New bounds for the free energy of directed polymers in dimension 1+1 and 1+2
aa r X i v : . [ m a t h - ph ] N ov NEW BOUNDS FOR THE FREE ENERGY OF DIRECTEDPOLYMERS IN DIMENSION (cid:0) AND (cid:0) HUBERT LACOIN
Abstract.
We study the free energy of the directed polymer in random environmentmodel in dimension 1 (cid:0) (cid:0)
2. For dimension one, we improve the statement ofComets and Vargas in [8] concerning very strong disorder by giving sharp estimates onthe free energy at high temperature. In dimension two, we prove that very strong disor-der holds at all temperatures, thus solving a long standing conjecture in the field.2000
Mathematics Subject Classification: 82D60, 60K37, 82B44Keywords: Free Energy, Directed Polymer, Strong Disorder, Localization, Fractional Mo-ment Estimates, Quenched Disorder, Coarse Graining. Introduction
The model.
We study a directed polymer model introduced by Huse and Henley(in dimension 1 (cid:0)
1) [18] with the purpose of investigating impurity-induced domain-wallroughening in the 2D-Ising model. The first mathematical study of directed polymers inrandom environment was made by Imbrie and Spencer [19], and was followed by numerousauthors [19, 3, 1, 23, 4, 6, 9, 5, 8, 26] (for a review on the subject see [7]). Directed polymersin random environment model, in particular, polymer chains in a solution with impurities.In our set–up the polymer chain is the graph tp i, S i qu ¤ i ¤ N of a nearest–neighbor pathin Z d , S starting from zero. The equilibrium behavior of this chain is described by ameasure on the set of paths: the impurities enter the definition of the measure as disorderedpotentials , given by a typical realization of a field of i.i.d. random variables ω (cid:16) t ω p i,z q ; i P N , z P Z d u (with associated law Q ). The polymer chain will tend to be attracted by largervalues of the environment and repelled by smaller ones. More precisely, we define theHamiltonian H N p S q : (cid:16) N ¸ i (cid:16) ω i,S i . (1.1)We denote by P the law of the simple symmetric random walk on Z d starting at 0 (inthe sequel P f p S q , respectively Qg p ω q , will denote the expectation with respect to P ,respectively Q). One defines the polymer measure of order N at inverse temperature β as µ p β q N p S q (cid:16) µ N p S q : (cid:16) Z N exp p βH N p S qq P p S q , (1.2)where Z N is the normalization factor which makes µ N a probability measure Z N : (cid:16) P exp p βH N p S qq . (1.3)We call Z N the partition function of the system. In the sequel, we will consider the caseof ω p i,z q with zero mean and unit variance and such that there exists B P p , such that λ p β q (cid:16) log Q exp p βω p , qq 8 , for 0 ¤ β ¤ B. (1.4) Finite exponential moments are required to guarantee that QZ N . The model can bedefined and it is of interest also with environments with heavier tails (see e.g. [26]) but wewill not consider these cases here.1.2. Weak, strong and very strong disorder.
In order to understand the role ofdisorder in the behavior of µ N , as N becomes large, let us observe that, when β (cid:16) µ N isthe law of the simple random walk, so that we know that, properly rescaled, the polymerchain will look like the graph of a d -dimensional Brownian motion. The main questionsthat arise for our model for β ¡ N , and how the polymer measure looks like whendiffusivity does not hold.Many authors have studied diffusivity in polymer models: in [3], Bolthausen remarkedthat the renormalized partition function W N : (cid:16) Z N {p QZ N q has a martingale propertyand proved the following zero-one law: Q " lim N Ñ8 W N (cid:16) * P t , u . (1.5)A series of paper [19, 3, 1, 23, 9] lead to Q " lim N Ñ8 W N (cid:16) * (cid:16) ñ diffusivity , (1.6)and a consensus in saying that this implication is an equivalence. For this reason, it isnatural and it has become customary to say that weak disorder holds when W N convergesto some non-degenerate limit and that strong disorder holds when W N tends to zero.Carmona and Hu [4] and Comets, Shiga and Yoshida [6] proved that strong disorderholds for all β in dimension 1 and 2. The result was completed by Comets and Yoshida[9]: we summarize it here Theorem 1.1.
There exists a critical value β c (cid:16) β c p d q P r , (depending of the law ofthe environment) such that (cid:13) Weak disorder holds when β β c . (cid:13) Strong disorder holds when β ¡ β c .Moreover: β c p d q (cid:16) for d (cid:16) , β c p d q P p , for d ¥ . (1.7)We mention also that the case β c p d q (cid:16) 8 can only occur when the random variable ω p , q is bounded.In [4] and [6] a characterization of strong disorder has been obtained in term of local-ization of the polymer chain: we cite the following result [6, Theorem 2.1] Theorem 1.2. If S p q and S p q are two i.i.d. polymer chains, we have Q " lim N Ñ8 W N (cid:16) * (cid:16) Q N ¥ µ b N (cid:1) p S p q N (cid:16) S p q N q (cid:16) 8+ (1.8) REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0)
Moreover if Q t lim N Ñ8 W N (cid:16) u (cid:16) there exists a constant c (depending on β and thelaw of the environment) such that for (cid:1) c log W N ¤ N ¸ n (cid:16) µ b n (cid:1) p S p q n (cid:16) S p q n q ¤ (cid:1) c log W N . (1.9)One can notice that (1.9) has a very strong meaning in term of trajectory localizationwhen W N decays exponentially: it implies that two independent polymer chains tend toshare the same endpoint with positive probability. For this reason we introduce now thenotion of free energy, we refer to [6, Proposition 2.5] and [9, Theorem 3.2] for the followingresult: Proposition 1.3.
The quantity p p β q : (cid:16) lim N Ñ8 N log W N , (1.10) exists Q -a.s., it is non-positive and non-random. We call it the free energy of the model,and we have p p β q (cid:16) lim N Ñ8 N Q log W N (cid:16) : lim N Ñ8 p N p β q . (1.11) Moreover p p β q is non-increasing in β . We stress that the inequality p p β q ¤ annealing bound. In view on(1.9), it is natural to say that very strong disorder holds whenever p p β q
0. One canmoreover define s β c p d q the critical value of β for the free energy i.e. : p p β q β ¡ s β c p d q . (1.12)Let us stress that, from the physicists’ viewpoint, s β c p d q is the natural critical point becauseit is a point of non-analyticity of the free energy (at least if s β c p d q ¡ s β c p d q ¥ β c p d q . It is widely believed that s β c p d q (cid:16) β c p d q , i.e.that there exists no intermediate phase where we have strong disorder but not very strongdisorder . However, this is a challenging question: Comets and Vargas [8] answered it indimension 1 (cid:0) s β c p q (cid:16)
0. In this paper, we make their result moreprecise. Moreover we prove that s β c p q (cid:16) Presentation of the results.
The first aim of this paper is to sharpen the result ofComets and Vargas on the 1 (cid:0) p p β q for small β . Our result is the following Theorem 1.4.
When d (cid:16) and the environment satisfies (1.4) , there exist constants c and β B (depending on the distribution of the environment) such that for all ¤ β ¤ β we have (cid:1) c β r (cid:0) p log β q s ¤ p p β q ¤ (cid:1) cβ . (1.13)We believe that the logarithmic factor in the lower bound is an artifact of the method.In fact, by using replica-coupling, we have been able to get rid of it in the Gaussian case. Theorem 1.5.
When d (cid:16) and the environment is Gaussian, there exists a constant c such that for all β ¤ . (cid:1) c β ¤ p p β q ¤ (cid:1) cβ . (1.14) HUBERT LACOIN
These estimates concerning the free energy give us some idea of the behavior of µ N for small β . Indeed, Carmona and Hu in [4, Section 7] proved a relation between p p β q and the overlap (although their notation differs from ours). This relation together withour estimates for p p β q suggests that, for low β , the asymptotic contact fraction betweenindependent polymers lim N Ñ8 N µ b N N ¸ n (cid:16) t S p q n (cid:16) S p q n u , (1.15)behaves like β .The second result we present is that s β c p q (cid:16)
0. As for the 1 (cid:0) p p β q for β close to zero. Theorem 1.6.
When d (cid:16) , there exist constants c and β such that for all β ¤ β , (cid:1) exp (cid:2)(cid:1) cβ (cid:10) ¤ p p β q ¤ (cid:1) exp (cid:2)(cid:1) cβ (cid:10) , (1.16) so that s β c p q (cid:16) , (1.17) and is a point of non-analyticity for p p β q . Remark 1.7.
After the appearance of this paper as a preprint, the proof of the aboveresult has been adapted by Bertin [2] to prove the exponential decay of the partition func-tion for
Linear Stochastic Evolution in dimension 2, a model that is a slight generalisationof directed polymer in random environment.
Remark 1.8.
Unlike in the one dimensional case, the two bounds on the free energyprovided by our methods do not match. We believe that the second moment method, thatgives the lower bound is quite sharp and gives the right order of magnitude for log p p β q .The method developped in [16] to sharpen the estimate on the critical point shift forpinning models at marginality adapted to the context of directed polymer should be ableto improve the result, getting p p β q ¤ (cid:1) exp p(cid:1) c ε β (cid:1)p (cid:0) ε qq for all β ¤ ε .1.4. Organization of the paper.
The various techniques we use have been inspired byideas used successfully for another polymer model, namely the polymer pinning on a defectline (see [24, 14, 10, 25, 15]).However the ideas we use to establish lower bounds differ sensibly from the ones leadingto the upper bounds. For this reason, we present first the proofs of the upper bound resultsin Section 2, 3 and 4. The lower bound results are proven in Section 5, 6 and 7.To prove the lower bound results, we use a technique that combines the so-called frac-tional moment method and change of measure. This approach has been first used forpinning model in [10] and it has been refined since in [25, 15]. In Section 2, we provea non-optimal upper bound for the free energy in the case of Gaussian environment indimension 1 (cid:0) (cid:0)
1, and in Section 4 we prove ourupper bound for the free energy in dimension 1 (cid:0) very strong disorder holds for all β . These sections are placed in increasing order of technical complexity, andtherefore, should be read in that order. REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0)
Concerning the lower–bounds proofs: Section 5 presents a proof of the lower bound ofTheorem 1.4. The proof combines the second moment method and a directed percolationargument. In Section 6 the optimal bound is proven for Gaussian environment, with aspecific Gaussian approach similar to what is done in [24]. In Section 7 we prove the lowerbound for arbitrary environment in dimension 1 (cid:0)
2. These three parts are completelyindependent of each other.2.
Some warm up computations
Fractional moment.
