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Fractional moment bounds and disorder relevance for pinning models
 COMMUN. MATH. PHYS
, 2007
"... We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(·) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n) = n −α−1 L(n), with L(·) slowly varying. The model undergoes a (de)locali ..."
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Cited by 36 (16 self)
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We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(·) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n) = n −α−1 L(n), with L(·) slowly varying. The model undergoes a (de)localization phase transition: the free energy (per unit length) is zero in the delocalized phase and positive in the localized phase. For α < 1/2 it is known that disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents [2, 24]. The same has been proven also for α = 1/2, but under the assumption that L(·) diverges sufficiently fast at infinity, an hypothesis that is not satisfied in the (1 + 1)dimensional wetting model considered in [13, 9], where L(·) is asymptotically constant. Here we prove that, if 1/2 < α < 1 or α> 1, then quenched and annealed critical points differ whenever disorder is present, and we give the scaling form of their difference for small disorder. In agreement with the socalled Harris criterion, disorder is therefore relevant in this case. In the marginal case α = 1/2, under the assumption that L(·) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed
Quenched and annealed critical points in polymer pinning models
"... Abstract. We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential u+Vn which the chain encounters when it visits a special state 0 at time n. The disorder (Vn) is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spe ..."
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Cited by 34 (6 self)
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Abstract. We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential u+Vn which the chain encounters when it visits a special state 0 at time n. The disorder (Vn) is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spends a positive fraction of its time at state 0, when u exceeds a critical value. We assume that for the Markov chain in the absence of the potential, the probability of an excursion from 0 of length n has the form n −c ϕ(n) with c ≥ 1 and ϕ slowly varying. Comparing to the corresponding annealed system, in which the Vn are effectively replaced by a constant, it was shown in [1], [4], [11] that the quenched and annealed critical points differ at all temperatures for 3/2 < c < 2 and c> 2, but only at low temperatures for c < 3/2. For high temperatures and 3/2 < c < 2 we establish the exact order of the gap between critical points, as a function of temperature. For the borderline case c = 3/2 we show that the gap is positive provided ϕ(n) → 0 as n → ∞, and for c> 3/2 with arbitrary temperature we provide an alternate proof of the result in [4] that the gap is positive, and extend it to c = 2. 1.
Marginal relevance of disorder for pinning models
, 2009
"... The effect of disorder on pinning and wetting models has attracted much attention in theoretical physics (e.g. [16, 11]). In particular, it has been predicted on the basis of the Harris criterion that disorder is relevant (annealed and quenched model have different critical points and critical exp ..."
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Cited by 29 (10 self)
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The effect of disorder on pinning and wetting models has attracted much attention in theoretical physics (e.g. [16, 11]). In particular, it has been predicted on the basis of the Harris criterion that disorder is relevant (annealed and quenched model have different critical points and critical exponents) if the return probability exponent α, a positive number that characterizes the model, is larger than 1/2. Weak disorder has been predicted to be irrelevant (i.e. coinciding critical points and exponents) if α < 1/2. Recent mathematical work (in particular [2, 10, 21, 22]) has put these predictions on firm grounds. In renormalization group terms, the case α = 1/2 is a marginal case and there is no agreement in the literature as to whether one should expect disorder relevance [11] or irrelevance [16] at marginality. The question is particularly intriguing also because the case α = 1/2 includes the classical models of twodimensional wetting of a rough substrate, of pinning of directed polymers on a defect line in dimension (3 + 1) or (1 + 1) and of pinning of an heteropolymer by a point potential in threedimensional space. Here we prove disorder relevance both for the general α = 1/2 pinning model and for the hierarchical version of the model proposed in [11], in the sense that we prove a shift of the quenched critical point with respect to the annealed one. In both cases we work with Gaussian disorder and we show that the shift is at least of order exp(−1/β 4) for β small, if β 2 is the disorder variance.
Pinning and wetting transition for (1+1)dimensional fields with Laplacian interaction
 Ann. Probab
"... Abstract. We consider a random field ϕ: {1,..., N} → R as a model for a linear chain attracted to the defect line ϕ = 0, i.e. the x–axis. The free law of the field is specified by the density exp ` − P i V (∆ϕi) ´ with respect to the Lebesgue measure on R N, where ∆ is the discrete Laplacian and w ..."
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Cited by 25 (4 self)
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Abstract. We consider a random field ϕ: {1,..., N} → R as a model for a linear chain attracted to the defect line ϕ = 0, i.e. the x–axis. The free law of the field is specified by the density exp ` − P i V (∆ϕi) ´ with respect to the Lebesgue measure on R N, where ∆ is the discrete Laplacian and we allow for a very large class of potentials V (·). The interaction with the defect line is introduced by giving the field a reward ε ≥ 0 each time it touches the x–axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative. We show that both models undergo a phase transition as the intensity ε of the pinning reward varies: both in the pinning (a = p) and in the wetting (a = w) case, there exists a critical value ε a c such that when ε> ε a c the field touches the defect line a positive fraction of times (localization), while this does not happen for ε < ε a c (delocalization). The two critical values are nontrivial and distinct: 0 < ε p c < ε w c < ∞, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at ε = ε p c is delocalized. On the other hand, the transition in the wetting model is of first order and for ε = ε w c the field is localized. The core of our approach is a Markov renewal theory description of the field. 1. Introduction and
EQUALITY OF CRITICAL POINTS FOR POLYMER DEPINNING TRANSITIONS WITH LOOP EXPONENT ONE
, 811
"... Abstract. We consider a polymer with configuration modeled by the trajectory of a Markov chain, interacting with a potential of form u + Vn when it visits a particular state 0 at time n, with {Vn} representing i.i.d. quenched disorder. There is a critical value of u above which the polymer is pinned ..."
