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On the depinning transition of the directed polymer in a random environment with a defect line
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On the depinning transition of the directed polymer in a random environment with a defect line
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ON THE DEPINNING TRANSITION OF THE DIRECTED POLYMER IN A RANDOM ENVIRONMENT WITH A DEFECT LINE by G okhan Yldrm A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) August 2013 Copyright 2013 G okhan Yldrm Acknowledgments I would like to thank my advisor Professor Kenneth S. Alexander for his guidance and support during my graduate studies. He was always very generous for his time and sharing his ideas with me. I would like to thank my dissertation committee members Professor Baxendale and Professor Goldstein. I had chance to take advanced courses on their research areas from them in addition to the basic probability theory and statistics courses. Their classes were excellent to obtain a solid background in probability theory. I am also thankful to Professor Stephan Haas who kindly accepted to be part of my dissertation committee. I am grateful to my master's thesis advisor Professor Alp Eden from Bogazici Univer- sity who played an important role in my decision to work on probability theory. Thanks to his guidance and support, I came to USC for my graduate studies. USC has been a great place to learn mathematics and do research. I also thank all professors and stas of USC mathematics department. Moe Vedadi, my closest friend at USC, has always been with me with his true friend- ship since we met rst day on USC campus. I am truly fortunate to meet him. ii Emre Aydogan has been and continues to be one of my best friends. His friendship and fantastic cheer has always enriched my life. I am deeply grateful to my family for their support throughout my life. I have realized that one of the most important things in life is having a loving family. I could not have achieved anything in my life without their love and support. iii Table of Contents Acknowledgments ii List of Figures vi Abstract vii Chapter 1: Introduction 1 1.1 Random Walk Pinning Model . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Partition Function and the Existence of the Free Energy . . . . . . 3 1.1.2 Variational Form of the Free Energy . . . . . . . . . . . . . . . . . 8 1.2 General Homogeneous Pinning Model . . . . . . . . . . . . . . . . . . . . 11 1.3 Disordered Pinning Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.3.1 Localization-Delocalization Transition . . . . . . . . . . . . . . . . 19 Chapter 2: Directed Polymers in a Random Environment with a Defect Line 23 2.1 Physical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Mathematical Formulation of the Problem . . . . . . . . . . . . . . . . . . 25 2.2.1 DPRE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.2 DPRE with defect line model . . . . . . . . . . . . . . . . . . . . . 27 2.3 Self-averaging of the Free Energy and Critical Point . . . . . . . . . . . . 30 2.3.1 The Constrained Model . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.2 The Free Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Chapter 3: Main Results 40 3.1 Statement of the Main Results and Overview of the Proofs . . . . . . . . 40 3.2 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.1 The Annealed Correlation Length . . . . . . . . . . . . . . . . . . 52 3.3 The Coarse Grained LatticeL CG . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.1 Assigning each site (I;J) in the coarse grained latticeL CG as open or closed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3.2 Second Moment Method and the Site Densities in L CG . . . . . . . 64 3.4 Lipschitz Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.5 Stochastic Domination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.6 Final Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 iv Bibliography 82 v List of Figures 1.1 1-D pinning model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 1-D disordered pinning model . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1 1 + 1 DPRE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 1 + 1 DPRE model with defect line. . . . . . . . . . . . . . . . . . . . . . 31 3.1 The Coarse Grained LatticeL CG . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 Site percolation onZ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3 The lowest open Lipschitz function F () . . . . . . . . . . . . . . . . . . . 72 3.4 A detour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.5 The innite good path G;1 from the site (0; 0). . . . . . . . . . . . . . . 78 vi Abstract We study the depinning transition of the 1 + 1 dimensional directed polymer in a random environment with a defect line model. The monomer locations of the polymer is modeled by the space-time trajectory of a one dimensional simple symmetric random walk. Random environment is introduced by assigning each site of Z 2 an independent and identically distributed normal random variable that interacts with the polymer when it visits that site. The defect line is incorporated to the model by having an additional constant potential u at the origin which gives a reward or penalty to the polymer as it visits the origin. There is a critical value of u above which polymer is pinned, placing a positive fraction of its monomers at zero with high probability. Our rst result is that the quenched free energy exists and the self-averaging holds, which implies that there is a nonrandom quenched critical point. To see the eect of disorder on the depinning transition, we compare the quenched free energy of the system as a function of u to the corresponding annealed system. Our main result is that the quenched and annealed free energies dier signicantly only in a very small neighborhood of the critical point and we show that the size of this neighborhood scales as at most as ! 0; where is the inverse temperature. vii Chapter 1 Introduction In chapter one, we introduce the polymer models and basic statistical mechanics concepts in this framework. We will review and summarize some of the main existing results, which we need to use in the following chapters, for random walk pinning models, general homogeneous and disordered pinning models. In chapter two, after reviewing the relevant physics literature on the directed polymer in a random environment with a defect line model, we introduce our problem in mathe- matical framework. We show that self-averaging occurs, meaning that the quenched free energy and critical point are nonrandom, o a null set. In chapter three, we present our main results and develop the necessary tools to prove them. 1.1 Random Walk Pinning Model In this section, we introduce the one dimensional random walk pinning model and basic statistical mechanics concepts in this framework. Let P x be the distribution of the one dimensional simple symmetric random walk, S =fS j ;j 0g; on Z starting at the 1 point x, and we use P for P 0 . We consider S =fS j ;j 0g as a path of random walk and all probability measures will be dened on the path space. Let L N () :=L N (S) = N X j=1 1 S j =0 be the local time of random walk at 0 up to time N: We introduce the following polymer (probability) measures on the path space in Boltzmann-Gibbs way as follows: d N;u dP (S) = 1 Z N;u e uL N (S) (1.1.1) where Z N;u =E P 0 (e uL N (S) ) (1.1.2) is the normalizing constant and called the partition function of the system. The exponential factor H N;u (S) :=uL N (S) is called the Hamiltonian of the system which is the energy functional and identies the (negative) energy of each micro-state. S N N 1 Figure 1.1: 1-D pinning model 2 One of the most fundamental result in statistical mechanics (based on the law of energy conservation and the basic postulate that all micro-states of the same energy level are equiprobable) is that when the system is in thermal equilibrium with its environment, the probability of nding the system in a given micro-state is given by Boltzmann-Gibbs distribution as we dened above. Note that under the usual random walk measureP 0 , each path of lengthN has equal probability but polymer measure N;u () favors paths which has more return to zero. Under the polymer measure N;u (), random walk paths are weighted proportional to their returns to zero in the rst N steps. 