Before going into the core of the proof, we want to presenthere the starting step that will be used repeatedly thourough Sections 2, 3 and 4. Wewant to find an upper–bound for the quantity p p β q (cid:16) lim N Ñ8 N Q log W N . (2.1)However, it is not easy to handle the expectation of a log, for this reason we will use thefollowing trick . Let θ P p , q , we have (by Jensen inequality) Q log W N (cid:16) θ Q log W θN ¤ θ log QW θN . (2.2)Hence p p β q ¤ lim inf N Ñ8 θN log QW θN . (2.3)We are left with showing that the fractional moment QW θN decays exponentially which isa problem that is easier to handle.2.2. A non optimal upper–bound in dimension (cid:0) . To introduce the reader to thegeneral method used in this paper, combining fractional moment and change of measure,we start by proving a non–optimal result for the free–energy, using a finite volume criterion.As a more complete result is to be proved in the next section, we restrict to the Gaussiancase here. The method used here is based on the one of [8], marorizing the free energy ofthe directed polymer by the one of multiplicative cascades. Let us mention that is has beeshown recently by Liu and Watbled [22] that this majoration is in a sense optimal, theyobtained this result by improving the concentration inequality for the free energy.The idea of combining fractional moment with change of measure and finite volumecriterion has been used with success for the pinning model in [10].
Proposition 2.1.
There exists a constant c such that for all β ¤ p p β q ¤ (cid:1) cβ p| log β | (cid:0) q (2.4) Proof of Proposition 2.1 in the case of Gaussian environment.
For β sufficiently small, wechoose n to be equal to Q C | log β | β U for a fixed constant C (here and thourough the paperfor x P R , r x s , respectively t x u will denote the upper, respectively the lower integer partof x ) and define θ : (cid:16) (cid:1) p log n q(cid:1) . For x P Z we define W n p x q : (cid:16) P exp (cid:3) n ¸ i (cid:16) r βω p i,S i q (cid:1) β { s(cid:11) t S n (cid:16) x u . (2.5) HUBERT LACOIN
Note that ° x P Z W n p x q (cid:16) W n . We use a statement which can be found in the proof ofTheorem 3.3. in [8]: log QW θnm ¤ m log Q ¸ x P Z r W n p x qs θ m P N . (2.6)This combined with (2.3) implies that p p β q ¤ θn log Q ¸ x P Z r W n p x qs θ . (2.7)Hence, to prove the result, it is sufficient to show that Q ¸ x P Z r W n p x qs θ ¤ e (cid:1) , (2.8)for our choice of θ and n .In order to estimate Q r W n p x qs θ we use an auxiliary measure r Q . The region where thewalk p S i q ¤ i ¤ n is likely to go is J n (cid:16) pr , n s (cid:2) r(cid:1) C ? n, C ? n sq X N (cid:2) Z where C is a bigconstant.We define r Q as the measure under which the ω i,x are still independent Gaussian variableswith variance 1, but such that r Qω i,x (cid:16) (cid:1) δ n p i,x qP J n where δ n (cid:16) {p n { ? C log n q . Thismeasure is absolutely continuous with respect to Q andd r Q d Q (cid:16) exp (cid:4)(cid:5)(cid:1) ¸p i,x qP J n (cid:18) δ n ω i,x (cid:0) δ n (cid:26)(cid:12)(cid:13) . (2.9)Then we have for any x P Z , using the H¨older inequality we obtain, Q (cid:17) W n p x q θ (cid:25) (cid:16) r Q (cid:18) d Q d r Q p W n p x qq θ (cid:26) ¤ (cid:3) r Q (cid:19)(cid:2) d Q d r Q (cid:10) (cid:1) θ (cid:27)(cid:11) (cid:1) θ (cid:1) r QW n p x q(cid:9) θ . (2.10)The first term on the right-hand side can be computed explicitly and is equal to (cid:3) Q (cid:2) d Q d r Q (cid:10) θ (cid:1) θ (cid:11) (cid:1) θ (cid:16) exp (cid:2) θδ n p (cid:1) θ q J n (cid:10) ¤ e, (2.11)where the last inequality is obtained by replacing δ n and θ by their values (recall θ (cid:16) (cid:1) p log n q(cid:1) ). Therefore combining (2.10) and (2.11) we get that Q ¸ x P Z p W n p x qq θ ¤ e ¸| x |¤ n (cid:1) r QW n p x q(cid:9) θ . (2.12)To bound the right–hand side, we first get rid of the exponent θ in the following way: ¸| x |¤ n n (cid:1) θ (cid:1) r QW n p x q(cid:9) θ ¤ n (cid:1) θ t x P Z , | x | ¤ n such that r QW n p x q ¤ n (cid:1) u(cid:0) ¸| x |¤ n t r QW n p x q¡ n (cid:1) u r QW n p x q n p (cid:1) θ q . (2.13) REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0) If n is sufficiently large ( i.e., β sufficiently small) the first term on the right-hand side issmaller than 1 { n so that ¸| x |¤ n (cid:1) r QW n p x q(cid:9) θ ¤ exp p q r QW n (cid:0) n . (2.14)We are left with showing that the expectation of W n with respect to the measure r Q issmall. It follows from the definition of r Q that r QW n (cid:16) P exp p(cid:1) βδ n t i | p i, S i q P J n uq , (2.15)and therefore r QW n ¤ P t the trajectory S goes out of J n u (cid:0) exp p(cid:1) nβδ n q . (2.16)One can choose C such that the first term is small, and the second term is equal toexp p(cid:1) βn { {? C log n q ¤ exp p(cid:1) C { { ? C q that can be arbitrarily small by choosing C large compared to p C q { . In that case (2.8) is satisfied and we have p p β q ¤ θn log e (cid:1) ¤ (cid:1) β C | log β | (2.17)for small enough β . (cid:3) Proof of the upper bound of Theorem 1.4 and 1.5
The upper bound we found in the previous section is not optimal, and can be improvedby replacing the finite volume criterion (2.8) by a more sophisticated coarse grainingmethod. The technical advantage of the coarse graining we use, is that we will not haveto choose the θ of the fractional moment close to 1 as we did in the previous section andthis is the way we get rid of the extra log factor we had. The idea of using this type ofcoarse graining for the copolymer model appeared in [25] and this has been a substantialsource of inspiration for this proof.We will prove the following result first in the case of Gaussian environment, and thenadapt the proof to general environment. Proof in the case of Gaussian environment.
Let n be the smallest squared integer biggerthan C β (cid:1) (if β is small we are sure that n ¤ C β (cid:1) ). The number n will be used inthe sequel of the proof as a scaling factor. Let θ θ (cid:16) { N (cid:16) nm (where m is meant to tend to infinity).Let I k denote the interval I k (cid:16) r k ? n, p k (cid:0) q? n q . In order to estimate QW θN wedecompose W N according to the contribution of different families path: W N (cid:16) ¸ y ,y ,...,y m P Z | W p y ,y ,...,y m q (3.1)where | W p y ,y ,...,y m q (cid:16) P exp (cid:19) N ¸ i (cid:16) (cid:2) βω i,S i (cid:1) β (cid:10) t S in P I yi , i (cid:16) ,...,m u(cid:27) . (3.2)Then, we apply the inequality p° a i q θ ¤ ° a θi (which holds for any finite or countablecollection of positive real numbers) to this decomposition and average with respect to Q to get, HUBERT LACOIN
PSfrag replacements O n n n n n n n n (cid:0)? n (cid:0) ? n (cid:0) ? n (cid:0) ? n (cid:1)? n (cid:1) ? n (cid:1) ? n Figure 1.
The partition of W nm into | W p y ,...,y m q is to be viewed as a coarse grain-ing. For m (cid:16) p y , . . . , y q (cid:16) p , (cid:1) , , , , (cid:1) , (cid:1) , q , | W p y ,...,y m q n corresponds to thecontribution to W N of the path going through the thick barriers on the figure. QW θnm ¤ ¸ y ,y ,...,y m P Z Q | W θ p y ,y ,...,y m q . (3.3)In order to estimate Q | W θ p y ,y ,...,y m q , we use an auxiliary measure as in the previous section.The additional idea is to make the measure change depend on y , . . . , y m .For every Y (cid:16) p y , . . . , y m q we define the set J Y as J Y : (cid:16) p km (cid:0) i, y k ? n (cid:0) z q , k (cid:16) , . . . , m (cid:1) , i (cid:16) , . . . , n, | z | ¤ C ? n ( , (3.4)where y is equal to zero. Note that for big values of n and m J Y (cid:18) C mn { (3.5)We define the measure r Q Y the measure under which the ω p i,x q are independent Gaussianvariables with variance 1 and mean r Q Y ω p i,x q (cid:16) (cid:1) δ n tp i,x qP J Y u where δ n (cid:16) n (cid:1) { C (cid:1) { .The law r Q Y is absolutely continuous with respect to Q and its density is equal tod r Q Y d Q p ω q (cid:16) exp (cid:4)(cid:5)(cid:1) ¸p i,x qP J Y (cid:16) δ n ω p i,x q (cid:0) δ n { (cid:24)(cid:12)(cid:13) . (3.6)Using H¨older inequality with this measure as we did in the previous section, we obtain Q (cid:17)| W θ p y ,y ,...,y m q(cid:25) (cid:16) r Q Y (cid:18) d Q d r Q Y | W θ p y ,y ,...,y m q(cid:26)¤ r Q Y (cid:3)(cid:19)(cid:2) d Q d r Q Y (cid:10) (cid:1) θ (cid:27)(cid:11) (cid:1) θ (cid:1) r Q Y | W p y ,...,y m q(cid:9) θ . (3.7)The value of the first term can be computed explicitly REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0)
PSfrag replacements O n n n n n n n n (cid:0)? n (cid:0) ? n (cid:0) ? n (cid:0) ? n (cid:1)? n (cid:1) ? n (cid:1) ? n Region where the environment is modified
Figure 2.
