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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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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.
Disorder relevance at marginality and critical point shift
 Ann. Inst. Henri Poincar e Probab. Stat
"... Abstract. Recently the renormalization group predictions on the effect of disorder on pinning models have been put on mathematical grounds. The picture is particularly complete if the disorder is relevant or irrelevant in the Harris criterion sense: the question addressed is whether quenched disord ..."
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Abstract. Recently the renormalization group predictions on the effect of disorder on pinning models have been put on mathematical grounds. The picture is particularly complete if the disorder is relevant or irrelevant in the Harris criterion sense: the question addressed is whether quenched disorder leads to a critical behavior which is different from the one observed in the pure, i.e. annealed, system. The Harris criterion prediction is based on the sign of the specific heat exponent of the pure system, but it yields no prediction in the case of vanishing exponent. This case is called marginal, and the physical literature is divided on what one should observe for marginal disorder, notably there is no agreement on whether a small amount of disorder leads or not to a difference between the critical point of the quenched system and the one for the pure system. In [14] we have proven that the two critical points differ at marginality of at least exp(−c/β4), where c> 0 and β2 is the disorder variance, for β ∈ (0, 1) and Gaussian IID disorder. The purpose of this paper is to improve such a result: we establish in particular that the exp(−c/β4) lower bound on the shift can be replaced by exp(−c(b)/βb), c(b)> 0 for b> 2 (b = 2 is the known upper bound and it is the result claimed in [8]), and we deal with very general distribution of the IID disorder variables. The proof relies on coarse graining estimates and on a fractional moment–change of measure argument based on multibody potential modifications of the law of the disorder.
NEW BOUNDS FOR THE FREE ENERGY OF DIRECTED POLYMER IN DIMENSION 1 + 1 AND 1 + 2
, 2009
"... We study the free energy of the directed polymer in random environment in dimension 1 + 1 and 1 + 2. For dimension 1, we improve the statement of Comets and Vargas in [7] concerning very strong disorder by giving sharp estimates on the free energy at high temperature. In dimension 2, we prove that ..."
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We study the free energy of the directed polymer in random environment in dimension 1 + 1 and 1 + 2. For dimension 1, we improve the statement of Comets and Vargas in [7] concerning very strong disorder by giving sharp estimates on the free energy at high temperature. In dimension 2, we prove that very strong disorder holds at all temperatures, thus solving a long standing conjecture.
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.
The martingale approach to disorder irrelevance for pinning models, Electron
 Comm. Probab
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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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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.
Annealed vs quenched critical points for a random walk pinning
"... Abstract We study a random walk pinning model, where conditioned on a simple random walk Y on Z d acting as a random medium, the path measure of a second independent simple random walk X up to time t is Gibbs transformed with Hamiltonian −L t (X, Y ), where L t (X, Y ) is the collision local time b ..."
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Abstract We study a random walk pinning model, where conditioned on a simple random walk Y on Z d acting as a random medium, the path measure of a second independent simple random walk X up to time t is Gibbs transformed with Hamiltonian −L t (X, Y ), where L t (X, Y ) is the collision local time between X and Y up to time t. This model arises naturally in various contexts, including the study of the parabolic Anderson model with moving catalysts, the parabolic Anderson model with Brownian noise, and the directed polymer model. It falls in the same framework as the pinning and copolymer models, and exhibits a localizationdelocalization transition as the inverse temperature β varies. We show that in dimensions d = 1, 2, the annealed and quenched critical values of β are both 0, while in dimensions d ≥ 4, the quenched critical value of β is strictly larger than the annealed critical value (which is positive).
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.
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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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.