieve this, we shall introduce three intermediate quantities

i

Ž . Ž .

F_{t,␧ , ␦} a, h is 1, 2, 3 and prove that

1 2 3

_˜

4.11 ⌿ , F , F , F , ⌿.

Ž

.

Ž .

The proof of 4.11 comes in four steps, organized as Sections 4.2᎐4.5. In order not to overburden notations, we shall often not explicitly express dependen-cies on a, ␧, ␦.

One of the crucial aspects of the proof is that the statement of Theorem 6

Ž Ž . . Ž .

does not allow for error factors of the form exp ␬ a, ␧, ␦ t with ␬ a, ␧, ␦ tending to zero as a,␧, ␦ x0. The reader should keep this in mind.

4.2. Coarse graining of the RW. We start by defining F1_{. Divide time into}

intervals of length ␧ra2_:

2 2

4.12 I s j y 1 ␧ra , j␧ra , jG 1.

Ž

.

j

Ž

Ž

.

Put ␴ s 0 and0

4.13 ␴ s inf j G ␴ q ␦r␧ : S s 0 for some i g I , kG 1.

Ž

.

k



ky1

Ž

.

i j

4

That is, ␴ , ␴ , . . . number the intervals in which the walk returns to the_{1 2}

Ž .

origin leaving gaps of at least ␦r␧ y 1 in the numbering. Define

2 4.14 I s I l 0, tra , kG 1,

Ž

.

k

ž

D

j

/

Ž

␴ky1-jF␴k  4  4 2

For 1F k - m_{tr a}2, we set s_ks 1 if the random walk is negative just prior

to its first zero in I , and s_{␴ k}s 0 otherwise. For k s m_{tr a}2, on the other

k

hand, we set s s 1 if the random walk is negative at tra2_{, and s s 0}

k k

otherwise. Let

4.15 Z ␻ s ␻ .

Ž

.

k

Ž

.

_Ý

i

igIk

We can now define our first intermediate quantity: 1 1 1 Ft ,␧ , ␦

Ž

a, h

.

s ⺕ log E exp y2 aH_t

_Ž

_Ž

t ,␧ , ␦

Ž

a, h

.

_.

_.

, mtr a2 1 < < Ht ,␧ , ␦

Ž

a, h

.

s

Ý

s Zk



k

Ž

␻ q ah I .

.

k

4

ks1 4.16

Ž

.

STEP 1. ⌿ , F1.

PROOF. The proof comes in six parts.

Ž .i We have recall 4.1 and 4.10 :w Ž . Ž .x 1 ⌿t ,␧ , ␦

Ž

a, h

.

s ⺕ log E exp y2 aH_t

Ž

Ž

t ,␧ , ␦

Ž

a, h

.

.

.

, 2 _m 2 tra tr a Ht ,␧ , ␦

Ž

a, h

.

s

_Ý

␰ ␻ q ah si

Ž

i

.

_Ý

_Ý

␰ ␻ q ah .i

Ž

i

.

is1 k_{s1 igIk} 4.17

Ž

.

w Ž . Ž .x

Remark that, by a trivial rescaling of the parameters see 4.12 ᎐ 4.14 , we have

4.18 H a,␬ h s H 2 2 2 ␬a, h for any␬ G 0,

Ž

.

t ,␧ , ␦

Ž

.

␬ t, ␬ ␧ , ␬ ␦

Ž

.

and the same for H1_{. Furthermore, for any h , h G 0,} 1 2 Ht ,␧ , ␦

Ž

a, h1

.

y Ht ,1␧ , ␦

Ž

a, h2

.

m_{tr a}2 m_{tr a}2 s a h y h

Ž

1 2

.

_Ý

_Ý

␰ qi

_Ý

_Ý

Ž

ah2q␻i

. Ž

␰ y s .i k

.

ks1 igIk ks1 igIk 4.19

Ž

.

In order to prove⌿ $ F1_{, we split Hs H}Ž I .q HŽ I I . _with

4.20 HŽ I . s H1 _{a, 1}q␳ hX s H

2 2 2

1 _{a 1q␳ , h}X _,

Ž

.

t ,␧ , ␦ t ,␧ , ␦

Ž

Ž

.

.

tŽ1q␳ . , ␧ Ž1q␳ . , ␦ Ž1q␳ .

Ž

Ž

.

.

Ž . Ž . X Ž I I .

and take the r.h.s. of 4.19 with h1s h, h s 1 q2 ␳ h as H . On the other hand, in order to prove F1_{$ ⌿, we split H}1_{s H}XŽ I ._{q H}XŽ II . _with

4.21 HXŽ I . s H a, 1q␳ hX s H 2 2 2 a 1q␳ , hX ,

Ž

.

t ,␧ , ␦ t ,␧ , ␦

Ž

Ž

.

.

tŽ1q␳ . , ␧ Ž1q␳ . , ␦ Ž1q␳ .

Ž

Ž

.

.

Ž . Ž . X XŽ II .

and take minus the r.h.s. of 4.19 with h1s 1 q␳ h , h s h as H2 . We Ž

shall prove that if we choose a, ␧, ␦ small enough in this order because of

. Ž .

Definition 2 , then also the requirement in 4.9 is met:

1 _X

y1 Ž I I .

4.22 lim sup log E ⺕ exp y2 a 1 q␳ H a, h, h F 0,

Ž

.

_t

Ž

Ž

Ž

.

t ,␧ , ␦

Ž

.

.

.

tª⬁

Ž .ii To prove 4.22 , we first carry out the expectation overŽ . ␻: y1 Ž I I . ⺕ exp y2 a 1 q

Ž

Ž

␳

.