This figure represent in a rough way the change of measure Q Y . The regionwhere the mean of ω p i,x q is lowered (the shadow region on the figure) corresponds to theregion where the simple random walk is likely to go, given that it goes through the thickbarriers. (cid:3) Q (cid:19)(cid:2) d Q d r Q Y (cid:10) θ (cid:1) θ (cid:27)(cid:11) (cid:1) θ (cid:16) exp (cid:2) J Y θδ n p (cid:1) θ q (cid:10) ¤ exp p m q , (3.8)where the upper bound is obtained by using the definition of δ n , (3.5) and the fact that θ (cid:16) { r Q Y | W p y ,...,y m q (cid:16) P exp p(cid:1) βδ n t i |p i, S i q P J Y uq t S kn P I yk , k Pr ,m su . (3.9)We define J : (cid:16) tp i, x q , i (cid:16) , . . . , n, | x | ¤ C ? n us J : (cid:16) tp i, x q , i (cid:16) , . . . , n, | x | ¤ p C (cid:1) q? n u . (3.10)Equation (3.9) implies that (recall that P x is the law of the simple random walk startingfrom x , and that we set y (cid:16) qr Q Y | W p y ,...,y m q ¤ m ¹ k (cid:16) max x P I P x exp p(cid:1) βδ n t i : p i, S i q P J uq t S n P I yk (cid:1) yk (cid:1) u . (3.11)Combining this with (3.1), (3.7) and (3.8) we havelog QW θN ¤ m (cid:19) (cid:0) log ¸ y P Z (cid:2) max x P I P x exp p(cid:1) βδ n t i : p i, S i q P J uq t S n P I y u(cid:10) θ (cid:27) . (3.12) If the quantity in the square brackets is smaller than (cid:1)
1, by equation (2.3) we have p p β q ¤ (cid:1) { n . Therefore, to complete the proof it is sufficient to show that ¸ y P Z (cid:2) max x P I P x exp p(cid:1) βδ n t i : p i, S i q P J uq t S n P I y u(cid:10) θ (3.13)is small. To reduce the problem to the study of a finite sum, we observe (using some wellknown result on the asymptotic behavior of random walk) that given ε ¡ R such that ¸| y |¥ R (cid:2) max x P I P x exp p(cid:1) βδ n t i : p i, S i q P J uq t S n P I y u(cid:10) θ ¤ ¸| y |¥ R max x P I p P x t S n P I y uq θ ¤ ε. (3.14)To estimate the remainder of the sum we use the following trivial bound ¸| y | R (cid:2) max x P I P x exp p(cid:1) βδ n t i : p i, S i q P J uq t S n P I y u(cid:10) θ ¤ R (cid:2) max x P I P x exp p(cid:1) βδ n t i : p i, S i q P J uq(cid:10) θ . (3.15)Then we get rid of the max in the sum by observing that if a walk starting from x makesa step in J , the walk with the same increments starting from 0 will make the same stepin s J (recall (3.10)).max x P I P x exp p(cid:1) βδ n t i : p i, S i q P J uq ¤ P exp (cid:0)(cid:1) βδ n i |p i, S i q P s J ((cid:8) . (3.16)Now we are left with something similar to what we encountered in the previous section P exp (cid:0)(cid:1) βδ n i : p i, S i q P s J ((cid:8) ¤ P t the random walk goes out of s J u (cid:0) exp p(cid:1) nβδ n q . (3.17)If C is chosen large enough, the first term can be made arbitrarily small by choosing C large, and the second is equal to exp p(cid:1) C (cid:1) { {? C q and can be made also arbitrarily smallif C is chosen large enough once C is fixed. An appropriate choice of constant and theuse of (3.16) and (3.17) can leads then to2 R (cid:2) max x P I P x exp p(cid:1) βδ n t i : p i, S i q P J uq(cid:10) θ ¤ ε. (3.18)This combined with (3.14) completes the proof. (cid:3) Proof of the general case.
In the case of a general environment, some modifications haveto be made in the proof above, but the general idea remains the same. In the change ofmeasure one has to change the shift of the environment in J Y (3.6) by an exponential tiltof the measure as followd r Q Y d Q p β q (cid:16) exp (cid:4)(cid:5)(cid:1) ¸p i,z qP J Y (cid:16) δ n ω p i,z q (cid:0) λ p(cid:1) δ n q(cid:24)(cid:12)(cid:13) . (3.19) REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0)
The formula estimating the cost of the change of measure (3.8) becomes (cid:3) Q (cid:2) d Q d r Q Y (cid:10) θ (cid:1) θ (cid:11) (cid:1) θ (cid:16) exp (cid:2) J Y (cid:18)p (cid:1) θ q λ (cid:2) θδ n (cid:1) θ (cid:10) (cid:0) θλ p(cid:1) δ n q(cid:26)(cid:10) ¤ exp p m q , (3.20)where the last inequality is true if β n is small enough if we consider that θ (cid:16) { λ p x q x Ñ (cid:18) x { ω has 0 mean and unit variance). The next thing we haveto do is to compute the effect of this change of measure in this general case, i.e. find anequivalent for (3.9). When computing r Q Y | W p y ,...,y m q , the quantity r Q Y exp p βω , (cid:1) λ p β qq (cid:16) exp r λ p β (cid:1) δ n q (cid:1) λ p(cid:1) δ n q (cid:1) λ p β qs (3.21)appears instead of exp p(cid:1) βδ n q . Using twice the mean value theorem, one gets that thereexists h and h in p , q such that λ p β (cid:1) δ n q (cid:1) λ p(cid:1) δ n q (cid:1) λ p β q (cid:16) δ n (cid:16) λ hδ n q (cid:1) λ β (cid:1) hδ n q(cid:24) (cid:16) (cid:1) βδ n λ hδ n (cid:0) h β q . (3.22)And as ω has unit variance lim x Ñ λ x q (cid:16)
1. Therefore if β and δ n are chosen smallenough, the right-hand side of the above is less than (cid:1) βδ n {
2. So that (3.9) can be replacedby r Q Y | W p y ,...,y m q ¤ P exp (cid:2)(cid:1) βδ n t i |p i, S i q P J Y u(cid:10) t S kn P I yk , k Pr ,m su . (3.23)The remaining steps follow closely the argument exposed for the Gaussian case. (cid:3) Proof of the upper bound in Theorem 1.6
In this section, we prove the main result of the paper: very strong disorder holds at alltemperature in dimension 2.The proof is technically quite involved. It combines the tools of the two previoussections with a new idea for the change a measure: changing the covariance structure ofthe environment. We mention that this idea was introduced recently in [15] to deal withthe marginal disorder case in pinning model. We choose to present first a proof for theGaussian case, where the idea of the change a measure is easier to grasp.Before starting, we sketch the proof and how it should be decomposed in different steps:(a) We reduce the problem by showing that it is sufficient to show that for some realnumber θ QW θN decays exponentially with N .(b) We use a coarse graining decomposition of the partition function by splitting itinto different contributions that corresponds to trajectories that stays in a largecorridor. This decomposition is similar to the one used in Section 3.(c) To estimate the fractional moment terms appearing in the decomposition, wechange the law of the environment around the corridors corresponding to eachcontribution. More precisely, we introduce negative correlations into the Gaussianfield of the environment. We do this change of measure in such a way that thenew measure is not very different from the original one.(d) We use some basic properties of the random walk in Z to compute the expectationunder the new measure. Proof for Gaussian environment.