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Cited by 18 (2 self)
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Abstract. We consider a polymer with configuration modeled by the trajectory of a Markov chain, interacting with a potential of form u + Vn when it visits a particular state 0 at time n, with {Vn} representing i.i.d. quenched disorder. There is a critical value of u above which the polymer is pinned by the potential. A particular case not covered in a number of previous studies is that of loop exponent one, in which the probability of an excursion of length n takes the form ϕ(n)/n for some slowly varying ϕ; this includes simple random walk in two dimensions. We show that in this case, at all temperatures, the critical values of u in the quenched and annealed models are equal, in contrast to all other loop exponents, for which these critical values are known to differ at least at low temperatures. 1.
Coarse graining, fractional moments and the critical slope of random copolymers
 Electron. J. Probab
"... Abstract. For a muchstudied model of random copolymer at a selective interface we prove that the slope of the critical curve in the weakdisorder limit is strictly smaller than 1, which is the value given by the annealed inequality. The proof is based on a coarsegraining procedure, combined with u ..."
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Cited by 15 (3 self)
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Abstract. For a muchstudied model of random copolymer at a selective interface we prove that the slope of the critical curve in the weakdisorder limit is strictly smaller than 1, which is the value given by the annealed inequality. The proof is based on a coarsegraining procedure, combined with upper bounds on the fractional moments of the partition function.
Hierarchical pinning models, quadratic maps and quenched disorder
, 2007
"... We consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by B. Derrida, V. Hakim and J. Vannimenius [3], which can be reinterpreted as an infinite dimensional dynamical system with random initial condition (the disorder). It is defined through a recurrence r ..."
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Cited by 14 (12 self)
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We consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by B. Derrida, V. Hakim and J. Vannimenius [3], which can be reinterpreted as an infinite dimensional dynamical system with random initial condition (the disorder). It is defined through a recurrence relation for the law of a random variable {Rn}n=1,2,..., which in absence of disorder (i.e., when the initial condition is degenerate) reduces to a particular case of the wellknown Logistic Map. The largen limit of the sequence of random variables 2 −n log Rn, a nonrandom quantity which is naturally interpreted as a free energy, plays a central role in our analysis. The model depends on a parameter α> 0, related to the geometry of the hierarchical lattice, and has a phase transition in the sense that the free energy is positive if the expectation of R0 is larger than a certain threshold value, and it is zero otherwise. It was conjectured in [3] that disorder is relevant (respectively, irrelevant or marginally relevant) if 1/2 < α < 1 (respectively, α < 1/2 or α = 1/2), in the sense that an arbitrarily small amount of randomness in the initial condition modifies the critical point with respect to that of the pure (i.e., nondisordered) model if α ≥ 1/2, but not if α < 1/2. Our main result is a proof of these conjectures for the case α ̸ = 1/2. We emphasize that for α> 1/2 we find the correct scaling form (for weak disorder) of the critical point shift.
Quenched large deviation principle for words in a letter sequence
, 2009
"... When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words ..."
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Cited by 10 (4 self)
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When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. In a companion paper the annealed and the quenched LDP are applied to the collision local time of transient random walks, and the existence of an intermediate phase for a class of interacting stochastic systems is established.
HIERARCHICAL PINNING MODEL WITH SITE DISORDER: DISORDER IS MARGINALLY RELEVANT
, 2008
"... We study a hierarchical disordered pinning model with site disorder for which, like in the bond disordered case [5, 8], there exists a value of a parameter b (enters in the definition of the hierarchical lattice) that separates an irrelevant disorder regime and a relevant disorder regime. We show ..."
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Cited by 9 (6 self)
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We study a hierarchical disordered pinning model with site disorder for which, like in the bond disordered case [5, 8], there exists a value of a parameter b (enters in the definition of the hierarchical lattice) that separates an irrelevant disorder regime and a relevant disorder regime. We show that for such a value of b the critical point of the disordered system is different from the critical point of the annealed version of the model. The proof goes beyond the technique used in [8] and it takes explicitly advantage of the inhomogeneous character of the Green function of the model.
Copolymers at selective interfaces: new bounds on the phase diagram
 J. Statist. Phys
"... Abstract. We investigate the phase diagram of disordered copolymers at the interface between two selective solvents, and in particular its weakcoupling behavior, encoded in the slope mc of the critical line at the origin. We focus on the directed walk case, which has turned out to be, in spite of t ..."
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Cited by 9 (3 self)
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Abstract. We investigate the phase diagram of disordered copolymers at the interface between two selective solvents, and in particular its weakcoupling behavior, encoded in the slope mc of the critical line at the origin. We focus on the directed walk case, which has turned out to be, in spite of the apparent simplicity, extremely challenging. In mathematical terms, the partition function of such a model does not depend on all the details of the Markov chain that models the polymer, but only on the time elapsed between successive returns to zero and on whether the walk is in the upper or lower half plane between such returns. This observation leads to a natural generalization of the model, in terms of arbitrary laws of return times: the most interesting case being the one of return times with power law tails (with exponent 1 + α, α = 1/2 in the case of the symmetric random walk). The main results we present here are: (1) the improvement of the known result 1/(1 + α) ≤ mc ≤ 1, as soon as α> 1 for what concerns the upper bound, and down to α ≈ 0.65 for the lower bound. (2) a proof of the fact that the critical curve lies strictly below the critical curve of the annealed model for every nonzero value of the coupling parameter. We also provide an argument that rigorously shows the strong dependence of the phase diagram on the details of the return probability (and not only on the tail behavior). Lower bounds are obtained by exhibiting a new localization strategy, while upper bounds are based on estimates of noninteger moments of the partition function.