1.1.1 Partition Function and the Existence of the Free Energy Denition 1.1.1. A sequencefa n g n1 is called sub-additive if it satises the inequality a m+n a m +a n for all m;n 1: For the proof of the following lemma, see the appendix 7 of [25]. Lemma 1.1.2 (Fekete's sub-additive lemma). For every sub-additive sequencefa n g n1 ; the limit lim n!1 an n exists and is equal to inf n1 an n 2 [1;1): Lemma 1.1.3. Let a n (u) =u + logE P 0 (e uLn ): Thenfa n (u)g n1 denes a sub-additive sequence for any u2R: 3 Proof. By Markov property of SSRW, we have E P 0 (e uL n+k ) = X x E P 0 (e uL n+k 1 Sn=x ) = X x E P 0 (e uLn 1 Sn=x )E Px (e uL k ) X x E P 0 (e uLn 1 Sn=x )e u E P 0 (e uL k ) =e u E P 0 (e uLn )E P 0 (e uL k ) Multiplying both sides by e u ; and then taking the log of both sides, we get a n+k (u)a n (u) +a k (u): By Lemma 1.1.2, lim n!1 a n (u) n exists for all u2R: The limit F(u) = lim n!1 logE P 0 (e uLn ) n is called the free energy of the system. Observe that for u 0; E P 0 (e uLn ) 1: Therefore, F(u) = lim n!1 logE P 0 (e uLn ) n 0 4 and also for any u2R; E P 0 (e uLn ) E P 0 (e uLn 1 >n ) E P 0 (e uLn ) P 0 ( >n) Therefore, F(u) = lim n!1 logE P 0 (e uLn ) n 0 since P 0 ( > n) Cn 1=2 as n!1; where = inffn 1 : S n = 0g is the rst return time of random walk to the origin. Recall that for two sequences (a n ) n1 and (b n ) n1 ; a n b n means that lim n!1 a n b n = 1: The following lemma gives an exponential order lower bound in terms of free energy for the partition function of the system of size N: Lemma 1.1.4. For all N 1, E P 0 (e uL N )e u e NF(u) : Proof. Let a n (u) =u + logE P 0 (e uLn ): By sub-additivity a N (u) N inf n1 a n (u) n = lim n!1 a n (u) n = F(u); 8N 1; 5 Therefore u + logE P 0 (e uL N )NF(u) By taking the exponential of both sides, we get the result. A similar upper bound is proved in [3]. Lemma 1.1.5. There exists a K > 0 such that 8j 1; E P 0 (e uL jM )Kje j where M = 1 F(u) is the correlation length. Here are some properties of the free energy function F(): For the proofs, see [25] and [27] by Giacomin. a) F(u) is 0 on (1; 0] and strictly increasing and positive on (0;1): b) F(u) is a convex function. c) F(u) is real analytic except at the origin. d) F(u)c 1 u 2 ; as u! 0 + for a positive constant c 1 : The estimate in d) shows that F(u) is notC 2 at the origin but by convexity, it isC 1 : In a standard statistical mechanics terminology this means that the system undergoes a second order phase transition, in the sense that the non-analyticity of the free energy comes from a singularity in the second derivative of the free energy [25]. 6 Note that F 0 (u) = lim n!1 1 N d du logE P 0 (e uL N ) = lim n!1 1 N E P 0 (L N e uL N ) E P 0 (e uL N ) = lim n!1 E N;u L N N where E N;u is the expectation with respect to the polymer measure N;u : We see that there is a drastic change in the system as u passes from non-positive to positive regime. Indeed, F 0 (u) is the expected fraction of the visits to zero by random walk under the polymer measure and it passes from zero to a positive value. This is an example of transition from delocalized to localized phase. We dene u c = supfu : F(u) = 0g as the critical point of the system. The parameter space is split into two regions: L :=fu :u>u c g is called the localized phase and D :=fu :uu c g is called the delocalized phase. 7 From properties of F(); we see that for 1-dimensional random walk pinning model u c = 0: 1.1.2 Variational Form of the Free Energy Let's recall some facts on SSRW onZ d : Let = inffn 1 :S n = 0g be the rst return time of random walk to the origin. We can also consider it as an excursion length. Then, as n!1; we have: d = 1; P (S n = 0) 1 p 2n ; P ( =n) K 1 n 3=2 d = 2; P (S n = 0) A 2 n ; P ( =n) K 2 n(logn) 2 d 3; P (S n = 0) A d n d=2 ; P ( =n) K d n d=2 Note that, in all cases P ( =n)n c (n) as n!1; where c 1; and (n) a slowly varying function, that is, (bn) (n) ! 1 as n!1; for all b> 0: Note that L N k () k N 8 where 1 = ; (1.1.3) k = inffi> k1 :S i = 0g; k 2 (1.1.4) time of the k th return. Therefore, for a> 0; P (L ak k) =P ( k ak) =P ( k ak) e ak Ee k =e ak (Ee ) k =e k( logM ()a) where M (t) = Ee t is the moment generating function of : Since the last inequality true for all > 0; we have P ( k ak)e k sup >0 ( logM ()a) Let's dene I (a) = sup >0 ( logM ()a) which is called the large deviation rate function related to : 9 By Large Deviation Theory, lim k!1 1 k logP ( k ak) =I (a) and also lim !0 + lim k!1 1 k logP ( k k 2 (a;a +)) =I (a) In a more general context, we have the following lemma. For the proof and more on Large Deviation Theory, see [18]. Lemma 1.1.6. For a recurrent Markov Chain lim N!1 1 N logP (L N N) = lim N!1 1 N logP ( N 1 N) =I ( 1 ) Also, lim !0 + lim N!1 1 N logP ( L N N 2 (; +)) =I ( 1 ) (1.1.5) The following lemma gives a variational formula for the free energy of the model which was proved in [2]. Lemma 1.1.7. F(u) = sup 2(0;1) (uI ( 1 )) 10 Proof. Let k> 1 be a large integer, F(u) = lim N!1 1 N logE P 0 (e uL N (S) ) = max 0jk1 lim N!1 1 N logE P 0 (e uL N (S) 1L N N 2( j k ; j+1 k ] ) max 0jk1 lim N!1 1 N log(e u(j+1)N=k P ( L N N j k )) max 0jk1 lim N!1 1 N log(e u(j+1)N=k e j k I (k=j)N ) sup 2(0;1) (uI ( 1 )) + u k If we let k go to innity, we have F(u) sup 2(0;1) (uI ( 1 )) The reverse inequality is proved in a similar way. 1.2 General Homogeneous Pinning Model Polymer pinning models describe the interaction between a directed polymer and a one dimensional defect line. In absence of interaction, the polymer spatial conguration is modeled by (n;X n ) n0 ; where (X n ) n0 is a Markov Chain in a state space with a xed element which we call 0; with law P: The defect line is Nf0g and each time the polymer touches the line, that is X n = 0; it gets an energy reward or penalty which is either positive or negative. The interaction between polymer and the defect line depends only on the law of the return time to 0 of the Markov chain. For this reason, 11 the general pinning model involves a renewal sequence =f 0 ; 1 ;:::g where 0 = 0 andf i i1 g i1 are independent identically distributed positive integer valued random variables. If the polymer is modeled by a Markov chain (X n ) n0 ; then =f 0 ; 1 ;:::g is the set of return times of (X n ) n0 to 0 with the assumption 0 = 0; andL N isj\(0;N]j: One of the reasons polymer pinning models are studied extensively by physicists and mathematicians is that they show a so called localization-delocalization phase transition. The most interesting cases for polymer pinning models involve a renewal sequence with power law, more precisely, K(n) := P( 1 =n) = L(n) n 1+ (1.2.1) where 0 and L is a slowly varying function. Recall that a function L : (0;1)! (0;1) is called slowly varying at innity if lim x!1 L(rx) L(x) = 1 for every r > 0: A slowly varying function diverges or vanishes at innity slower than any polynomial. Recall that a renewal sequence is called recurrent if P n1 K(n) = 1 and transient if P n1 K(n)< 1: In pinning model literature, it is usually assumed that the renewal sequence is recurrent and hence it contains innitely many points P-a.s.. 12 Observe that in the case = 0; the assumption P n1 K(n) = 1 implies that L() tends to zero as n approaches to innity. Note also that E( 1 ) = X n1 nK(n) is nite for > 1 and innite for < 1: Therefore, occupies a positive fraction of N; and called positive recurrent if E( 1 )<1; whereas the density of in N is zero and it is called null recurrent if 1 <1 a.s. and E( 1 ) = +1: As we have seen in the previous section, asymptotically this model covers random walk pinning models in any dimension. The polymer measure is dened in Boltzmann-Gibbs way by dP N;u dP () = 1 Z N;u e uj\(0;N]j 1 N2 (1.2.2) where Z N;u = E e uj\(0;N]j 1 N2 is called the partition function of the constrained system. Remark 1.2.1. The constraint 1 N2 in the denition of the polymer measure is boundary condition. The free model is dened in the following way: dP f N;u dP () = 1 Z f N;u e uj\(0;N]j (1.2.3) where Z f N;u = E e uj\(0;N]j : The constrained model 1.2.2 is more manageable but for most of the results there is no dierence between the models. See [25]. 13 Remark 1.2.2. Note that the assumption that the renewal sequence is recurrent is harm- less because it only changes the exponential factor: Let = P n1 K(n) < 1; then dene ^ := K(n)=: It is easy to see that ^ is recurrent and that the polymer measure and partition function of the model with K() is the same as the model with ^ provided that u is replaced by u + log : By sub-additivity argument, it is easy to show that the following limit exists: F(u) = lim N!1 1 N log E e uj\(0;N]j 1 N2 (1.2.4) The function F() is called the free energy of the system. Let l N = j\ (0;N]j N Note that F 0 (u) = lim N!1 log 1 N d du log E e uj\(0;N]j 1 N2 = lim N!1 E N;u (l N ) F 0 (u) is the expected fraction of the visits to zero by the polymer under the polymer measure in the limit as N tends to innity. We dene u c = supfu : F(u) = 0g 14 The critical point u c is either zero or positive which depends on whether the renewal is recurrent, or transient as we see in Theorem 1.2.3. For the proof of the following theorem which summarizes the main results in this section, see [25] and [27]. Theorem 1.2.3. For homogeneous pinning model , u c = log X n1 K(n) (1.2.5) and as u&u c F(u) (uu c ) max(1=;1) ^ L(1=(uu c )); (1.2.6) where ^ L is a slowly varying function associated with L: Therefore, the transition is of k th order if 2 (1=k; 1=(k 1)): The order of the transition for = 1=k;k 1; is either k th or (k + 1) th order, and this depends on the slowly varying function L(n) that denes K(n): In statistical physics literature, the specic heat exponent is introduced as = 2 lim u&uc log F(u) log(uu c ) provided the limit exists. From Theorem 1.2.3, we see that = 2 max(1=; 1): (1.2.7) 15 Note that > 0 as soon as > 1=2: In the next section, we will see that when the disorder is present, the exponent being greater than 1=2 or less than 1=2 will be important for the disorder relevance. 