We fix n to be the smallest squared integer bigger thanexp p C { β q for some large constant C to be defined later, for small β we have n ¤ exp p C { β q . The number n will be used in the sequel of the proof as a scaling factor.For y (cid:16) p a, b q P Z we define I y (cid:16) r a ? n, p a (cid:0) q? n (cid:1) s (cid:2) r b ? n, p b (cid:0) q? n (cid:1) s so that I y are disjoint and cover Z . For N (cid:16) nm , we decompose the normalized partition function W N into different contributions, very similarly to what is done in dimension one (i.e.decomposition (3.3)), and we refer to the figure 2 to illustrate how the decompositionlooks like: W N (cid:16) ¸ y ,...,y m P Z | W p y ,...,y m q (4.1)where | W p y ,...,y m q (cid:16) P exp (cid:3) N ¸ i (cid:16) (cid:16) βω i,S i (cid:1) β { (cid:24)(cid:11) t S in P I yi , i (cid:16) ,...,m u . (4.2)We fix θ p° a i q θ ¤ ° a θi (which holds for any finite orcountable collection of positive real numbers) to get QW θN ¤ ¸ y ,...,y m P Z Q | W θ p y ,...,y m q . (4.3)In order to estimate the different terms in the sum of the right–hand side in (4.3), wedefine some auxiliary measures r Q Y on the the environment for every Y (cid:16) p y , y , . . . , y m q P Z d (cid:0) with y (cid:16)
0. We will choose the measures Q Y absolutely continuous with respect to Q . We use H¨older inequality to get the following upper bound: Q | W θ p y ,...,y m q ¤ (cid:3) Q (cid:2) d Q d r Q Y (cid:10) θ (cid:1) θ (cid:11) (cid:1) θ (cid:1) r Q Y | W p y ,...,y m q(cid:9) θ . (4.4)Now, we describe the change of measure we will use. Recall that for the 1-dimensionalcase we used a shift of the environment along the corridor corresponding to Y . The readercan check that this method would not give the exponential decay of W N in this case.Instead we change the covariance function of the environment along the corridor on whichthe walk is likely to go by introducing some negative correlation.We introduce the change of measure that we use for this case. Given Y (cid:16) p y , y , . . . , y m q we define m blocks p B k q k Pr ,m s and J Y their union (here and in the sequel, | z | denotes the l norm on Z ): B k : (cid:16) p i, z q P N (cid:2) Z : r i { n s (cid:16) k and | z (cid:1) ? ny k (cid:1) | ¤ C ? n ( ,J Y : (cid:16) m ¤ k (cid:16) B k . (4.5)We fix the covariance the field ω under the law r Q Y to be equal to r Q Y (cid:0) ω i,z ω i,z C Y p i,z q , p j,z : (cid:16) tp i,z q(cid:16)p j,z V p i,z q , p j,z if D k P r , m s such that p i, z q and p j, z
1q P B k tp i,z q(cid:16)p j,z otherwise, (4.6) REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0) where V p i,z q , p j,z : (cid:16) p i, z q (cid:16) p j, z t| z (cid:1) z C ?| j (cid:1) i |u C C n ? log n | j (cid:1) i | otherwise. (4.7)We define p V : (cid:16) p V p i,z q , p j,z i,z q , p j,z B . (4.8)One remarks that the so-defined covariance matrix C Y is block diagonal with m identicalblocks which are copies of I (cid:1) p V corresponding to the B k , k P r , m s , and just ones on thediagonal elsewhere. Therefore, the change of measure we describe here exists if and onlyif I (cid:1) p V is definite positive.The largest eigenvalue for p V is associated to a positive vector and therefore is smallerthan max p i,z qP B ¸p j,z B (cid:7)(cid:7) V p i,z q , p j,z C C ? log n . (4.9)For the sequel we choose n such that the spectral radius of p V is less than p (cid:1) θ q{ I (cid:1) p V is positive definite. With this setup, r Q Y is well defined.The density of the modified measure r Q Y with respect to Q is given byd r Q Y d Q p ω q (cid:16) ? det C Y exp (cid:2)(cid:1) t ω pp C Y q(cid:1) (cid:1) I q ω (cid:10) , (4.10)where t ωM ω (cid:16) ¸p i,z q , p j,z N (cid:2) Z ω p i,z q M p i,z q , p j,z ω p j,z , (4.11)for any matrix M of p N (cid:2) Z q with finite support.Then we can compute explicitly the value of the second term in the right-hand side of(4.4) (cid:3) Q (cid:2) d Q d r Q Y (cid:10) θ (cid:1) θ (cid:11) (cid:1) θ (cid:16) gffe det C Y det (cid:1) C Y (cid:1) θ (cid:1) θI (cid:1) θ (cid:9) (cid:1) θ . (4.12)Note that the above computation is right if and only if C Y (cid:1) θI is a definite positive matrix.Since its eigenvalues are the same of those of p (cid:1) θ q I (cid:1) p V , this holds for large n thanksto (4.9). Using again the fact that C Y is composed of m blocks identical to I (cid:1) p V , we getfrom (4.12) (cid:3) Q (cid:2) d Q d r Q (cid:10) θ (cid:1) θ (cid:11) (cid:1) θ (cid:16) (cid:3) det p I (cid:1) p V q det p I (cid:1) p V {p (cid:1) θ qq (cid:1) θ (cid:11) m { . (4.13)In order to estimate the determinant in the denominator, we compute the Hilbert-Schmidtnorm of p V . One can check that for all n } p V } (cid:16) ¸p i,z q , p j,z B V p i,z q , p j,z
1q ¤ . (4.14)We use the inequality log p (cid:0) x q ¥ x (cid:1) x for all x ¥ (cid:1) { p V {p (cid:1) θ q is bounded by 1 { det (cid:19) I (cid:1) p V (cid:1) θ (cid:27) (cid:16) exp (cid:3) Trace (cid:3) log (cid:3) I (cid:1) p V (cid:1) θ (cid:11)(cid:11)(cid:11) ¥ exp (cid:3)(cid:1) } p V } p (cid:1) θ q (cid:11)¥ exp (cid:2)(cid:1) p (cid:1) θ q (cid:10) . (4.15)For the numerator, Trace p V (cid:16) p I (cid:1) p V q ¤
1. Combining this with(4.13) and (4.15) we get (cid:3) Q (cid:2) d Q d r Q Y (cid:10) θ (cid:1) θ (cid:11) (cid:1) θ ¤ exp (cid:2) m p (cid:1) θ q (cid:10) . (4.16)Now that we have computed the term corresponding to the change of measure, we estimate | W p y ,...,y m q under the modified measure (just by computing the variance of the Gaussianvariables in the exponential, using (4.6)) : r Q Y | W p y ,...,y m q (cid:16) P r Q Y exp (cid:3) N ¸ i (cid:16) (cid:2) βω i,S i (cid:1) β (cid:10)(cid:11) t S kn P I yk , k (cid:16) ,...,m u(cid:16) P exp (cid:4)(cid:6)(cid:6)(cid:5) β ¸ ¤ i, j ¤ Nz,z Z (cid:1) C Y p i,z q , p j,z
1q (cid:1) tp i,z q(cid:16)p j,z t S i (cid:16) z,S j (cid:16) z t S kn P I yk , k (cid:16) ,...,m u . (4.17)Replacing C Y by its value we get that r Q Y | W p y ,...,y m q (cid:16) P exp (cid:4)(cid:6)(cid:6)(cid:5)(cid:1) β ¸ ¤ i (cid:24) j ¤ N ¤ k ¤ m tpp i,S i q , p j,S j qqP B k , | S i (cid:1) S j |¤ C ?| i (cid:1) j |u C C n ? log n | j (cid:1) i | (cid:12)ÆÆ(cid:13) t S kn P I yk , k (cid:16) ,...,m u . (4.18)Now we do something similar to (3.11): for each “slice” of the trajectory p S i q i Prp m (cid:1) q k,mk s ,we bound the contribution of the above expectation by maximizing over the startingpoint (recall that P x denotes the probability distribution of a random walk starting at x ). Thanks to the conditioning, the starting point has to be in I y k . Using the translationinvariance of the random walk, this gives us the following ( _ stands for maximum): r Q Y | W p y ,...,y m q ¤ m ¹ i (cid:16) k max x P I P x (cid:18) exp (cid:3)(cid:1) β ¸ ¤ i (cid:24) j ¤ n t| S i |_| S j |¤ C ? n, | S i (cid:1) S j |¤ C ?| i (cid:1) j |u C C n ? log n | j (cid:1) i | (cid:11) t S n P I yk (cid:1) yk (cid:1) u(cid:26) . (4.19)For trajectories S of a directed random-walk of n steps, we define the quantity REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0) G p S q : (cid:16) ¸ ¤ i (cid:24) j ¤ n t| S i |_| S j |¤ C ? n, | S i (cid:1) S j |¤ C ?| i (cid:1) j |u C C n ? log n | j (cid:1) i | . (4.20)Combining (4.19) with (4.16), (4.4) and (4.3), we finally get QW θN ¤ exp (cid:2) m p (cid:1) θ q (cid:10) (cid:19)¸ y P Z max x P I (cid:2) P x exp (cid:2)(cid:1) β G p S q(cid:10) t S n P I y u(cid:10) θ (cid:27) m . (4.21)The exponential decay of QW θN (with rate n ) is guaranteed if we can prove that ¸ y P Z max x P I (cid:2) P x exp (cid:2)(cid:1) β G p S q(cid:10) t S n P I y u(cid:10) θ (4.22)is small. The rest of the proof is devoted to that aim.We fix some ε ¡
0. Asymptotic properties of the simple random walk, guarantees thatwe can find R (cid:16) R ε such that ¸| y |¥ R max x P I (cid:2) P x exp (cid:2)(cid:1) β G p S q(cid:10) t S n P I y u(cid:10) θ ¤ ¸| y |¥ R max x P I p P x t S n P I y uq θ ¤ ε. (4.23)To estimate the rest of the sum, we use the following trivial and rough bound ¸| y | R max x P I (cid:18) P x exp (cid:2)(cid:1) β G p S q(cid:10) t S n P I y u(cid:26) θ ¤ R (cid:18) max x P I P x exp (cid:2)(cid:1) β G p S q(cid:10)(cid:26) θ (4.24)Then we use the definition of G p S q to get rid of the max by reducing the width of thezone where we have negative correlation:max x P I P x exp (cid:2)(cid:1) β G p S q(cid:10) ¤ P exp (cid:3)(cid:1) β ¸ ¤ i (cid:24) j ¤ n t| S i |_| S j |¤p C (cid:1) q? n, | S i (cid:1) S j |¤ C ?| i (cid:1) j |u C C n ? log n | j (cid:1) i | (cid:11) . (4.25)We define s B : (cid:16) tp i, z q P N (cid:2) Z : i ¤ m, | z | ¤ p C (cid:1) q? n u . We get from the above thatmax x P I P x exp (cid:2)(cid:1) β G p S q(cid:10) ¤ P t the RW goes out of s B u(cid:0) P exp (cid:3)(cid:1) β ¸ ¤ i (cid:24) j ¤ n t| S i (cid:1) S j |¤ C ?| i (cid:1) j |u C C n ? log n | j (cid:1) i | (cid:11) (4.26)One can make the first term of the right-hand side arbitrarily small by choosing C large,in particular on can choose C such that P " max i Pr ,n s | S n | ¥ p C (cid:1) q? n * ¤ p ε { R q θ . (4.27)To bound the other term, we introduce the quantity D p n q : (cid:16) ¸ ¤ i (cid:24) j ¤ n n ? log n | j (cid:1) i | , (4.28)and the random variable X , X : (cid:16) ¸ ¤ i (cid:24) j ¤ n t| S i (cid:1) S j |¤ C ?| i (cid:1) j |u n ? log n | j (cid:1) i | . (4.29)For any δ ¡
0, we can find C such that P p X q ¥ p (cid:1) δ q D p n q . We fix C such that thisholds for some good δ (to be chosen soon), and by remarking that 0 ¤ X ¤ D p n q almostsurely, we obtain (using Markov inequality) P t X ¡ D p n q{ u ¥ (cid:1) δ. (4.30)Moreover we can estimate D p n q getting that for n large enough D p n q ¥ a log n. (4.31)Using (4.30) and (4.31) we get P exp (cid:3)(cid:1) β ¸ ¤ i (cid:24) j ¤ n t| S i (cid:1) S j |¤ C ?| i (cid:1) j |u C C n ? log n | j (cid:1) i | (cid:11) (cid:16) P exp (cid:2)(cid:1) β C C X (cid:10)¤ δ (cid:0) exp (cid:2)(cid:1) β ? log n C C (cid:10) . (4.32)Due to the choice of n we have made (recall n ¥ exp p C { β q ), the second term is lessthan exp (cid:1)(cid:1) β C { {p C C q(cid:9) . We can choose δ , C and C such that, the right-handside is less that p ε { R q θ . This combined with (4.27), (4.26), (4.24) and (4.23) allow us toconclude that ¸ y P Z max x P I (cid:2) P x exp (cid:2)(cid:1) β G p S q(cid:10) t S n P I y u(cid:10) θ ¤ ε (4.33)So that with a right choice for ε , (4.21) implies QW θN ¤ exp p(cid:1) m q . (4.34)Then (2.3) allows us to conclude that p p β q ¤ (cid:1) { n . (cid:3) Proof for general environment.