Ht ,␧ , ␦

Ž

a, h, h⬘

.

.

2 mtr a X 2 y1 s exp y2 a 1 q

Ž

␳

.

Ž

hy 1 q

Ž

␳ h

.

.

Ý

Ý

␰i k_{s1 igIk} 2 mtr a X 2 y1 = exp y2 a 1 q

Ž

␳

.

Ž

1q␳ h

.

Ý

Ý

Ž

␰ y si k

.

k_{s1 igIk} 4.23

Ž

.

2 mtr a y1

= exp

Ý

Ý

log cosh 2 a 1



Ž

q␳

.

Ž

␰ y si k

.

4

ks1 igI_k 2 2 mtr a mtr a 2 < < 2 F exp Aa

Ý

Ý

␰ y s y Bai k

Ý

Ý

␰i ks1 igIk ks1 igIk Ž X .

for some constants A, B) 0 which depend on h, h ,␳ but not on t, a, ␧, ␦ . The crucial point is that the second summand in the exponent is able to kill the first summand for arbitrary A, B) 0, provided the parameters a, ␧, ␦ are chosen appropriately. Thus, to complete the proof of ⌿ $ F1_{, it remains to}

show that

2 2

mtr a mtr a

1

2 < < 2

4.24 lim sup log E exp Aa ␰ y s y Ba ␰ F 0.

Ž

.

_Ý

_Ý

i k

_Ý

_Ý

i

t

ž

/

tª⬁ ks1 igIk ks1 igIk

This is a problem about simple random walk and its zeroes. The only difference between HŽ I I . _{and H}XŽ II . _{is that the second summand on the r.h.s.}

comes with a minus and h , h interchanged. However, this obviously leads1 2

Ž . Ž .

to the same type of estimate as 4.23 . Therefore 4.24 proves Step 1 completely.

Žiii To prove 4.24 , we introduce the standard return times of the random. Ž . walk:



4

T0s 0, Tls inf i ) Tly1: Sis 0 , lG 1, l_{tr a}2s min l: T G tra



_l 2

4

4.25

Ž

.

and the excursion times

4.26 ␶ s T y T , 1F l - l 2, ␶ s tra2 y T .

Ž

.

l l ly1 tr a ltr a2

Ž

.

ltr a2y1

We further define ␩ s 1 if the sign of the lth excursion is negative, andl ␩ s 0 otherwise. Then, obviously, we can write the second summand in thel

mtr a2 < <

Next we estimate the first summand Ýks1 Ýig I_k ␰ y s in terms of the samei k quantities. Put t0s 0, and let t be the first zero of the random walk in thek

Ž 2. 2 Ž x

interval I_␴ 1F k - m_{tr a} , and t_{m 2}s tra . On the time interval t_ky1, t_k

k tr a

the random walk makes a number of excursions, and s just depends on thek sign of the last one; that is, sks 1 if and only if this is negative. By construction, only this last excursion can have length greater than or equal to Ž␦r␧ ␧ra s ␦ra see 4.12 and 4.13 . It follows that if i is not in an.Ž 2. 2 w Ž . Ž .x excursion of length less than ␦ra2 _{and i does not belong to one of the}

intervals I , then_␴

k

4.28 ␰ s s for the k with ig I .

Ž

.

i k k

Ž < < 2.

From these considerations we obtain recall that I_␴ s␧ra

k m_{tr a}2 l_{tr a}2 ␦ ␧ < < 2 4.29 ␰ y s F ␶ 1 ␶ - q m .

Ž

.

_Ý

_Ý

i k

_Ý

l

½

l _a2

5

tr a _a2 ks1 igIk ls1 Ž . Ž . Ž .

Combining 4.27 and 4.29 we see that, in order to prove 4.24 , it now suffices to show that

2

ltr a

1 ␦

2

lim sup log E exp Aa

_Ý

␶ 1 ␶ -_l

½

_{l 2}

5

ž

t a tª⬁ ls1 4.30

Ž

.

2 ltr a 2 2 qA␧ mtr a y Ba

Ý

␶ ␩l l

/

F 0 ls1 for appropriate a,␧, ␦.

Živ As the. ␩ ’s are independent of the ␶ ’s 0 or 1 with probability 1r2l l Ž

. 2

each , we can integrate out the former and replaceyBa Ýl l␶ ␩ in the r.h.s. ofl

1 1 2

Ž4.30 by Ý log. l Ž2q exp yBa2 Ž ␶ . We next claim thatl..

l_{tr a}2

1 1 1 2

2

A␧ m

_Ž

_{tr a} y 1 q

_.

₂

_Ý

log

_Ž

₂q exp yBa₂

_Ž

␶_l

_.

_.

F 0 4.31

Ž

.

ls1

for 0-␧ - ␧ ␦ .0

Ž

.

Ž x Ž 2.

To see why, pick any of the intervals t_ky1, t_k 1F k - m_{tr a} . If any of the

Ž x 2

excursions on t_ky1, t_k has length greater than or equal to ␦ra , then for the

l indexing this excursion we have

1 1 2 1 1

4.32 log q exp yBa␶ F log q exp yB␦

Ž

.

Ž

2 2

Ž

l

.

.

Ž

2 2

Ž

.

.

and hence

1 1 1 2

4.33 A␧ q log q exp yBa ␶ F 0 for 0 -␧ - ␧ ␦ .

Ž

.

2

Ž

2 2

Ž

l

.

.

0

Ž

.

Therefore

Žk.

1 1 1 2

4.34 A␧ q log q exp yBa␶ F 0 for 0 -␧ - ␧ ␦ ,

Ž

.

2

Ý

Ž

2 2

_Ž

l

_.

.

0

Ž

.