16 1.3 Disordered Pinning Model In previous section, we have seen that the phase transition of the homogeneous pinning model can be of any order, from rst order to innite, depending on the distribution K() of the renewal sequence : In this section, we summarize the results on the eect of disorder on the pinning/depinning transition. The main questions of physical interest are whether for given ;u the polymer is \pinned", meaning that it places a positive fraction of its monomers at 0 for large N with high probability. the location and nature of the depinning transition as and u vary the eect of disorder, as seen by comparing the transition to the annealed case Let ! =f! i g i0 be a collection of i.i.d. random variables with law P; which plays the role of quenched randomness in the system. We assume that E(! 1 ) = 0; andE(! 2 1 ) = 1 whereE is the expectation with respect toP: In this disordered pinning model, the Hamiltonian of the system and the quenched polymer measure is dened to be in the following form: H N; () = N X j=1 (u +! j ) j 17 d ;u;q N dP () = 1 Z ;u;q N e H N; () 1 N where j := 1 j2 ;u2R; 0: Superscript q denotes the quenched randomness and the normalizing sum is called the quenched partition function: Z ;u;q N = E e H N; () 1 N S N N 1 Figure 1.2: 1-D disordered pinning model: disorder is only on the defect line. : potential. : disorder. There have been a great progress in the mathematical understanding of the role of the quenched randomness in the polymer models in last ten years. In particular, most of the physicists' predictions on the eect of disorder on pinning/depinning transition, the location of the critical points, and the path properties of the polymer in the two thermodynamic phases are justied rigorously, see [3], [19], [24] and [26]. Expectation with respect to the polymer measure will be denoted by E ;u;q N () 18 The following theorem shows that the thermodynamic limit of free energy exists and does not depend on the realization of the randomness ! : Theorem 1.3.1. If E(j! 1 j)<1; then the limit F q (;u) := lim N!1 1 N logZ ;u;q N = lim N!1 1 N E logZ ;u;q N exists for every 0;u2R and it is P-a.s. constant. For the proof see [25]. F q (;u) is convex in (;u); non-decreasing, continuous everywhere and dierentiable almost everywhere in u as a consequence of convexity. 1.3.1 Localization-Delocalization Transition Observe that F (;u) 1 N E log E e P N j=1 (u+! j )n 1 1 =N = u N + 1 N logK(N) so this simple argument shows that F (;u) 0: SinceF (;u) is non-decreasing for a given; the localization/delocalization critical point is dened to be u q c () := supfu :F q (;u) = 0g (1.3.1) 19 and the the function !u q c () is called the critical line. The parameter space (;u) is split into two regions: L :=f(;u) : 0;u>u q c ()g is called the localized phase and D :=f(;u) : 0;uu q c ()g is called the delocalized phase. The critical point u c () is strictly decreasing as a function of and since u c () is a concave function, u q c ()!1 as !1: This means that even the disorder on average is repulsive, the defect line can pin the polymer. See [2] and [25]. The annealed free energy and the annealed critical point are dened in the following way: F a (;u) := lim N!1 1 N logEZ ;u;q N (1.3.2) u a c () := supfu :F a (;u) = 0g (1.3.3) It is an easy application of Jensen's inequality that F q (;u)F a (;u) (1.3.4) 20 therefore u a c ()u q c () for all 0: (1.3.5) Note also that u a c () =u q c (0) logM() (1.3.6) where M() =E(e ! 1 ); in particular logM() = 2 2 for the Gaussian disorder. Thanks to a series of papers [3, 4, 5, 19, 23, 24, 26, 28, 47, 55, 56] by Alexander, Zygouras and Derrida, Giacomin, Lacoin, Toninelli the eect of disorder on the depinning transition is well understood in most of the cases under Gaussian disorder. We can summarize their main results as follows: (i) for every > 0; u a c ()6=u q c () if is large. (ii) for every > 1=2 and > 0; u a c ()6=u q c (): (iii) for every < 1=2; u a c () =u q c () if is small. (iv) for = 0 and > 0; u a c () =u q c (): (v) for = 1=2 and P n1 1 nL(n) 2 <1; u a c () =u q c () if is small. In particular, they showed that (vi) for 1=2<< 1; there exists a constantc and slowly varying function ~ L associated with L such that for all small c 1 2 21 ~ L( 1 )<u q c ()u a c ()<c 2 21 ~ L( 1 ) 21 (vii) for = 1; c 1 2 ~ L( 1 )<u q c ()u a c () (viii) for > 1; c 1 2 <u q c ()u a c ()<c 2 The case = 1=2 is marginal and not fully understood. It is believed that, see [?] for every > 0; u a c ()<u q c () as long as X n1 1 nL(n) 2 =1: For the proofs of statements (i) and (ii), see [3], [19] and [25]. Statements (iii) and (v) are proved by using second moment method by Alexander [3] and by using replica-coupling approach by Toninelli [56] for Gaussian disorder; and then by using a Martingale approach by Lacoin for general disorder case [47]. Statements (vi) and (vii) are proved in [2], [3], [4], [19]. 22 Chapter 2 Directed Polymers in a Random Environment with a Defect Line 2.1 Physical Motivation The directed polymer in a random environment is a model in the theory of disordered systems. The 1 + 1 dimensional version of the model rst appeared in statistical physics literature by Huse and Henley [36] in 1985 as a model for the interface in two-dimensional Ising models with random exchange interaction. This was followed by other physics papers by Huse, Henley and Fisher [37], Kardar [45], Kardar and Zhang [44]. Since then it has been used to describe a variety of phenomena: formation of magnetic domains in spin- glasses [36], vortex lines in superconductors [50], turbulence in viscous incompressible uids (Burger turbulence) [10] and KPZ equation [43]. The statistical mechanics of an elastic object in a random environment has been the focus of many studies in the theoretical physics literature for more than a two decades [36], [44], [52]. In particular, understanding the competition between extended and point defects and their role on the pinning phenomena has been an important quest for physicists working on high-temperature superconductors. In physics literature, an early study of pinning/depinning transition of directed polymer in random environment(point 23 defect) with a defect line(extended defect) was done by Kardar [42], who investigated numerically the pinning of 1 + 1 dimensional directed polymer in a random environment to the defect line. The polymer was found to depin from the line defect, if the pinning potential of the defect line is smaller than a certain threshold value. Critical behavior associated with the depinning transition was later investigated by Zapatocky and Halpin- Healy [57]. The results of Kardar, and Zapatocky and Halpin-Healy were challenged by the work of Tang and Lyuksyutov [54], who argued that a directed polymer is always localized, although weakly, to the defect line in 1 + 1 dimensions, and claim that the depinning transition only exists above 1+1 dimensions. Their conclusion was supported by Balents and Kardar [7], who performed numerical simulations and developed a func- tional renormalization group analysis. On the other hand, analysis of directed polymer itself by Kolomeisky and Strately [46], have led to a dierent conclusion when dier- ent renormalization group Ansatz were used. In a later work by Hwa and Natterman [39], a systematic analysis of the competition in pinning between point and line defects for a directed polymer was investigated. By using the known results for the directed polymer in a random environment in 1 + 1 dimension in the absence of defect line, they constructed a renormalization group analysis directly in 1 + 1 dimension, which is the critical dimension for this problem. Their results support the conclusion of Tang and Lyuksyutov, that the polymer is always pinned at and below 1 + 1 dimensions. All these physics results need to be justied rigorously and our work is toward understanding the depining transition of directed polymer in a random environment with a defect line. 24 2.2 Mathematical Formulation of the Problem 2.2.1 DPRE model The d-dimensional integer lattice version can be formulated as follows: Let P be the distribution of the simple symmetric random walk S =fS j ;j 0g on Z d starting at the origin, d 1: We shall represent the polymer chain as a graphf(j;S j )g n j=1 in NZ d ; so that the polymer lives in (1 +d) dimensional discrete lattice and stretches in the direction of the rst coordinate. Each point (j;S j )2NZ d stands for the position of the j-th monomer in this picture. The random environment is described by V =fv(i;x) : i 1;x2 Z d g which is an independent and identically distributed collection of random variables dened on a probability space (;G;Q) such that E Q [exp(v(i;x))] <1; for all 2 R: Random environment represents the bulk disorder or impurities in the system. In this dissertation, we assume that the disorder distribution is Gaussian with zero mean and unit variance. We dene the Hamiltonian of the system which gives the energy of a polymer f(j;S j )g n j=1 by H N (S) = N X j=1 v(j;S j ) and the quenched polymer