The case of general environment does not differ very muchfrom the Gaussian case, but one has to a different approach for the change of measurein (4.4). In this proof, we will largely refer to what has been done in the Gaussian case,whose proof should be read first.Let K be a large constant. One defines the function f K on R as to be f K p x q (cid:16) (cid:1) K t x ¡ exp p K qu . Recall the definitions (4.5) and (4.7), and define g Y function of the environment as REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0) g Y p ω q (cid:16) exp (cid:4)(cid:5) m ¸ k (cid:16) f K (cid:4)(cid:5) ¸p i,z q , p j,z B k V p i,z q , p j,z ω i,z ω j,z . Multiplying by g Y penalizes by a factor exp p(cid:1) K q the environment for which there is tomuch correlation in one block. This is a way of producing negative correlation in theenvironment. For the rest of the proof we use the notation U k : (cid:16) ¸p i,z q , p j,z B k V p i,z q , p j,z ω i,z ω j,z (4.35)We do a computation similar to (4.4) to get Q (cid:17)| W θ p y ,...,y m q(cid:25) ¤ (cid:1) Q (cid:17) g Y p ω q(cid:1) θ (cid:1) θ (cid:25)(cid:9) (cid:1) θ (cid:1) Q (cid:17) g Y p ω q| W p y ,...,y n q(cid:25)(cid:9) θ . (4.36)The block structure of g Y allows to express the first term as a power of m . Q (cid:17) g Y p ω q(cid:1) θ (cid:1) θ (cid:25) (cid:16) (cid:2) Q (cid:18) exp (cid:2)(cid:1) θ (cid:1) θ f K p U q(cid:10)(cid:26)(cid:10) m . (4.37)Equation (4.14) says that Var Q p U q ¤ . (4.38)So that P U ¥ exp p K q( ¤ exp p(cid:1) K q , (4.39)and hence Q (cid:18) exp (cid:2)(cid:1) θ (cid:1) θ f K p U q(cid:10)(cid:26) ¤ (cid:0) exp (cid:2)(cid:1) K (cid:0) θ (cid:1) θ K (cid:10) ¤ , (4.40)if K is large enough. We are left with estimating the second term Q (cid:17) g Y p ω q| W p y ,...,y n q(cid:25) (cid:16) P Qg Y p ω q exp (cid:3) nm ¸ i (cid:16) r βω i,S i (cid:1) λ p β qs(cid:11) t S kn P I yk , k (cid:16) ...m u . (4.41)For a fixed trajectory of the random walk S , we consider s Q S the modified measure on theenvironment with density d s Q S d Q : (cid:16) exp (cid:3) nm ¸ i (cid:16) r βω i,S i (cid:1) λ p β qs(cid:11) . (4.42)Under this measure s Q S ω i,z (cid:16) z (cid:24) S i Qωe βω , (cid:1) λ p β q : (cid:16) m p β q if z (cid:16) S i . (4.43)As ω , has zero-mean and unit variance under Q , (1.4) implies m p β q (cid:16) β (cid:0) o p β q aroundzero and that Var s Q S ω i,z ¤ p i, z q if β is small enough. Moreover s Q S is a productmeasure, i.e. the ω i,z are independent variables under Q S . With this notation (4.41)becomes P s Q S r g Y p ω qs t S kn P I yk , k (cid:16) ,...,m u . (4.44) As in the Gaussian case, one wants to bound this by a product using the block structure.Similarly to (4.19), we use translation invariance to get the following upper bound m ¹ k (cid:16) max x P I P x s Q S exp p f K p U qq t S n P I yk (cid:1) yk (cid:1) u . (4.45)Using this in (4.36) with the bound (4.40) we get the inequality QW θN ¤ m p (cid:1) θ q(cid:4)(cid:5) ¸ y P Z (cid:18) max x P I P x s Q S exp p f K p U qq t S n P I y u(cid:26) θ (cid:12)(cid:13) m . (4.46)Therefore to prove exponential decay of QW θN , it is sufficient to show that ¸ y P Z (cid:18) max x P I P x s Q S exp p f K p U qq t S n P I y u(cid:26) θ (4.47)is small. As seen in the Gaussian case ( cf. (4.23),(4.24)), the contribution of y far fromzero can be controlled and therefore it is sufficient for our purpose to checkmax x P I P x s Q S exp p f K p U qq ¤ δ, (4.48)for some small δ . Similarly to (4.26), we force the walk to stay in the zone where theenvironment is modified by writingmax i P I P x s Q S exp p f K p U qq ¤ P t max i Pr ,n s | S i | ¥ p C (cid:1) q? n u(cid:0) max x P I P x s Q S exp p f K p U qq t| S n (cid:1) S |¤p C (cid:1) q? n u . (4.49)The first term is smaller than δ { C is large enough. To control the second term, wewill find an upper bound for P x s Q S exp p f K p U qq t max i Pr ,n s | S i (cid:1) S |¤p C (cid:1) q? n u , (4.50)which is uniform in x P I .What we do is the following: we show that for most trajectories S the term in f K hasa large mean and a small variance with respect to Q S so that f K p . . . q (cid:16) (cid:1) K with large s Q S probability. The rest will be easy to control as the term in the expectation is at mostone.The expectation of U under s Q S is equal to m p β q ¸ ¤ i,j ¤ n V p i,S i q , p j,S j q . (4.51)When the walk stays in the block B we have (using definition (4.29)) ¸ ¤ i,j ¤ n V p i,S i q , p j,S j q (cid:16) C C X. (4.52)The distribution of X under P x is the same for all x P I . It has been shown earlier (cf.(4.30) and (4.31)), that if C is chosen large enough, P " m p β q C C X ¤ ? log n C C * ¤ δ . (4.53) REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0) As m p β q ¥ β { β is small, if C is large enough (recall n ¥ exp p C { β q ), this togetherwith (4.52) gives. P x " m p β q s Q S p U q ¤ p K q ; max i Pr ,n s | S i (cid:1) S | ¤ p C (cid:1) q? n * ¤ δ . (4.54)To bound the variance of U under s Q S , we decompose the sum U (cid:16) ¸p i,z q , p j,z B V p i,z q , p j,z ω i,z ω j,z (cid:16) m p β q ¸ ¤ i,j ¤ n V p i,S i q , p j,S j q(cid:0) m p β q ¸ ¤ i ¤ n p j,z B V p i,S i q , p j,z ω j,z m p β q t z S j uq(cid:0) ¸p i,z q , p j,z B V p i,z q , p j,z ω i,z (cid:1) m p β q t z (cid:16) S i uqp ω j,z (cid:1) m p β q t z S j uq . (4.55)And hence (using the fact that p x (cid:0) y q ¤ x (cid:0) y ).Var s Q S U ¤ m p β q ¸p j,z B (cid:3) ¸ ¤ i ¤ n V p i,S i q , p j,z (cid:0) ¸pp i,z q , p j,z B V p i,z q , p j,z , (4.56)where we used that Var s Q S ω i,z ¤ β small enough). The last term isless than 8 thanks to (4.14), so that we just have to control the first one. Independentlyof the choice of p j, z we have the bound ¸ ¤ i ¤ n V p i,S i q , p j,z
1q ¤ ? log nC C n . (4.57)Moreover it is also easy to check that ¸p j,z B ¸ ¤ i ¤ n V p i,S i q , p j,z
1q ¤ C nC ? log n , (4.58)(these two bounds follow from the definition of V p i,z q , p j,z : (4.7)). Therefore ¸p j,z B (cid:3) ¸ ¤ i ¤ n V p i,S i q , p j,z ¤ (cid:20)(cid:21) ¸p j,z B ¸ ¤ i ¤ n V p i,S i q , p j,z max p j,z qP B ¸ ¤ i ¤ n V p i,S i q , p j,z
1q ¤ . (4.59)Injecting this into (4.56) guaranties that for β small enoughVar s Q S U ¤ . (4.60)With Chebyshev inequality, if K has been chosen large enough and s Q S U ¥ p K q , (4.61)we have s Q S U ¤ exp p K q( ¤ δ { . (4.62) Hence combining (4.62) with (4.54) gives P x s Q S " U ¤ exp p K q ; max i Pr ,n s | S i (cid:1) S | ¤ p C (cid:1) q? n * ¤ δ { . (4.63)We use this in (4.49) to getmax x P I P x s Q S exp p f K p U qq ¤ δ (cid:0) e (cid:1) K . (4.64)So that our result is proved provided that K has been chosen large enough. (cid:3) Proof of the lower bound in Theorem 1.4
In this section we prove the lower bound for the free-energy in dimension 1 in arbitraryenvironment. To do so we apply the second moment method to some quantity related tothe partition function, and combine it with a percolation argument. The idea of the proofwas inspired by a study of a polymer model on hierarchical lattice [21] where this type ofcoarse-graining appears naturally.
Proposition 5.1.
There exists a constant C such that for all β ¤ we have p p β q ¥ (cid:1) Cβ pp log β q (cid:0) q . (5.1)We use two technical lemmas to prove the result. The first is just a statement aboutscaling of the random walk, the second is more specific to our problem. Lemma 5.2.
There exists an a constant c RW such that for large even squared integers n , P t S n (cid:16) ? n, S i ? n for i n u (cid:16) c RW n (cid:1) { (cid:0) o p n (cid:1) { q . (5.2) Lemma 5.3.
For any ε ¡ we can find a constant c ε and β such that for all β ¤ β ,for every even squared integer n ¤ c ε {p β | log β |q we have Var Q (cid:19) P (cid:3) exp (cid:3) n (cid:1) ¸ i (cid:16) p βω i,S i (cid:1) λ p β qq(cid:11) (cid:7)(cid:7)(cid:7)(cid:7) S n (cid:16) ? n, S i ? n for i n (cid:11)(cid:27) ε. (5.3) Proof of Proposition 5.1 from Lemma 5.2 and 5.3.