measure is dened in the usual Boltzmann-Gibbs way by d ;q N dP (S) = 1 Z ;q N e H N (S) (2.2.1) 25 where > 0 is the inverse temperature and Z ;q N =E P h e H N (S) i is the normalizing constant which is called the quenched partition function. S N N 0 1 2 3 4 5 6 7 8 9 10 11 1 Figure 2.1: 1 + 1 DPRE model Note that quenched polymer measure and partition function are random quantities on the environment space (;G;Q); and the superscript q refers to this quenched ran- domness. Note that when = 0; the polymer measure 2.2.1 is the usual random walk measure and under the random walk measure, paths exhibit diusive behavior. The rst rigorous mathematical work on directed polymers was done by Imbrie and Spencer [41] in 1988 and they proved that in dimension d 3 with small enough ; the end point of the polymer scales as n 1=2 ; i.e. the polymer is diusive. Bolthausen [9], by 26 normalizing the partition function W ;q n = Z ;q n =E Q [Z ;q n ] placed the polymer model in the framework of martingales, and observed that the almost sure limit of the rescaled partition function W 1 = lim n!1 W ;q n is subject to a dichotomy: there are only two possibilities for the positivity of the limit; Q(W 1 > 0) = 1 or Q(W 1 = 0) = 1 because the eventfW 1 = 0g is a tail event. He also improved the result of Imbre and Spencer to a central limit theorem for the end point of the walk which means that in d 3 entropy dominates at high enough temperature, namely the polymer behaves essentially as if it does not live in bulk disorder. In the literature [11, 15, 16], a natural manner for measuring the disorder due to the random environment is to call Q(W 1 > 0) = 1 as weak disorder andQ(W 1 = 0) = 1 as strong disorder. Comets and Yoshida [17], showed that there exists a critical value c = c (d;v)2 [0;1] with c = 0; for d = 1; 2 and 0< c 1 ford 3 such thatQ(W 1 > 0) = 1 If2f0g[ (0; c ) andQ(W 1 = 0) = 1 if > c : The case c =1 can only occur if the environment random variable v(i;x) is a.s. bounded. 2.2.2 DPRE with defect line model The usual set up for directed polymer in a random environment which we described above doesn't contain a defect line. In some pinning models, disorder is present only at the defect lineNf0g; so there is no bulk disorder. In our model, we assumed that the polymer lives in bulk disorder and when it visits the origin, or the axis in space time, it encounters an additional deterministic potential, u2 R; which plays the role of the 27 strength of the defect line. To incorporate the eect of defect line to the model, we dene the Hamiltonian of the system and the quenched polymer measure in the following way: H u N (S) = N X j=1 (v(j;S j ) +u1 S j =0 ) (2.2.2) = H N (S) +L N (S); (2.2.3) d ;u;q N dP (S) = 1 Z ;u;q N e H u N (S) (2.2.4) where > 0 is the inverse temperature and Z ;u;q N =E P h e H u N (S) i is the the quenched partition function. Here P is the distribution of the SSRW when S 0 = 0 a.s.. In this mathematical framework, the random environment V corresponds to \background point defects" and the potential u at the origin in space-time picture is the analog of \the extended defect" or \the defect line" in physics literature. The quenched free energy is dened by f q (;u) := lim N!1 1 N logZ ;u;q N where this limit is taken Q a.s.. The existence and non-randomness of this limit will be proved in the next section. The eect of the quenched disorder on the phase transition is quantied by comparing the quenched model to the corresponding annealed model, which is obtained by averaging 28 the quenched Boltzmann-Gibbs weight over the disorder to give the annealed Boltzmann- Gibbs weight, that is, the annealed polymer measure is dened by d ;u;a N dP (S) = 1 Z ;u;a N E Q h e H u N (S) i (2.2.5) where Z ;u;a N =E P h E Q h e H u N (S) ii is the the annealed partition function. Note that by Fubini's theorem and the environ- ment distribution being Gaussian, E P h E Q h e H u N (S) ii =e 2 2 N E P h e uL N i : The annealed free energy is dened by f a (;u) := lim N!1 1 N logZ ;u;a N = lim N!1 1 N log e 2 2 N E P [e uL N ] = 2 2 + F(u); where F() is the free energy of the 1-D SSRW pinning model. By Jensen's inequality, f q (;u)f a (;u) for all 0 and u2R: (2.2.6) 29 We dene the quenched critical point as u q c () := inffu> 0 :f q (;u)>f q (; 0)g and the annealed critical point as u a c () := inffu> 0 :f a (;u)>f a (; 0)g: Since the quenched free energy is nonrandom, u q c () is constant Q-a.s.. Note that since the critical point u c for 1-D SSRW random walk pinning model is zero, u c = 0, u a c () = 0 for all > 0: Therefore, we have u a c ()u q c () for all 0: (2.2.7) 2.3 Self-averaging of the Free Energy and Critical Point In this section, we will rst prove the existence of the quenched free energy for the constrained model and then by using concentration of measure property of Gaussian processes, we relate the quenched free energy of the constrained model with the quenched free energy of the unconstrained model. We will mostly follow the methods developed in the papers [11] and [12]. We will use the following Gaussian concentration lemma: 30 S N N 0 1 2 3 4 5 6 7 8 9 10 11 1 Figure 2.2: 1 + 1 DPRE model with defect line. : potential. : disorder. Proposition 2.3.1. Let F :R N !R be a Lipschitz function with constant A; that is, jF (x)F (y)jAjjxyjj; x;y2R N ; wherejjjj denotes the Euclidean norm onR N : Then, if Z = (Z 1 ;Z 2 ;:::;Z N ) is a vector of i.i.d. standard normal random variables, we have P(jF (Z) E(F (Z))j>t) exp ( t 2 2A 2 ) for any t> 0: For the proof of the above proposition, see [40]. For the sake of the notational convenience, we will not write ;u;q in the partition function in some places. Let's introduce some notations: 31 Z N (x) =Z N (x;V ) :=Z ;u;q N;x =E Px h e P N j=1 (v(j;S j )+u1 S j =0 ) i Z N (x;y) =Z N (x;y;V ) :=E Px h e P N j=1 (v(j;S j )+u1 S j =0 ) 1 S N =y i where P x is the SSRW measure when S 0 =x a.s. Let n be the time shift operator of order n on the environment V : for all x2Z d and k;n 1; ( n v)(k;x) =v(k +n;x): The proof of the following lemma is an easy application of the Markov property of SSRW: Lemma 2.3.2. For every x;y2Z d and every integer N;M; we have Z N+M (x) = X y Z N (x;y;V )Z M (y; N V ) (2.3.1) Z N+M (x;z) = X y Z N (x;y;V )Z M (y;z; N V ) (2.3.2) We use the notation N x to mean that there is a SSRW path from 0 to x in N steps. The following is also standard. Lemma 2.3.3. For any x2Z d with N x; E Q [logZ N (0;x;V )] 1 2 E Q [logZ 2N (0; 0;V )]: 32 Proof. From Lemma 2.3.2, for any x2Z d with N x; we have Z 2N (0; 0;V )Z N (0;x;V )Z N (x; 0; N V ) (2.3.3) By taking expectation and using the i.i.d. property of V; we get E Q [logZ 2N (0; 0;V )]E Q [logZ N (0;x;V )] +E Q [logZ N (x; 0; N V )] (2.3.4) = 2E Q [logZ N (0;x;V )] (2.3.5) 2.3.1 The Constrained Model Due to periodicity of SSRW, we will assume that N;M are even integers in this subsection. In the constrained model, polymer measure and the corresponding partition function are dened in the following way: d ;u;q;c N dP (S) = 1 Z ;u;q;c N e P N j=1 (v(j;S j )+u1 S j =0 ) 1 S N =0 where > 0 is the inverse temperature and Z ;u;q;c N =E P h e P N j=1 (v(j;S j )+u1 S j =0 ) 1 S N =0 i is the the quenched constrained partition function. 33 By the Markov property of SSRW, we have Z M+N (0; 0)E P 0 h e P M+N j=1 (v(j;S j )+u1 S j =0 ) 1 S N =0 1 S M+N =0 i E P 0 h e P N j=1 (v(j;S j )+u1 S j =0 ) 1 S N =0 i E P 0 h e P M j=1 ( N v(j;S j )+u1 S j =0 ) 1 S M =0 i Z N (0; 0;V )Z M (0; 0; N V ) By taking logarithm and then expectation, we get E Q [logZ N+M (0; 0;V )]E Q [logZ N (0; 0;V )] +E Q [logZ M (0; 0; N V )] (2.3.6) =E Q [logZ N (0; 0;V )] +E Q [logZ M (0; 0;V )] (2.3.7) Therefore,fE Q [logZ N (0; 0;V )]g N1 forms a super-additive sequence and hence by Sub- additive Lemma, the following limit exists: lim N!1 1 N E Q [logZ N (0; 0;V )] = sup N1 1 N E Q [logZ N (0; 0;V )] Note that by using Jensen's inequality, Fubini's theorem and L N N=2; we have E Q [logZ N (0; 0;V )] logE Q [Z N (0; 0;V )] logE P [e 2 2 N+uL N ] log e 2 +u 2 N 34 Therefore, for all N 1; 1 N E Q [logZ N (0; 0;V )] 2 +u 2 (2.3.8) Since the distribution of the random environment V =fv(i;x) : i 1; x2 Z d g is invariant under time shift operators n for all n 1; by Sub-additive Ergodic Theorem [21], the quenched free energy of the constrained model exists andQ-a.s. constant: f q;c (;u) = lim N!1 1 N logZ N (0; 0;V ) = lim N!1 1 N E Q [logZ N (0; 0;V )]: Being almost sure constant is called the self averaging property of the free energy. 2.3.2 The Free Model In this section we prove the existence of the free energy of the unconstrained model. Let's consider the set =f(i;x) : 1iN;i xg; where i x means that there is a SSRW path from 0 to x in i steps. Let M =jj By ordering the elements of the set ; we can use it as an index set for R M : For a given two SRRW paths ; of length N; let's dene the following vectors v ;w inR M as follows: a (i;x) = 1 (i)=x We dene a function F :R M !R by F (z) = logE P h e P N i=1 (z i;S i +u1 S i =0 ) i 35 = logE P h e P (i;x)2 (z i;x 1 S i =x +u1 S i =0 ) i = logE P h e (a S z+ P (i;x)2 u1 S i =0 ) i : By the Cauchy-Schwartz's inequality, we get j(a S z + X (i;x)2 u1 S i =0 ) (a S z 0 + X (i;x)2 u1 S i =0 )jjja S jjjjzz 0 jj p Njjzz 0 jj Therefore, F is a Lipschitz function with constant at most p N: By using the concentration result from Proposition 2.3.1, we have E Q [e 1 p N j logZ N (x)E Q Z N (x)j ] = Z 1 0 Q(j logZ N (x)E Q Z N (x)j p N logt)dt 1 + Z 1 1 e (logt) 2 2 2 dt =K()<1 Therefore, we have the following proposition: Proposition 2.3.4. a) For any t> 0; Q(j logZ ;u;q N E Q Z ;u;q N j>t)e t 2 2N 2 b) For any t> 0 and N x; Q(j logZ ;u;q N (x)E Q Z ;u;q N (x)j>t)e t 2 2N 2 36 c) There exists a constant K =K()> 0 such that E Q [e 1 p N j logZ ;u;q N (x)E Q Z ;u;q N (x)j ]K; N 1; N x It is easy to see that the partition function of the unconstrained system of length N is bounded below by that of the constrained system of the same size, therefore E Q [logZ ;u;q N (0; 0)]E Q [logZ ;u;q N ]: We will now show that we can bound from above the expected value of the logarithm of the partition function of the system of lengthN by the expected value of the logarithm of the partition function of the constrained system of length 2N plus a term of order o(N) as the system size tends to innity. Let = 1 p N and consider, E Q [logZ N ] 1 logE Q [Z N ] = 1 logE Q " X x:N x Z N (0;x) ! # 1 logE Q " X x:N x Z N (0;x) # = 1 logE Q " X x:N x e (logZ N (0;x)E Q [logZ N (0;x)]) e E Q [logZ N (0;x)] # 1 log e 2 E Q [logZ 2N (0;0)] E Q " X x:N x e (logZ N (0;x)E Q [logZ N (0;x)]) #! 1 log e 2 E Q [logZ 2N (0;0)] X x:N x E Q h e (logZ N (0;x)E Q [logZ N (0;x)]) i !! 37 1 log K(2N + 1) d e 2 E Q [logZ 2N (0;0)] d log (K(2N + 1)) + 1 2 E Q [logZ 2N (0; 0)] In the rst inequality we used Jensen's inequality and in the second inequality we used the fact that for non-negative numbers a 1 ;a 2 ;:::;a n and 0<< 1; n X i=1 a i ! n X i=1 a i : In the third and fth inequalities, we used Lemma 2.3.3 and partc) of Proposition 2.3.4, respectively. Hence, we have E Q [logZ N (0; 0)]E Q [logZ N ]d p N log (K(2N + 1)) + 1 2 E Q [logZ 2N (0; 0)] Dividing each term by N and then letting N!1; we get f q;c (;u)f q (;u)f q;c (;u)Q a.s. Therefore, the quenched free energy of the original model exists and equals to the quenched free energy of the constrained model: Theorem 2.3.5. For every > 0 and u2R; f q (;u) = lim N!1 1 N logZ ;u;q N = lim N!1 1 N E Q [logZ ;u;q;c N ] (2.3.9) 38 existsQ-a.s. 39 Chapter 3 Main Results 3.1 Statement of the Main Results and Overview of the Proofs In this chapter, we present our main results, and develop the necessary tools to prove them. We use coarse graining procedure to introduce a coarse grained lattice and a site percolation model on it by covering the rst quadrant of the plane by rectangular boxes of side length proportional to the annealed correlation length. These rectangular boxes are associated with the vertices of the coarse grained lattice. Then by second moment method each site of the coarse grained lattice is identied as open if the disorder in the associated box is favorable for the polymer to stay in with a probability at least p; which can be made as close to 1 as we wish by tuning the parameters of the model, and otherwise the site is identied as closed. We then use recent results of [20] and [31] for Lipschitz percolation to identify an innite open path from the origin for which the fraction of links which lie on thex-axis(defect line) can be made as close to 1 as we wish by letting the site percolation density be close to 1. Coarse graining procedure produces ak-dependent percolation model but by using stochastic domination [49], we can transfer the results of independent model to the dependent one. 40 Theorem 3.1.1. Consider the 1 + 1 dimensional directed polymer in a random environ- ment with a defect line model which has the Hamiltonian dened as in ( 2.2.2). Suppose thatV =fv(i;x) :i 1;x2Z d g is a collection of independent and identically distributed standard normal random variables. Then, given 0<< 1; there exists a K =K() as follows: Provided that and u are suciently small and uK; we have f a (;u)f q (;u) (1)f a (;u): In particular, we have u a c ()u q c ()u a c () +K(): Remark 3.1.2. Note that u a c () = 0 for all 0: 41 3.2 Some Preliminaries In this section, we assume thatC L ;c s are two positive constants such thatC L 1> 3c s > 0: For two random walk paths S 1 = (S 1 0 ;S 1 1 ; ) and S 2 = (S 2 0 ;S 2 1 ; ); dene B N (S 1 ;S 2 ) = N X i=1 1 S 1 i =S 2 i (3.2.1) as the overlap of two SSRW paths up to time N: Recall also that L N :=L N (S) = N X i=1 1 S i =0 : (3.2.2) Lemma 3.2.1. For any N 1; k 1; u 0; and x2Z, we have E Px e uL kN E P 0 e u(L N +1) k Proof. Note that for any y2Z; by Markov property of SSRW E Py e uL N E P 0 e u(L N +1) : (3.2.3) Therefore, E Px e uL kN = X y E Px e uL kN 1 S (k1)N =y = X y E Px e uL (k1)N 1 S (k1)N =y E Py e uL N 42 E P 0 e u(L N +1) E Px e uL (k1)N : By iterating over k; we get E Px e uL kN E P 0 e u(L N +1) k : (3.2.4) Note that for any x2Z; E P 2 x;x e uB N =E P 2 0;0 e uB N (3.2.5) and for x6=x 0 ; E P 2 x;x 0 e uB N E P 2 0;0 e u(B N +1) (3.2.6) where P 2 x;x 0 is the product measure P x P x 0: Therefore, as a consequence of Lemma 3.2.1, we also have the following lemma: Lemma 3.2.2. For any N 1; k 1;u 0; and x;x 0 2Z, we have E P 2 x;x 0 e uB kN E P 2 0;0 e u(B N +1) k : Lemma 3.2.3. For any N 1;u 0; and x 1; E P x+1 e uL N E Px e uL N (3.2.7) 43 Proof. Let x = inffn 1 :S n =xg: E P x+1 e uL N = N X k=1 E P x+1 e uL N 1 x=k +E P x+1 e uL N 1 x>N = N X k=1 E Px e uL Nk P x+1 ( x =k) +P x+1 ( x >N) E Px e uL N P x+1 ( x N) +P x+1 ( x >N) E Px e uL N : Lemma 3.2.4. Let C L 1> 3c s > 0 be two positive integers. Then lim inf N!1 inf jxjcs p N P x max 1iN jS i j 3C L p N;jS N jc s p N > 0; lim inf N!1 inf jxjcs p N P x max 1iN jS i j 3C L p N;jS N C L p Njc s p N > 0 lim inf N!1 inf jxjcs p N P x max 1iN jS i j 3C L p N;jS N +C L p Njc s p N > 0 Proof. By Donsker invariance principle, lim N!1 P 0 max 1iN jS i j 3C L p N;jS N jc s p N] = P max 0t1 jB t j 3C L ;jB 1 jc s where (B t ) 0t1 is a 1-dimensional Brownian motion with B 0 = 0 P-a.s.. Therefore, there exists an N 0 =N 0 (C L ;c s ) 1 such that for all NN 0 ; P 0 max 1iN jS i j 3C L p N;jS N jc s p N 1 2 P max 0t1 jB t j 3C L ;jB 1 jc s 44 Note that for 0xc s p N; P x max 1iN jS i j 3C L p N;jS N jc s p N P x max 1iN jS i xj 2C L p N;xc s p NS N x = P 0 max 1iN jS i j 2C L p N;c s p NS N 0 and forc s p Nx 0; P x max 1iN jS i j 3C L p N;jS N jc s p N P 0 max 1iN jS i j 2C L p N; 0S N c s p N = P 0 max 1iN jS i j 2C L p N;c s p NS N 0 Due to invariance principle, there exists an N 1 = N 1 (C L ;c s ) 1 such that for all NN 1 P 0 max 1iN jS i j 2C L p N;c s p NS N 0 1 2 P max 0t1 jB t j 2C L ;c s B 1 0 Hence, for all NN 1 and forjxjc s p N; P x max 1iN jS i j 3C L p N;jS N jc s p N 1 2 P max 0t1 jB t j 2C L ;c s B 1 0 : This completes the proof of the rst claim in the theorem. 45 For the last two claims, note that for 0xc s p N; P x max 1iN jS i j 3C L p N;jS N C L p Njc s p N P x max 1iN jS i xj 2C L p N;x + (C L c s ) p NS N x +C L p N = P 0 max 1iN jS i j 2C L p N; (C L c s ) p NS N C L p N and forc s p Nx 0; P x max 1iN jS i j 3C L p N;jS N C L p Njc s p N P 0 max 1iN jS i j 2C L p N;C L p NS N (C L +c s ) p N Again by invariance principle, there exists an N 2 =N 2 (C L ;c s ) 1 such that for all NN 2 and forjxjc s p N; P x max 1iN jS i j 3C L p N;jS N C L p Njc s p N 1 2 P max 0t1 jB t j 2C L ;C L B 1 C L +c s By a similar argument, there exists an N 3 = N 3 (C L ;c s ) 1 such that for all N N 3 and forjxjc s p N; P x max 1iN jS i j 3C L p N;jS N +C L p Njc s p N 46 1 2 P max 0t1 jB t j 2C L ;(C L +c s )B 1 C L +c s The main ideas for the proof of the following lemma is due to Prof. S.R.S. Varadhan. We are grateful to him. Lemma 3.2.5. Let C L 1 > 3c s > 0: Then there exists a constant 0 < 0 = 0 (C L ;c s )< 1; such that for all suciently large N andjxjc s p N; E Px e uL N 1 N 0 E Px e uL N where N =fS :jS 0 jc s p N; max 1iN jS i jC L p N;jS N jc s p Ng Proof. Let's dene the following probability measure on the path space: N;x;u (A) := E Px [e uL N 1 A ] E Px [e uL N ] : Let W (n;x) =E Px [e uLn ] =E Px [e [V (S 1 )++V (Sn)] ] where V () =u1 0 (): The process with the distribution N;x;u () is a non-stationary Markov process with transition probabilities from x to x 1 at time k given by (x;y;k;N;u) as follows: (x;y;k;N;u) = 8 > > > > < > > > > : e V (y) 2 W (Nk1;y) W (Nk;x) if k<N 1 2 if kN. 47 where y =x 1: Note that, for y =x 1 and k<N; (x;y;k;N;u) = N;x;u (S k+1 =yjS k =x) = N;x;u (S k+1 =y;S k =x) N;x;u (S k =x) = E Px [e uL N 1 S k =x 1 S k+1 =y ] E Px [e uL N 1 S k =x ] = E Px [e uL k 1 S k =x ]E Px [e uL Nk 1 S 1 =y ] E Px [e uL k 1 S k =x]E Px [e uL Nk ] = 1 2 E Px [e uL k 1 S k =x ]e V (y) E Py [e uL Nk1 ] E Px [e uL k 1 S k =x ]E Px [e uL Nk ] = e V (y) 2 W (Nk 1;y) W (Nk;x) and for kN; (x;y;k;N;u) =P x (S 1 =y) = 1 2 : Note that W (n + 1;x) = 1 2 e V (x+1) W (n;x + 1) + 1 2 e V (x1) W (n;x 1) and by Lemma 3.2.1 and Lemma 3.2.3, we have for x 1; W (n;x + 1)W (n;x) and W (n; 1)e u W (n; 0); and for x1; W (n;x)W (n;x + 1) and W (n;1)e u W (n; 0): 48 Therefore, for x 1;k 1; (x;x 1;k;N;u) 1 2 ; and for x1;k 1; (x;x + 1;k;N;u) 1 2 : Hence, the N;x;u chain S u N can be coupled