Let n be some fixed integer and define „ W : (cid:16) P exp (cid:3) n (cid:1) ¸ i (cid:16) p βω i,S i (cid:1) λ p β qq(cid:11) t S n (cid:16)? n, S i ? n for 0 i n u , (5.4)which corresponds to the contribution to the partition function W n of paths with fixedend point ? n staying within a cell of width ? n , with the specification the environment onthe last site is not taken in to account. „ W depends only of the value of the environment ω in this cell (see figure 3).One also defines the following quantities for p i, y q P N (cid:2) Z : „ W p y,y (cid:0) q i : (cid:16) P ? ny (cid:17) e ° n (cid:1) k (cid:16) r βω in (cid:0) k,Sk (cid:1) λ p β qs t S n (cid:16)?p y (cid:0) q n, S i (cid:1) y ? n ? n for 0 i n u(cid:25) , „ W p y,y (cid:1) q i : (cid:16) P ? ny (cid:17) e ° n (cid:1) k (cid:16) r βω in (cid:0) k,Sk (cid:1) λ p β qs t S n (cid:16)p y (cid:1) q? n, (cid:1)? n S i (cid:1) y ? n i n u(cid:25) . (5.5) REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0)
PSfrag replacementsO n ? n Figure 3.
We consider a resticted partition function „ W by considering only paths goingfrom one to the other corner of the cell, without going out. This restriction will giveus the independence of random variable corresponding to different cells which will becrucial to make the proof works. which are random variables that have the same law as „ W . Moreover because of indepen-dence of the environment in different cells, one can see that (cid:1)„ W p y,y (cid:8) q i ; p i, y q P N (cid:2) Z such that i (cid:1) y is even (cid:9) , is a family of independent variables.Let N (cid:16) nm be a large integer. We define Ω (cid:16) Ω N as the set of pathΩ : (cid:16) t S : i P r , m s , | S in (cid:1) S p i (cid:1) q n | (cid:16) ? n, j P r , n (cid:1) s , S p i (cid:1) q n (cid:0) j P (cid:0) S p i (cid:1) q n , S in (cid:8)u , (5.6)where the interval (cid:0) S i p n (cid:1) q , S in (cid:8) is to be seen as (cid:0) S in , S i p n (cid:1) q(cid:8) if S in S i p n (cid:1) q , and S : (cid:16) s (cid:16) p s , s , . . . , s m q P Z m (cid:0) : s (cid:16) | s i (cid:1) s i (cid:1) | (cid:16) , i P r , m s( . (5.7)We use the trivial bound W N ¥ P (cid:19) exp (cid:0) nm ¸ i (cid:16) p βω i,S i (cid:1) λ p β qq(cid:8) t S P Ω u(cid:27) , (5.8)to get that W N ¥ ¸ s P S m (cid:1) ¹ i (cid:16) „ W p s i ,s i (cid:0) q i exp (cid:1) βω p i (cid:0) q n,s i (cid:0) ? n (cid:1) λ p β q(cid:9) . (5.9)(the exponential term is due to the fact the „ W does not take into account to site in thetop corner of each cell).The idea is of the proof is to find a value of n (depending on β ) such that we aresure that for any value of m we can find a path s such that along the path the valuesof p„ W p s i ,s i (cid:0) q i q are not to low (i.e. close to the expectation of „ W ) and to do so, it seemsnatural to seek for a percolation argument.Let p c be the critical exponent for directed percolation in dimension 1 (cid:0) C and β such that for all n ¤ C β | log β | and β ¤ β . Q t„ W ¥ Q „ W { u ¥ p c (cid:0) . (5.10)We choose n to be the biggest squared even integer that is less than C β | log β | . (in particularhave n ¥ C β | log β | if β small enough).As shown in figure 4, we associate to our system the following directed percolationpicture. For all p i, y q P N (cid:2) Z such that i (cid:1) y is even: (cid:13) If „ W p y,y (cid:8) q i ¥ p { q Q „ W , we say that the edge linking the opposite corners of thecorresponding cell is open. (cid:13) If „ W p y,y (cid:8) q i p { q Q „ W , we say that the same edge is closed.Equation (5.10) and the fact the considered random variables are independent assures thatwith positive probability there exists an infinite directed path starting from zero.PSfrag replacements O n n n n n n n n (cid:0)? n (cid:0) ? n (cid:0) ? n (cid:0) ? n (cid:1)? n (cid:1) ? n (cid:1) ? n Figure 4.
This figure illustrates the percolation argument used in the proof. To eachcell is naturally associated a random variable „ W y,y (cid:8) i , and these random variables arei.i.d. When „ W y,y (cid:8) i ¥ { Q „ W we open the edge in the corresponding cell (thick edgeson the picture). As this happens with a probability strictly superior to p c , we have apositive probability to have an infinite path linking 0 to infinity. When there exists an infinite open path is linking zero to infinity exists, we can definethe highest open path in an obvious way. Let p s i q mi (cid:16) denotes this highest path. If m islarge enough, by law of large numbers we have that with a probability close to one, m ¸ i (cid:16) (cid:16) βω ni, ? ns i (cid:1) λ p β q(cid:24) ¥ (cid:1) mλ p β q . (5.11)Using this and and the percolation argument with (5.9) we finally get that with apositive probability which does not depend on m we have W nm ¥ (cid:17)p { q e (cid:1) λ p β q Q „ W (cid:25) m . (5.12)Taking the log and making m tend to infinity this implies that REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0) p p β q ¥ n (cid:16)(cid:1) λ p β q (cid:1) log 2 (cid:0) log Q „ W (cid:24) ¥ (cid:1) cn log n. (5.13)For some constant c , if n is large enough (we used Lemma 5.2 to get the last inequality.The result follows by replacing n by its value. (cid:3) Proof of Lemma 5.2.
Let n be square and even. T k , k P Z denote the first hitting time of k by the random walk S (when k (cid:16) P t S n (cid:16) ? n, S i ? n, for all 1 i n u(cid:16) n (cid:1) ¸ k (cid:16) P t T ? n { (cid:16) k, S j ¡ j n and T ? n (cid:16) n u(cid:16) P t T ? n { n, S j ? n for all j n and T (cid:16) n qu , (5.14)where the second equality is obtained with the strong Markov property used for T (cid:16) T ? n { ,and the reflexion principle for the random walk. The last line is equal to P t max k Pr ,n s S k P r? n { , ? n q| T (cid:16) n u P t T (cid:16) n u . (5.15)We use here a variant of Donsker’s Theorem, for a proof see [20, Theorem 2.6]. Lemma 5.4.
The process t ÞÑ " S r nt s? n (cid:7)(cid:7)(cid:7)(cid:7) T (cid:16) n * , t P r , s (5.16) converges in distribution to the normalized Brownian excursion in the space D pr , s , R q . We also know that (see for example [13, Proposition A.10]) for n even P p T (cid:16) n q (cid:16)a { πn (cid:1) { (cid:0) o p n (cid:1) { q . Therefore, from (5.15) we have P t S n (cid:16) ? n, S i ? n, for all 1 i n u(cid:16) a { πn (cid:1) { P (cid:18) max t Pr , s e t P p { , q(cid:26) (cid:0) o p n (cid:1) { q . (5.17)Where e denotes the normalized Brownian excursion, and P its law. (cid:3) Proof of Lemma 5.3.
Let β be fixed and small enough, and n be some squared even integerwhich is less than c ε {p β | log β |q . We will fix the value c ε independently of β later in theproof, and always consider that β is sufficiently small. By a direct computation thevariance of P (cid:19) exp (cid:3) n (cid:1) ¸ i (cid:16) r βω i,S i (cid:1) λ p β qs(cid:11) (cid:7)(cid:7)(cid:7)(cid:7) S n (cid:16) ? n, S i ? n for 0 i n (cid:27) (5.18)is equal to P b (cid:19) exp (cid:3) n (cid:1) ¸ i (cid:16) γ p β q t S p q i (cid:16) S p q i u(cid:11) (cid:7)(cid:7)(cid:7)(cid:7) A n (cid:27) (cid:1) . (5.19)where A n (cid:16) ! S p j q n (cid:16) ? n, S p j q i ? n for 0 i n, j (cid:16) , ) , (5.20)and γ p β q (cid:16) λ p β q(cid:1) λ p β q (recall that λ p β q (cid:16) log Q exp p βω p , qq ), and S p j q n , j (cid:16) , P b . From this it follows that if n issmall the result is quite straight–forward. We will therefore only be interested in the caseof large n (i.e. bounded away from zero by a large fixed constant).We define τ (cid:16) p τ k q k ¥ (cid:16) t S p q i (cid:16) S p q i , i ¥ u the set where the walks meet (it canbe written as an increasing sequence of integers). By the Markov property, the randomvariables τ k (cid:0) (cid:1) τ k are i.i.d. , we say that τ is a renewal sequence .We want to bound the probability that the renewal sequence τ has too many returnsbefore times n (cid:1)
1, in order to estimate (5.19). To do so, we make the usual computationswith Laplace transform.From [11, p. 577] , we know that1 (cid:1) P b exp p(cid:1) xτ q (cid:16) ° n P N exp p(cid:1) xn q P t S p q n (cid:16) S p q n u . (5.21)Thanks to the the local central limit theorem for the simple random walk, we know thatfor large n P t S p q n (cid:16) S p q n u (cid:16) ? πn (cid:0) o p n (cid:1) { q . (5.22)So that we can get from (5.21) that when x is close to zerolog P b exp p(cid:1) xτ q (cid:16) (cid:1)? x (cid:0) o p? x q . (5.23)We fix x such that log P exp p(cid:1) xτ q ¤ ? x { x ¤ x . For any k ¤ n we have P b t| τ X r , n (cid:1) s| ¥ k u (cid:16) P b t τ k ¤ n (cid:1) u ¤ exp pp n (cid:1) q x q P b exp p(cid:1) τ k x q¤ exp (cid:16) nx (cid:0) k log P b exp p(cid:1) xτ q(cid:24) . (5.24)For any k ¤ X n ? x \ (cid:16) k one can choose x (cid:16) p k { n q ¤ x in the above and use thedefinition of x to get that P b t| τ X r , n (cid:1) s| ¥ k u ¤ exp (cid:0)(cid:1) k {p n q(cid:8) . (5.25)In the case where k ¡ k we simply bound the quantity by P b t| τ X r , n (cid:1) s| ¥ k u ¤ exp (cid:0) k {p n q(cid:8) ¤ exp p(cid:1) nx { q . (5.26)By Lemma (5.2), if n is large enough P b A n ¥ { c RW n (cid:1) . (5.27)A trivial bound on the conditioning gives us P b (cid:0)| τ X r , n (cid:1) s| ¥ k (cid:7)(cid:7) A n (cid:8) ¤ min (cid:0) , c (cid:1) RW n exp (cid:0)(cid:1) k {p n q(cid:8)(cid:8) if k ¤ k ,P b (cid:0)| τ X r , n (cid:1) s| ¥ k (cid:7)(cid:7) A n (cid:8) ¤ c (cid:1) RW n exp p(cid:1) nx { q otherwise . (5.28)We define k : (cid:16) r π b n log p c (cid:1) RW n qs . The above implies that for n large enough we have REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0) P b (cid:0)| τ X r , n (cid:1) s| ¥ k (cid:7)(cid:7) A n (cid:8) ¤ k ¤ k ,P b (cid:0)| τ X r , n (cid:1) s| ¥ k (cid:7)(cid:7) A n (cid:8) ¤ exp (cid:0)(cid:1) k {p n q(cid:8) if k ¤ k ¤ k ,P b (cid:0)| τ X r , n (cid:1) s| ¥ k (cid:7)(cid:7) A n (cid:8) ¤ exp p(cid:1) nx { q otherwise . (5.29)Now we are ready to bound (5.19). Integration by part gives, P b (cid:16) exp p γβ | τ X r , n (cid:1) s|q (cid:7)(cid:7) A n (cid:24) (cid:1) (cid:16) γ p β q » 8 exp p γ p β q x q P b (cid:0)| τ X r , n (cid:1) s| ¥ x (cid:7)(cid:7) A n (cid:8) d x. (5.30)We split the right-hand side in three part corresponding to the three different bounds wehave in (5.28): x P r , k s , x P r k , k s and x P r k , n s . It suffices to show that each part isless than ε { γ p β q » k exp p γ p β q x q P b (cid:0)| τ X r , n (cid:1) s| ¥ x (cid:7)(cid:7) A n (cid:8) d x ¤ γ p β q k exp p γ p β q k q . (5.31)One uses that n ¤ c ε β | log β | and γ p β q (cid:16) β (cid:0) o p β q to get that for β small enough and n large enough if c ε is well chosen we have k γ p β q ¤ β a n log n ¤ ε { , (5.32)so that γ p β q k exp p γ p β q k q ¤ ε { γ p β q » k k exp p γ p β q x q P b (cid:0)| τ X r , n (cid:1) s| ¥ x (cid:7)(cid:7) A n (cid:8) d x ¤ γ p β q » 8 exp (cid:0) γ p β q x (cid:1) x {p n q(cid:8) d x (cid:16) » 8 exp (cid:2) x (cid:1) x nγ p β q (cid:10) d x. (5.33)Replacing n by its value, we see that the term that goes with x in the exponential can bemade arbitrarily large, provided that c ε is small enough. In particular we can make theleft-hand side less than ε { γ p β q » nk exp p γ p β q x q P b (cid:0)| τ X r , n (cid:1) s| ¥ x (cid:7)(cid:7) A n (cid:8) d x ¤ γ p β q » n exp p γ p β q x (cid:1) nx { q d x (cid:16) n exp p(cid:1)r γ p β q (cid:1) x { s n q . (5.34)This is clearly less than ε { n is large and β is small. (cid:3) Proof of the lower bound of Theorem 1.5
In this section we use the method of replica-coupling that is used for the disorderedpinning model in [24] to derive a lower bound on the free energy. The proof here is anadaptation of the argument used there to prove disorder irrelevance.
The main idea is the following: Let W N p β q denotes the renormalized partition functionfor inverse temperature β . A simple Gaussian computation givesd Q log W N p? t q d t (cid:7)(cid:7)(cid:7)(cid:7) t (cid:16) (cid:16) (cid:1) P b N ¸ i (cid:16) t S p q i (cid:16) S p q i u . (6.1)Where S p q and S p q are two independent random walk under the law P b . This impliesthat for small values of β (by the equality of derivative at t (cid:16) Q log W N p β q (cid:19) (cid:1) log P b exp (cid:3) β { N ¸ i (cid:16) t S p q N (cid:16) S p q N u(cid:11) . (6.2)This tends to make us believe that p p β q (cid:16) (cid:1) lim N Ñ8 log P b exp (cid:3) β { N ¸ i (cid:16) t S p q N (cid:16) S p q N u(cid:11) . (6.3)However, things are not that simple because (6.2) is only valid for fixed N , and one needssome more work to get something valid when N tends to infinity. The proofs aims to useconvexity argument and simple inequalities to be able to get the inequality p p β q ¥ (cid:1) lim N Ñ8 log P b exp (cid:3) β N ¸ i (cid:16) t S p q N (cid:16) S p q N u(cid:11) . (6.4)The fact that convexity is used in a crucial way make it quite hopeless to get the otherinequality using this method. Proof.
Let use define for β fixed and t P r , s Φ N p t, β q : (cid:16) N Q log P exp (cid:3) N ¸ i (cid:16) (cid:18)? tβω i,S i (cid:1) tβ (cid:26)(cid:11) , (6.5)and for λ ¥ N p t, λ, β q : (cid:16) N Q log P b exp (cid:3) N ¸ i (cid:16) (cid:17)? tβ p ω i,S p q i (cid:0) ω i,S p q i q (cid:1) tβ (cid:0) λβ t S p q i (cid:16) S p q i u(cid:25)(cid:11) . (6.6)One can notice that Φ N p , β q (cid:16) N p , β q (cid:16) p N p β q (recall the definition of p N (1.11)), so that Φ N is an interpolation function. Via the Gaussian integration by parformula Qωf p ω q (cid:16) Qf ω q , (6.7)valid (if ω is a centered standard Gaussian variable) for every differentiable functions suchthat lim | x |Ñ8 exp p(cid:1) x { q f p x q (cid:16)
0, one finds
REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0) dd t Φ N p t, β q (cid:16) (cid:1) β N N ¸ j (cid:16) ¸ z P Z Q (cid:4)(cid:5) P exp (cid:1)° Ni (cid:16) (cid:17)? tβω i,S i (cid:1) tβ (cid:25)(cid:9) t S j (cid:16) z u P exp (cid:1)° Ni (cid:16) (cid:17)? tβω i,S i (cid:1) tβ (cid:25)(cid:9) (cid:12)(cid:13) (cid:16) (cid:1) β N Q (cid:1) µ p? tβ q n (cid:9)b (cid:19) N ¸ i (cid:16) t S p q i (cid:16) S p q i u(cid:27) . (6.8)This is (up to the negative multiplicative constant (cid:1) β {
2) the expected overlap fraction oftwo independent replicas of the random–walk under the polymer measure for the inversetemperature ? tβ . This result has been using Itˆo formula in [4, Section 7].For notational convenience, we define H N p t, λ, S p q , S p qq (cid:16) N ¸ i (cid:16) (cid:18)? tβ p ω i,S p q i (cid:0) ω i,S p q i q (cid:1) tβ (cid:0) λβ ! S p q i (cid:16) S p q i )(cid:26) . (6.9)We use Gaussian integration by part again, for Ψ N :dd t Ψ N p t, λ, β q (cid:16) β N N ¸ j (cid:16) Q P b exp (cid:0) H N p t, λ, S p q , S p qq(cid:8) t S p q j (cid:16) S p q j u P b exp (cid:0) H N p t, λ, S p q , S p qq(cid:8)(cid:1) β N N ¸ j (cid:16) ¸ z P Z Q (cid:4)(cid:6)(cid:6)(cid:5) P b (cid:2) t S p q j (cid:16) z u (cid:0) t S p q j (cid:16) z u(cid:10) exp (cid:0) H N p t, λ, S p q , S p qq(cid:8) P b exp (cid:0) H N p t, λ, S p q , S p qq(cid:8) (cid:12)ÆÆ(cid:13) ¤ β N N ¸ j (cid:16) Q P b exp (cid:0) H N p t, λ, S p q , S p qq(cid:8) t S p q j (cid:16) S p q j u P b exp (cid:0) H N p t, λ, S p q , S p qq(cid:8) (cid:16) dd λ Ψ N p t, λ, β q . (6.10)The above implies that for every t P r , s and λ ¥ N p t, λ, β q ¤ Ψ N p , λ (cid:0) t, β q . (6.11)Comparing (6.8) and (6.10), and using convexity and monotonicity of Ψ N p t, λ, β q withrespect to λ , and the fact that Ψ N p t, , β q (cid:16) Φ N p t, β q one gets (cid:1) dd t φ N p t, β q (cid:16) dd λ Ψ N p t, λ, β q(cid:7)(cid:7)(cid:7)(cid:7) λ (cid:16) ¤ Ψ N p t, (cid:1) t, β q (cid:1) Φ N p t, β q (cid:1) t ¤ Ψ N p , , β q (cid:1) Φ N p t, β q , (6.12)where in the last inequality we used p (cid:1) t q ¥ N p , β q (cid:16) p N p β q we get p N p β q ¥ p (cid:1) e q Ψ N p , , β q . (6.13)On the right-hand side of the above we recognize something related to pinning models.More precisely Ψ N p , , β q (cid:16) N log Y N , (6.14)where Y N (cid:16) P b exp (cid:3) β N ¸ i (cid:16) ! S p q N (cid:16) S p q N )(cid:11) (6.15)is the partition function of a homogeneous pinning system of size N and parameter 2 β with underlying renewal process the sets of zero of the random walk S p q (cid:1) S p q . This isa well known result in the study of pinning model ( we refer to [13, Section 1.2] for anoverview and the results we cite here) thatlim N Ñ8 N log Y N (cid:16) f p β q , (6.16)where f denotes the free energy of the pinning model. Moreover, it is also stated f p h q h Ñ (cid:0)(cid:18) h { . (6.17)Then passing to the limit in (6.14) ends the proof of the result for any constant strictlybigger that 4. (cid:3) Proof the lower bound in Theorem 1.6
The technique used in the two previous sections could be adapted here to prove theresults but in fact it is not necessary. Because of the nature of the bound we want to provein dimension 2 (we do not really track the best possible constant in the exponential), itwill be sufficient here to control the variance of W n up to some value, and then theconcentration properties of log W n to get the result. The reader can check than using thesame method in dimension 1 does not give the right power of β .First we prove a technical result to control the variance of W n which is the analog of(5.3) in dimension 1. Recall that γ p β q : (cid:16) λ p β q (cid:1) λ p β q with λ p β q : (cid:16) log Q exp p βω p , qq . Lemma 7.1.