to the P x chain S N in a such a way that jS u n jjS n j for all n 1: Therefore, N;x;u ( N )P x ( N ) By Lemma 3.2.4, we have a lower bound for P x ( N ): Lemma 3.2.6. Let 0<< 1 be given. Then, for suciently large N andjxjc s p N with x6= 0; E Px e uL N 1 2 P(Z c s p )e (1)NF(u) Proof. Let x = inffn 1 :S n =xg: For a given 0 << 1; there exists an N 0 =N 0 (c s ;) such that for all NN 0 and for 0<xc s p N; P x ( 0 N) = P 0 ( x N) 49 P 0 (S N c s p N) = P 0 (S N c s p p N) 1 2 P(Z c s p ) whereZ denotes the standard normal random variable under the probability measure P. Note that since 1-D SSRW is symmetric, the same bound would be true forc s p N x< 0: Therefore, for suciently large N andjxjc s p N; E Px e uL N N X k=x e u E P 0 e uL Nk P x ( 0 =k) N X k=x e u e u e (Nk)F(u) P x ( 0 =k) N X k=x e (1)NF(u) P x ( 0 =k) = e (1)NF(u) P x ( 0 N) 1 2 P(Z c s p )e (1)NF(u) In the rst inequality we used the Markov property of SSRW and in the second inequality we used Lemma 1.1.4. Remark 3.2.7. By Lemma 1.1.4, we have E P 0 e uL N e u e NF(u) : 50 Therefore, for suciently small u> 0; E P 0 e uL N 1 2 P(Z c s p )e (1)NF(u) : We need to use some facts on the excursion length distribution of (p;q)-walks. First, a denition: Denition 3.2.8. Let X 1 ;X 2 ; be a sequence of independent and identically dis- tributed random variables with P(X 1 = b) = P(X 1 = b) = p=2 2 (0; 1=2) and P(X 1 = 0) =q> 0 where p +q = 1 and b is a positive integer. The partial sum process S = (S n ) n0 ; where S 0 = 0 and S n = P n i=1 X i ; is called a (p;q)-walk. Remark 3.2.9. Let S N =S 1 N S 2 N ; where S 1 N ;S 2 N are SSRWs. Note that ( S N ) N1 is a (1=2; 1=2)-walk with b = 2, and B N (S 1 ;S 2 ) =L N ( S): Let = inffn 1 :S n = 0g: For the proofs of the following propositions, see [25]. Proposition 3.2.10. For any (p;q)-walk, p2 (0; 1); we have P( =n) r p 2 n 3=2 as n!1: 51 Remark 3.2.11. For (1; 0)-walk, the corresponding asymptotic is the following: P( = 2n) r 1 4 n 3=2 as n!1: 3.2.1 The Annealed Correlation Length Recall that the annealed free energy of the system is dened as f a (;u) := lim N!1 1 N logE P 0 E Q e P N j=1 (v(j;S j )+u1 S j =0 ) = lim N!1 1 N log e 2 2 N E P 0 e uL N (S) = 2 2 + F(u) where F() is the free energy of the 1-D SSRW pinning model which we reviewed in Chapter 1: The annealed correlation length M will be dened as M :=M(u) = c 1 F(u) Note that M!1 as u! 0 since F(u)c 1 u 2 as u! 0: Next proposition will be important when we deal with a system of length of multiple of one correlation length. 52 Proposition 3.2.12. Let 0<a< 1 be given. Then there exists a constant K =K(a)> 0 such that for suciently small and u; and uK(a); we have E P 2 0;0 e 2 2 (B M (S 1 ;S 2 )+1) 1 a: We will prove Proposition 3.2.12 in the rest of this section. Recall that a random variableX is said to have a Geometric distribution with param- eter p2 (0; 1); if P(X =k) = (1p) k1 p where k = 1; 2; 3; : Then P(X >k) = (1p) k and the moment generating function of X is given by the following formula: E(e tX ) = pe t 1 (1p)e t (3.2.8) provided that t< log(1p): Let E i denote the length of the i th excursion of S =S 1 S 2 from 0. Then P (B N + 1>k) P ( max 1ik E i N) = (1P (E 1 >N)) k for all N;k 1: 53 By Remark 3.2.11, P (E 1 > N) q 1 N 1=2 as N ! 1, therefore for a xed 0<K 4 < 1; there exists N 4 =N 4 (K 4 ) such that for all NN 4 P (B N + 1>k) (1K 4 r 1 N 1=2 ) k for all k 1: (3.2.9) Therefore, under the random walk measure P; B M + 1 is stochastically dominated by a Geometric random variable G M with parameter p M =K 4 r 1 M 1=2 for the annealed correlation length M being suciently large, which is equivalent to u being suciently small. Therefore, we will have for u suciently small, E P 2 0;0 e 2 2 (B M (S 1 ;S 2 )+1) 1 p M e 2 2 1 (1p M )e 2 2 1 (3.2.10) provided that 2 2 < log(1p M ); which is equivalent to the condition p M > 1e 2 2 : (3.2.11) The condition 3.2.11 is necessary for the existence of the moment generating function of G M for 2 2 , and with the condition that u> 0 be suciently small, it implies that > 0 must be small as well. 54 From 3.2.10, for a given 0<a< 1; we would like to have p M e 2 2 1 (1p M )e 2 2 1a (3.2.12) which is equivalent to p M a + 1 a (1e 2 2 ) (3.2.13) Since 1e 2 2 2 2 and a< 1; the condition p M 4 2 a (3.2.14) will be sucient for 3.2.11 and 3.2.12. From part (d) of the properties of the free energy of the random walk pinning model on page 6, we have M (u) 2 as u! 0: Therefore, for a xed 0<K 5 < 1; we will have (u) 2 K 5 M for suciently small u: By using the denition of p M =K 4 q 1 M 1=2 , we get p M K 4 K 1=2 5 r 1 u: (3.2.15) Therefore, for and u suciently small, to satisfy the condition 3.2.14, K 4 K 1=2 5 r 1 u 4 2 a (3.2.16) 55 will be sucient. This is equivalent to u K (3.2.17) which will be a sucient condition for 3.2.12 where K :=K(a) = 4 p =aK 4 K 1=2 5 : 56 3.3 The Coarse Grained Lattice L CG In this section, we introduce a coarse grained lattice L CG :=f(I;J)2Z 2 :I 0; 0JIg: Let C L 1 > 3c s > 0 be two positive integers and N = k 0 M be a multiple of the annealed correlation lengthM such thatN and p N are integers. We use capital letters (I;J) for a site in the coarse grained lattice which corresponds to the segment R(I;J) :=f(k;l)2Z 2 :k =IN; (JC L c s ) p Nl (JC L +c s ) p Ng in the original latticeZ 2 : We call R(I;J) the window-(I;J) inZ 2 : The box starting from the window-(I;J) is the following region inZ 2 : B(I;J) := [IN; (I + 1)N] [(JC L 3C L ) p N; (JC L + 3C L ) p N] We say that there is a link between sites (I;J) and (K;L) if K =I + 1 andjLJj 1: 57 0 N 2N 3N 4N 5N 6N 7N 1 Figure 3.1: The Coarse Grained LatticeL CG . A path (I;J)!(K;L) from site (I;J) to site (K;L) in the coarse grained lattice L CG is a sequence of sites (I 1 ;J 1 ) = (I;J); (I 2 ;J 2 ); ; (I N ;J N ) = (K;L) such that I i+1 =I i + 1; andjJ i J i+1 j 1; 1i<N: (I;J)!(K;L) (I i ) will denote the second coordinate J i of a site (I i ;J i ) in the path (I;J)!(K;L) : We will use the notation (I;J) for (0;0)!(I;J) : For a given two paths 1 := 1 (I;J)!(K;L) and 2 := 2 (I;J)!(K;L) from site (I;J) to site (K;L); we say that 1 is closer to the x-axis than 2 if 1 (I i ) 2 (I i ) for each II i K: 58 Assume that each site (I;J)2L CG is dened as open or closed in a well-dened way. Then we introduce the following denitions: A path (I;J)!(K;L) from site (I;J) to site (K;L) is called open if its all sites are open. A path (I;J)!(K;L) from site (I;J) to site (K;L) is called maximal if it has the maximum number of open sites among all paths from site (I;J) to site (K;L). A path from site (I;J) to site (K;L) is called optimal if it is the maximal path which is closest to the x-axis. There is exactly one optimal path for given sites (I;J) and (K;L) and we denote it by opt (I;J)!(K;L) : 1 (I;J) will denote a general innite open path from the site (I;J). An innite open path from a site (I;J); which is closest to thex-axis among all such paths, is called the innite good path from the site (I;J); and we denote it by G;1 (I;J) : If there is an innite good path from a site (I;J), then it must be unique. G;1 will denote the innite good path from the site (0; 0): For 0 I K; G;1 I!K will denote the nite segment of the path G;1 between the sites with rst coordinates I and K; respectively. G;1 I will be used for G;1 0!I : Note that if the site (I 0 ;J 0 ) is on the innite good path from (0; 0); then opt (I 0 ;J 0 ) will be G;1 I 0 : For a given path (I;J) inL CG ; we identify a subset of the SSRW paths of lengthIN in the following way: 59 (I;J) IN :=the set of SSRW paths of lengthIN from (0; 0) to the window-(I;J) which pass through all the windowsR(I 0 ;J 0 ) which correspond to sites (I 0 ;J 0 ) in the path (I;J) and stay in the boxes B(I 0 ;J 0 ) starting from the windows R(I 0 ;J 0 ). And for 0IK; G;1 I!K ; we dene G;1 I!K (KI)N :=the set of SSRW paths of length (KI)N from the window-(I; G;1 (I)) to the window-(K; G;1 (K)) which pass through all the windows R(I 0 ;J 0 ) which corre- spond to the sites (I 0 ;J 0 ) in the path G;1 in between (I; G;1 (I)) and (K; G;1 (K)) and stay in the boxes B(I 0 ;J 0 ) starting from the windows R(I 0 ;J 0 ). The corresponding partition function restricted to the subset (I;J) IN of SSRW paths will be denoted by Z ;u;q IN ( (I;J) IN ) :=E P 0 e P IN i=1 (v(i;S i )+u1 S i =0 ) 1 (I;J) IN We dene random probability measures on the window R(I;J) associated with a given path (I;J) in the following way: For x2R(I;J); R(I;J) (I;J) (x) := Z ;u;q IN ( (I;J) IN ;S IN =x) Z ;u;q IN ( (I;J) IN ) ; (3.3.1) Let's consider the following subsets of SSRW paths: up N :=f(S 0 ; ;S N ) :jS 0 jc s p N;jS N C L p Njc s p N;jS i j 3C L p N; 1iNg; forward N :=f(S 0 ; ;S N ) :jS 0 jc s p N;jS N jc s p N;jS i j 3C L p N; 1iNg; 60 and down N :=f(S 0 ; ;S N ) :jS 0 jc s p N;jS N +C L p Njc s p N;jS i j 3C L p N; 1iNg: The sets up N;R(I;J) ; forward N;R(I;J) and down N;R(I;J) will refer to the up, forward, down set of random walk paths which start at the window R(I;J), stay in the boxB(I;J), and end up at the window R(I + 1;J +l); l = +1; 0;1; respectively. Let q (I;J) := R(I;J) opt (I;J) be the random probability distribution on the window-(I;J) which comes from the optimal path opt (I;J) ; and let q (0;0) := 0 be the point mass at zero. We dene the partition functions over the up, forward, down SSRW paths between the window-(I;J) and the window-(I + 1;J +l);I 0; J 1; l = +1; 0;1 with initial distribution q (I;J) as follows: Z ;u;q N; q (I;J) ( g N;R(I;J) ) :=E P q (I;J) e P N k=1 v(IN+k;S IN+k )+u1 S IN+k =0 1 g N;R(I;J) ! ; where g N;R(I;J) is up N;R(I;J) ; forward N;R(I;J) or down N;R(I;J) for l = +1; 0;1; respectively. 