For any ε , one can find a constant c ε ¡ and β ¡ such that for any β ¤ β , for any n ¤ exp (cid:0) c ε { β (cid:8) we have Var Q W n ¤ ε. (7.1) Proof.
A straight–forward computation shows that the the variance of W n is given byVar Q W n (cid:16) P b exp (cid:3) γ p β q n ¸ i (cid:16) t S p q i (cid:16) S p q i u(cid:11) (cid:1) . (7.2)where S p i q , i (cid:16) , n , it will be enough to prove the result for n large. For technical convenience we choose to prove the result for n (cid:16)u exp p(cid:1) c ε { γ p β qqu (recall γ p β q (cid:16) λ p β q (cid:1) λ p β q ) which does not change the result since γ p β q (cid:16) β (cid:0) o p β q .The result we want to prove seems natural since we know that p° ni (cid:16) t S p q i (cid:16) S p q i uq{ log n converges to an exponential variable (see e.g. [12]), and γ p β q (cid:18) c ε log n . However, con-vergence of the right–hand side of (7.2) requires the use of the dominated convergenceTheorem, and the proof of the domination hypothesis is not straightforward. It could beextracted from the proof of the large deviation result in [12], however we include a fullproof of convergence here for the sake of completeness. REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0)
We define τ (cid:16) p τ k q k ¥ (cid:16) t S p q i (cid:16) S p q i , i ¥ u the set where the walks meet (it canbe written as an increasing sequence). By the Markov property, the random variables τ k (cid:0) (cid:1) τ k are i.i.d. .To prove the result, we compute bounds on the probability of having too many pointbefore n in the renewal τ . As in the 1 dimensional case, we use Laplace transform to doso. From [11, p. 577] , we know that1 (cid:1) P b exp p(cid:1) xτ q (cid:16) ° n P N exp p(cid:1) xn q P t S p q n (cid:16) S p q n u . (7.3)The local central limit theorem says that for large nP b t S p q n (cid:16) S p q n u (cid:18) πn . (7.4)Using this into (7.3) we get that when x is close to zerolog P b exp p(cid:1) xτ q (cid:18) (cid:1) π | log x | . (7.5)We use the following estimate P b t| τ X r , n s| ¥ k u (cid:16) P b t τ k ¤ n u ¤ exp p nx q P b exp p(cid:1) τ k x q(cid:16) exp (cid:16) nx (cid:0) k log P b exp p(cid:1) xτ q(cid:24) . (7.6)Let x be such that for any x ¤ x , log P b exp p(cid:1) xτ q ¥ (cid:1) {| log x | . For k such that k {p n log p n { k qq ¤ x , we replace x by k {p n log p n { k qq in (7.6) to get P b t| τ Xr , n s| ¥ k u ¤ exp (cid:2) k log p n { k q (cid:1) k log r k {p n log n { k qs (cid:10) ¤ exp (cid:2)(cid:1) k log p n { k q (cid:10) , (7.7)where the last inequality holds if k { n is small enough. We fix k (cid:16) δn for some small δ .We get that P b t| τ X r , n s| ¥ k u ¤ exp (cid:2)(cid:1) k log p n { k q (cid:10) if k ¤ k P b t| τ X r , n s| ¥ k u ¤ exp (cid:2)(cid:1) k log p n { k q (cid:10) (cid:16) exp (cid:2)(cid:1) δn log p { δ q (cid:10) if k ¥ k . (7.8)We are ready to bound (7.2). We remark that using integration by part we obtain P exp p γ p β q| τ X r , n s|q (cid:1) (cid:16) » n γ p β q exp p γ p β q x q P b p τ X r , n s| ¥ x q d x. (7.9)To bound the right–hand side, we use the bounds we have concerning τ : (7.8). We haveto split the integral in three parts.The integral between 0 and 1 can easily be made less than ε { β small.Using n ¤ exp p c ε { γ p β qq , we get that » δn γ p β q exp p γ p β q x q P b p τ X r , n s| ¥ x q d x ¤ » δn γ p β q exp (cid:2) γ p β q x (cid:1) x log p n { x q (cid:10) d x ¤ » δn γ p β q exp (cid:2) γ p β q x (cid:1) γ p β q βxc ε (cid:10) ¤ c ε (cid:1) c ε . (7.10)This is less that ε { c ε is chosen appropriately. The last part to bound is » nδn γ p β q exp p γ p β q x q P b p τ X r , n s| ¥ x q ¤ nγ p β q exp (cid:2) γ p β q n (cid:1) δn log 1 { δ (cid:10) ¤ ε { , (7.11)where the last inequality holds if n is large enough, and β is small enough. (cid:3) Proof of the lower bound in Theorem 1.6.
By a martingale method that one can find aconstant c such that Var Q log W n ¤ C n, n ¥ , β ¤ . (7.12)(See [6, Proposition 2.5] and its proof for more details).Therefore Chebyshev inequality gives Q "(cid:7)(cid:7)(cid:7)(cid:7) n log W n (cid:1) n Q log W n (cid:7)(cid:7)(cid:7)(cid:7) ¥ n (cid:1) { * ¤ C n (cid:1) { . (7.13)Using Lemma 7.1 and Chebyshev inequality again, we can find a constant C such thatfor small β and n (cid:16) r exp p C { β qs we have Q t W n { u ¤ { . (7.14)This combined with (7.13) implies that (cid:1) log 2 n ¤ n (cid:1) { (cid:0) Q n log W n ¤ n (cid:1) { (cid:0) p p β q . (7.15)Replacing n by its value we get p p β q ¥ (cid:1) n (cid:1) { (cid:1) log 2 n ¥ (cid:1) exp p(cid:1) C { β q . (7.16) (cid:3) Acknowledgements:
The author is very grateful to Giambattista Giacomin for nu-merous suggestions and precious help for the writing of this paper, to Francesco Caravennafor the proof of Lemma 5.2 and to Fabio Toninelli and Francis Comets for enlighteningdiscussions. The author also acknowledges the support of ANR, grant POLINTBIO.
References [1] S. Albeverio. and X. Zhou,
A martingale approach to directed polymers in a random environment , J.Theoret. Probab. (1996) 171–189.[2] P. Bertin, Free energy for Linear Stochastic Evolutions in dimension two , preprint (2009).[3] E. Bolthausen,
A note on diffusion of directed polymer in a random environment , Commun. Math.Phys. (1989) 529–534.[4] P. Carmona and Y. Hu,
On the partition function of a directed polymer in a random Gaussianenvironment , Probab. Theor. Relat. Fields
Strong disorder implies strong localization for directed polymers in a randomenvironment , ALEA (2006) 217–229. REE ENERGY OF DIRECTED POLYMERS IN DIMENSION 1 (cid:0) (cid:0) [6] F. Comets, T. Shiga and N. Yoshida,
Directed Polymers in a random environment: strong disorderand path localization , Bernouilli Probabilistic Analysis of Directed Polymers in a RandomEnvironment: a Review , Adv. Stud. Pure Math. (2004) 115-142.[8] F. Comets and V. Vargas, Majorizing multiplicative cascades for directed polymers in random media ,ALEA (2006) 267–277.[9] F. Comets and N. Yoshida, Directed polymers in a random environment are diffusive at weak disorder ,Ann. Probab. Fractional moment bounds and disorderrelevance for pinning models , Commun. Math. Phys. (2009) 867–887.[11] Feller W.,
An Introduction to Probability Theory and Its Applications, Volume II , John Wiley & Sons,Inc. New York (1966).[12] N. Gantert and O. Zeitouni,
Large and moderate deviations for local time of a reccurent Markov chainon Z , Ann. Inst. H. Poincar Probab. Statist. (1998), 687–704.[13] G. Giacomin, Random polymer models , IC press, World Scientific, London (2007).[14] G. Giacomin, H. Lacoin and F. L. Toninelli,
Hierarchical pinning models, quadratic maps and quencheddisorder , To appear in Probab. Theory. Rel. Fields, arXiv:0711.4649 [math.PR].[15] G. Giacomin, H. Lacoin and F.L. Toninelli,
Marginal relevance of disorder for pinning models , toappear in Commun. Pure Appl. Math. , arXiv:0811.0723 [math-ph].[16] G. Giacomin, H. Lacoin and F. L. Toninelli,
Disorder relevance at marginality and critical point shift ,preprint (2009) arXiv:0906.1942v1 [math-ph].[17] G. Grimmett,
Percolation
Second Edition, Grundlehren der Mathematischen Wissenschaften ,Springer-Verlag, Berlin (1999).[18] D.A. Huse and C.L. Henley,
Pinning and roughening of domain wall in Ising systems due to randomimpurities , Phys. Rev. Lett. (1985) 2708–2711.[19] J.Z. Imbrie and T. Spencer, Diffusion of directed polymer in a random environment , J. Stat. Phys. An invariance principle for random walk conditioned by a late return to zero , Ann.Probab. (1976) 115-121 .[21] H. Lacoin and G. Moreno Directed polymer on hierarchical lattice with site disorder , preprint (2009)arXiv: arXiv:0906.0992v1 [math.PR].[22] Q. Liu and F. Watbled
Exponential inequalities for martingales and asymptotic properties of the freeenergy of directed polymers in random environment , preprint (2008) arXiv:0812.1719v1.[23] R. Song and X.Y. Zhou,
A remark on diffusion of directed polymers in random environment , J. Stat.Phys. A replica coupling-approach to disordered pining models
Commun. Math. Phys. (2008) 389-401.[25] F.L. Toninelli,
Coarse graining, fractional moments and the critical slope of random copolymers , Toappear in Electron. Journal Probab. arXiv:0806.0365 [math.PR].[26] V. Vargas,
Strong localization and macroscopic atoms for directed polymer , Probab. Theor. Relat.Fields,
Universit´e Paris Diderot and Laboratoire de Probabilit´es et Mod`eles Al´eatoires (CNRSU.M.R. 7599), U.F.R. Math´ematiques, Case 7012 (Site Chevaleret), 75205 Paris cedex 13,France
E-mail address ::