3.3.1 Assigningeachsite (I;J)inthecoarsegrainedlatticeL CG asopen or closed. Let's consider the following ltrations: F I :=(fv(i;x) : 1iIN;x2Zg); I 1: 61 Note that the measures q (I;J) areF I -measurable for all J 0. In the following, rst we will identify two non-random constants U on and U o which will be lower bounds for partition functions associated to on and o axis coarse grained lattice links such that U on 1 2 E Q Z ;u;q N; q (I;0) ( forward N;R(I;0) )jF I ; Qa:s: for each I 0; and U o minf 1 2 E Q Z ;u;q N; q (I;0) ( up N;R(I;0) )jF I ; 1 2 E Q Z ;u;q N; q (I;J) ( g N;R(I;J) )jF I g Qa:s: for each I 0;J 1; where g N;R(I;J) is any one of up N;R(I;J) ; forward N;R(I;J) or down N;R(I;J) : Note that for I 1; by Lemma 3.2.5 and Lemma 3.2.6, for suciently large N; and suciently small u; Q-a.s. E Q Z ;u;q N; q (I;0) ( forward N;R(I;0) )jF I = X x2R(I;0) q (I;0) (x)E Q E Px " e P N k=1 v(IN+k;S IN+k )+u1 S IN+k =0 1 forward N;R(I;0) #! = X x2R(I;0) q (I;0) (x)e 2 2 N E Px h e P N k=1 u1 S IN+k =0 1 forward N;R(I;0) i X x2R(I;0) q (I;0) (x)e 2 2 N 0 1 2 P(Z c s p )e (1)NF(u) 0 1 2 P(Z c s p )e 2 2 (1)N e (1)NF(u) = 0 1 2 P(Z c s p )e ( 2 2 +F(u))(1)N = 0 1 2 P(Z c s p )e f a (;u)(1)N 62 Therefore, let's dene U on := 0 1 4 P(Z c s p )e f a (;u)(1)N (3.3.2) And for I 0;J 1; for suciently large N; we have Q-a.s. E Q Z ;u;q N; q (I;J) ( g N;R(I;J) )jF I = X x2R(I;J) q (I;J) (x)E Q E Px " e P N k=1 v(IN+k;S IN+k )+u1 S IN+k =0 1 g N;R(I;J) #! X x2R(I;J) q (I;J) (x)e 2 2 N E Px h 1 g N;R(I;J) i 1 2 e 2 2 N P(A C L ;cs ) where A C L ;cs =fmax 0t1 jB t j 2C L ;(C L +c s )B 1 C L +c s g: Therefore, let's dene U o := 1 4 e 2 2 N P(A C L ;cs ): (3.3.3) We will now specify the rule which assigns each site (I;J) in the coarse grained lattice L CG as open or closed. The site (I;J) = (0; 0) is called open if Z ;u;q N ( up N )U o and Z ;u;q N ( forward N )U on : Assume that all the sites (K;L); for 0K <I and 0LK are dened as open or closed. 63 Then, conditionally onF I ; the site (I; 0) is called open if Z ;u;q N; q (I;0) ( up N;R(I;0) )U o and Z ;u;q N; q (I;0) ( forward N;R(I;0) )U on ; and the sites (I;J); 0<JI are called open if Z ;u;q N; q (I;J) ( up N;R(I;J) )U o ; Z ;u;q N; q (I;J) ( forward N;R(I;J) )U o and Z ;u;q N; q (I;J) ( down N;R(I;J) )U o : Note that since the (random) probability measures q (I;J) 's on the windows R(I;J)'s depend on the optimal path from (0; 0) to (I;J); they depend on the sites being open or closed dened in previous steps. LetfX (I;J) : (I;J)2L CG g be the indicator random variables such that X (I;J) = 8 > > < > > : 1 if the site (I;J) is open; 0 if the site (I;J) is closed. 3.3.2 Second Moment Method and the Site Densities in L CG The method which we describe now is generally called as second moment method in probability literature; for some applications, see [3], [48]. Let X be a random variable with nite mean and variance, and let 0<< 1: By the Chebyshev's Inequality, we have P (X (1)EX) =P (XEXEX) P (jXEXjEX) 64 1 2 Var(X) (EX) 2 Similarly, P (X (1 +)EX) 1 2 Var(X) (EX) 2 Hence, P ((1)EXX (1 +)EX) =P (X (1 +)EX)P (X (1)EX) = 1P (X (1 +)EX)P (X (1)EX) 1 2 2 Var(X) (EX) 2 1 provided that Var(X) (EX) 2 2 2 : (3.3.4) Now, we will use the second moment method and control the probability of a site (I;J)2L CG being open with respect to the rule dened in the subsection 3.3.1. Let's rst consider a site (I; 0) on the x-axis. We will use the fact that there are non-random constants U on and U o as we dened in 3.3.2 and 3.3.3 respectively, such that U on 1 2 E Q Z ;u;q N; q (I;0) ( forward N;R(I;0) )jF I Qa:s:; and Q-a.s. 65 U o minf 1 2 E Q Z ;u;q N; q (I;0) ( up N;R(I;0) )jF I ; 1 2 E Q Z ;u;q N; q (I;J) ( g N;R(I;J) )jF I g where g N;R(I;J) is any one of up N;R(I;J) ; forward N;R(I;J) or down N;R(I;J) : Then, by the second moment method Q Z ;u;q N; q (I;0) ( up N;R(I;0) )U o jF I 4 Var Q Z ;u;q N; q (I;0) ( up N;R(I;0) )jF I E Q Z ;u;q N; q (I;0) ( up N;R(I;0) ) jF I 2 Qa:s: and similarly Q Z ;u;q N; q (I;0) ( forward N;R(I;0) )U on jF I 4 Var Q Z ;u;q N; q (I;0) ( forward N;R(I;0) )jF I E Q Z ;u;q N; q (I;0) ( forward N;R(I;0) )jF I 2 Qa:s: Therefore, for the site (I; 0); Q-a.s. Q(X (I;0) = 1jF I ) = 1Q Z ;u;q N; q (I;0) ( up N;R(I;0) )U o or Z ;u;q N; q (I;0) ( forward N;R(I;0) )U on jF I 1 provided that Q-a.s. Var Q Z ;u;q N; q (I;0) ( forward N;R(I;0) )jF I E Q Z ;u;q N; q (I;0) ( forward N;R(I;0) )jF I 2 8 and Var Q Z ;u;q N; q (I;0) ( up N;R(I;0) )jF I E Q Z ;u;q N; q (I;0) ( up N;R(I;0) )jF I 2 8 : 66 Similarly, for a site (I;J) for J 1; we will have Q-a.s. Q(X (I;J) = 1jF I ) 1 provided that Q-a.s. Var Q Z ;u;q N; q (I;J) ( up N;R(I;J) )jF I E Q Z ;u;q N; q (I;J) ( up N;R(I;J) )jF I 2 12 ; Var Q Z ;u;q N; q (I;J) ( forward N;R(I;J) )jF I E Q Z ;u;q N; q (I;J) ( forward N;R(I;J) )jF I 2 12 and Var Q Z ;u;q N; q (I;J) ( down N;R(I;J) )jF I E Q Z ;u;q N; q (I;J) ( down N;R(I;J) )jF I 2 12 : We will now control the ratio of the variance and the square of the mean for the partition function over the up, forward and down sets of SSRW paths. Recall that H N (S) = P N i=1 v(i;S i ): Note that for given random walk paths S 1 and S 2 ; by using the fact that the distri- bution of the environment is Gaussian, we have E Q e H N (S 1 )+uL N (S 1 ) e H N (S 2 )+uL N (S 2 ) = e uL N (S 1 ) e uL N (S 2 ) e (2) 2 2 B N (S 1 ;S 2 ) e 2 2 2(NB N (S 1 ;S 2 )) = e uL N (S 1 ) e uL N (S 2 ) e 2 B N (S 1 ;S 2 ) e 2 N Let g N;R(I;J) denote any one of the up, forward, down sets of paths which starts from the window-(I;J): 67 Let's consider the following ratio: T (I;J) N;q = T (I;J) N;q (;u;k 0 ) := Var Q Z ;u;q N; q (I;J) ( g N;R(I;J) )jF I E Q Z ;u;q N; q (I;J) ( g N;R(I;J) )jF I 2 = E Q Z ;u;q N; q (I;J) ( g N;R(I;J) ) 2 jF I E Q Z ;u;q N; q (I;J) ( g N;R(I;J) )jF I 2 E Q Z ;u;q N; q (I;J) ( g N;R(I;J) )jF I 2 First, we will deal with the numerator. Using the facts thatN =k 0 M; (d1) 2 d 2 1 for d 1; the Cauchy-Schwartz inequality, and Lemma 3.2.1 and Lemma 3.2.2, we get Q-a.s. E Q Z ;u;q N; q (I;J) ( g N;R(I;J) ) 2 jF I E Q Z ;u;q N; q (I;J) ( g N;R(I;J) )jF I 2 = e 2 N X x;x 0 2R(I;J) q (I;J) (x) q (I;J) (x 0 ) E P 2 x;x 0 e 2 B N (S 1 ;S 2 ) 1 e uL N (S 1 ) e uL N (S 2 ) 1 g; 2 N;R(I;J) e 2 N X x;x 0 2R(I;J) q (I;J) (x) q (I;J) (x 0 )E P 2 x;x 0 e 2 B N (S 1 ;S 2 ) 1 e uL N (S 1 ) e uL N (S 2 ) e 2 N X x;x 0 2R(I;J) h q (I;J) (x) q (I;J) (x 0 ) E P 2 x;x 0 e 2 2 B N (S 1 ;S 2 ) 1 1=2 E Px e 2uL N (S 1 ) 1=2 E P x 0 e 2uL N (S 2 ) 1=2 i e 2 N E P 2 0;0 e 2 2 (B M (S 1 ;S 2 )+1) k 0 1 1=2 E P 0 e 2u(L M +1) k 0 = e 2 N E P 2 0;0 e 2 2 (B M (S 1 ;S 2 )+1) 1 + 1 k 0 1 1=2 E P 0 e 2u(L M +1) k 0 68 and for the denominator, Q-a.s. E Q Z ;u;q N; q (I;J) ( g N;R(I;J) )jF I = X x2R(I;J) q (I;J) (x)E Q E Px " e P N k=1 v(IN+k;S IN+k )+u1 S IN+k =0 1 g N;R(I;J) #! X x2R(I;J) q (I;J) (x)e 2 2 N E Px 1 g N;R(I;J) e 2 2 N K 6 where K 6 = 1 2 P(A C L ;cs ): Note that by Lemma 1.1.5, E P 0 e 2uL M Ke F(2u)M and F(2u)M! 4c 1 as u! 0; since F(u)c 1 u 2 as u! 0; and M =c 1 =F(u): Therefore, there exist a K 7 > 1 such that E P 0 e 2u(L M +1) K 7 for suciently small u: As a result, for any (I;J)2L CG ; we get Q-a.s. T (I;J) N;q 1 K 2 6 E P 0 e 2u(L M +1) k 0 E P 2 0;0 h e 2 2 (B M (S 1 ;S 2 )+1) 1 i + 1 k 0 1 1=2 K 2 6 K k 0 7 E P 2 0;0 h e 2 2 (B M (S 1 ;S 2 )+1) 1 i + 1 k 0 1 1=2 69 Hence, for any given 0<< 1; we can choose a in Proposition 3.2.12 as follows: a = K 4 6 2 K 2k 0 7 + 1 1=k 0 1: (3.3.5) Note that 0 < K 6 < 1 and K 7 > 1; therefore for the above choice of a, a will be less than 1 for all k 0 1: 70 3.4 Lipschitz Percolation Lipschitz percolation, the existence of open Lipschitz surfaces, was rst introduced and studied in the papers [20] and [31]. In this section, we brie y summarize and adapt some of their results for dimension d = 2; to use in our context. The site percolation model in Z 2 is obtained by designating each site x2 Z 2 open with probabilityp; otherwise closed, with dierent sites receiving independent states. The corresponding probability measure on the sample space =f0; 1g Z 2 will be denoted by P p ; and expectation byE p : LetZ + 0 =f0; 1; 2; 3;:::g: A function F :Z!Z + 0 is called Lipschitz if for any x;y2Z withjxyj = 1; we have jF (x)F (y)j 1: A Lipschitz functionF :Z!Z + 0 is called open if for eachx2Z; the site (x;F (x))2 Z 2 is open. Remark 3.4.1. In the original papers [20] and [31], it was assumed that F () 1; for our convenience we assume that F () 0: It does not change the results. LetLIPF be the event that there exists an open Lipschitz functionF :Z!Z + 0 : The eventLIPF is invariant under the translation ofZ 2 by the unit vector (1; 0): Therefore, 71 P p (LIPF ) = 0 or 1: Since LIPF is also an increasing event, there exists a p L 2 [0; 1] such that P p (LIPF ) = 8 > > < > > : 0 if p<p L ; 1 if p>p L : It was proved in [20] that 0<p L < 1: y x 1 Figure 3.2: Site percolation onZ 2 : : open, : closed y x 1 Figure 3.3: The lowest open Lipschitz function F () 72 One of the main results of [20] is that the random eld (F (x) :x2Z) is stationary and ergodic under each translation ofZ: For any familyF of Lipschitz functions, the lowest function F (x) = inffF (x) :F2Fg is also Lipschitz. If there exist an open Lipschitz function, then there exists a lowest open Lipschitz function, and it will be again denoted by F: Theorem 3.4.2 (Theorem 2, [31]). Let F be the lowest open Lipschitz function. For p>p L ; there exists =(p)> 0 such that P p (F (0)>n)e (n+1) ; n> 0: Remark 3.4.3. By Theorem 3.4.2, we can conclude that with positive probability there exists an innite good path starting from (0; 0) in the coarse grained lattice L CG , if each site (I;J)2L CG is assigned independently open with probability p > p L or closed otherwise. Note that since the law ofF (x) is the same for allx2Z 2 ; the choice of the origin in the above theorem is arbitrary. LetS be the set of allx2Z for whichF (x)> 0: Recall that by denition we assumed that F 0: Let S 0 be the vertex set of component containing 0 in the sub-graph of the nearest neighbor lattice ofZ induced by S: We dene S 0 =; if 0 = 2S: 73 The following theorem is a particular case of Theorem 3 of [31] for d = 2: Theorem 3.4.4. There exists p 0 L < 1 such that for p>p 0 L exp (n)P p (jS 0 j>n) exp ( n); n 1; where =(p) and = (p) are positive and nite. Since F () is stationary and ergodic, by Ergodic Theorem lim N!1 1 N N X i=1 1 (F (i1)=F (i)=0) =P p (F (0) =F (1) = 0); P p a.s. Let l s 2Z such that F (l s ) = 0 and F (l s + 1)6= 0; and let l f = inffl>l s + 1 :F (l) = 0g: We call the sequence of vertices (l s ;F (l s )); (l 1 ;F (l 1 ); (l 2 ;F (l 2 ): ; (l f ;F (l f )) a detour from (l s ; 0) to (l f ; 0), where l 1 =l s + 1 and l i+1 =l i + 1: (l s ,0) (l f ,0) 1 Figure 3.4: A detour 74 Note that P p (F (0) =F (1) = 0) = 1P p (the link between (0; 0) and (1; 0) is not a part of the graph of F ()) For k;l 0; let D k;l be the event that there is a detour from (k; 0) to (l; 0): For a given > 0 and 0 < p 0 < 1; we can choose l 0 ;l 1 suciently large so that exp ( (p 0 )(l 0 +l 1 + 1))<=2: Forp>p 0 and close to 1, we can make (l 0 + 1)l 1 (l 0 +l 1 + 1)(1p)<=2: Since (p) is increasing in p; by choosing p close to 1, we get P p (the link between (0; 0) and (1; 0) is not in F ()) X k0;l1 P p (D k;l ) X 0kl 0 1ll 1 P p (D k;l ) +P p (there is a detour over (0; 0) of length>l 0 +l 1 + 1) (l 0 + 1)l 1 (l 0 +l 1 )(1p) + exp ( (p)(l 0 +l 1 + 1)) In the third inequality, we used the fact that there is a detour from (k; 0) to (l; 0) implies that at least one of the sites in between must be closed. We also used Theo- rem 3.4.4 in the same inequality. 75 3.5 Stochastic Domination Denition 3.5.1. A collection of random variables (X s ) s2Z is called k-dependent if for each pair of subsets A;BZ such thatjabj>k for each a2A;b2B; the families of random variables (X s ) s2A and (X s ) s2B are independent. The next theorem is a particular case of the main theorem of the paper [49] by Liggett, Schonmann and Stacey. Theorem 3.5.2. Let (X s ) s2Z be a collection of 0-1 valuedk-dependent random variables, and suppose that there exists a p2 (0; 1) such that for each s2Z P(X s = 1)p: Then if p>p SD (k) = 1 k k (k + 1) k+1 ; then (X s ) s2Z is dominated from below by a product random eld with density 0<(p)< 1: Furthermore, (p)! 1 as p! 1: Let's consider the following ltrations: F I :=(fv(i;x) : 1iIN;x2Zg); I 1; and letL I CG :=f(I;J)2L CG : 0JIg: 76 Note that for each I 1; conditionally onF I fX (I;J) : (I;J)2L I CG g is a 3-dependent collection of random variables, and from section 3.3.2, we know that for a given > 0, for suciently small u> 0 and > 0 with uK(); we have Q(X (I;J) = 1jF I ) 1 Qa:s: for each I 1;J 0: Therefore, by Theorem 3.5.2 there exists a collection of independent and identically distributed 0-1 valued random variablesfY (I;J) : (I;J)2L CG g withQ(Y (I;J) = 1) =(p) and Q(X (I;J) Y (I;J) jF I ) = 1 Qa:s: (3.5.1) where p = p(u); and (p)! 1 as p! 1: Note also that p can be made greater than 1 for suciently small u> 0 and > 0 with uK(): By taking the expectation of both sides with respect to Q in 3.5.1, we get Q(X (I;J) Y (I;J) ) = 1 for all (I;J)2L CG : 77 3.6 Final Steps In section 3.3.2, we showed that for any 2 (0; 1), the site density p CG ofL CG can be made greater than 1 for suciently small u > 0 and > 0 with u K(). Therefore, we can assume that p CG > max(p L ;p 0 L ;p SD (3)) and close to 1: 0 N 2N 3N 4N 5N 6N 7N 1 Figure 3.5: The innite good path G;1 from the site (0; 0). Dark rectangles denote open sites and white ones denote closed sites in the coarse grained lattice L CG . By Remark 3.4.3, with positive probability the innite good path G;1 from the site (0; 0) exists. Let I L := the number of links of G;1 L on the x axis: Then from section 3.4, we know that the following limit exists Q-a.s. and constant: =(u) := lim L!1 I L L : 78 Furthermore, ! 1 as p CG ! 1 thanks to the argument at the end of the section 3.4. Recall that U o = 1 4 e 2 2 N P(A C L ;cs ) = 1 e 2 2 N where 1 = 1 4 P(A C L ;cs ); and U on = 0 1 4 P(Z c s p )e f a (;u)(1)N = 2 e f a (;u)(1)N where 2 = 0 1 4 P(Z cs p ); and Z is a standard normal random variable. For a standard normal random variable Z, the tail behavior is P(Z >x) 1 p 2x e x 2 =2 as x!1: Therefore, we have 2 0 4 p 2c s p e c 2 s =2 as ! 0: (3.6.1) Let's dene 3 =( log 2 + (1) log 1 ): Note that 3 > 0; since ; 1 ; 2 2 (0; 1); and 3 =O(1=) as ! 0: (3.6.2) Let q LN := G;1 L LN ; where q denotes the quenched randomness of the environment. 79 Note that 1 LN logZ ;u;q LN 1 LN logZ ;u;q LN ( q LN ) and Z ;u;q LN ( q LN ) = L Y I=1 Z ;u;q IN ( q IN ) Z ;u;q (I1)N ( q (I1)N ) (3.6.3) = L Y I=1 Z ;u;q N; q (I1; G;1 (I1)) ( G;1 I1!I N ) (3.6.4) where Z ;u;q 0 := 1: Remark 3.6.1. For a given realization of the environment, when we designate each site (I;J) in the coarse grained lattice L CG as open or closed, we used random probability measures q (I;J) := R(I;J) opt (I;J) on the window-(I;J) which comes from the optimal path opt (I;J) as we dened in section 3.3. In that case, since the optimal path for a site on the innite good path will be the nite part of that innite good path, the random probability measures in the product 3.6.3 will be the same measures as we used initially to dene sites as open or closed. Since there will be an innite good path from (0; 0)2L CG with positive probability, there exists a c> 0 such that for all L 1; Q 1 LN logZ ;u;q LN 1 LN logU I L on U LI L o c 80 By using the fact that the quenched free energy f q (;) has self-averaging property, N = Mk 0 ; M = c 1 =F(u) and f a (;u) = F(u) + 2 =2 and letting L!1; we get Q-a.s. f q (;u) 1 N logU on + (1) 1 N logU o = 1 N log 2 e f a (;u)(1)N + (1) 1 N log 1 e 2 2 N = (1)f a (;u) 1 N 3 + (1) 2 2 = (1)f a (;u) (F(u) + 2 2 2 2 ) 1 c 1 k 0 3 + (1) 2 2 = (1)f a (;u)f a (;u) 1 c 1 k 0 3 + 2 2 1 + 1 c 1 k 0 3 For suciently small > 0 andu> 0 withuK(), we can make 1; and by choosing k 0 of orderb 2 c, the second term can be made greater thanf a (;u); and the third term stays bounded and positive since 3 = O(1=) as ! 0. Therefore, we get f q (;u) (1 3)f a (;u): This completes the proof of Theorem 3.1.1. 81 Bibliography [1] Albeverio, S. and Zhou, X. (1996) A martingale approach to directed polymers in a random environment. J. 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Abstract (if available)
Abstract
We study the depinning transition of the 1+1 dimensional directed polymer in a random environment with a defect line model. The monomer locations of the polymer is modeled by the space-time trajectory of a one dimensional simple symmetric random walk. Random environment is introduced by assigning each site of Z² an independent and identically distributed normal random variable that interacts with the polymer when it visits that site. The defect line is incorporated to the model by having an additional constant potential u at the origin which gives a reward or penalty to the polymer as it visits the origin. There is a critical value of u above which polymer is pinned, placing a positive fraction of its monomers at zero with high probability. ❧ Our first result is that the quenched free energy exists and the self-averaging holds, which implies that there is a nonrandom quenched critical point. To see the effect of disorder on the depinning transition, we compare the quenched free energy of the system as a function of u to the corresponding annealed system. Our main result is that the quenched and annealed free energies differ significantly only in a very small neighborhood of the critical point and we show that the size of this neighborhood scales as at most β as β → 0, where β is the inverse temperature.
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Yıldırım, Gökhan
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On the depinning transition of the directed polymer in a random environment with a defect line
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