Results 1  10
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14
DISORDERED PINNING MODELS AND COPOLYMERS: BEYOND Annealed Bounds
, 2008
"... We consider a general model of a disordered copolymer with adsorption. This includes, as particular cases, a generalization of the copolymer at a selective interface introduced by T. Garel et al. [16], pinning and wetting models in various dimensions, and the PolandScheraga model of DNA denaturati ..."
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Cited by 32 (4 self)
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We consider a general model of a disordered copolymer with adsorption. This includes, as particular cases, a generalization of the copolymer at a selective interface introduced by T. Garel et al. [16], pinning and wetting models in various dimensions, and the PolandScheraga model of DNA denaturation. We prove a new variational upper bound for the free energy via an estimation of noninteger moments of the partition function. As an application, we show that for strong disorder the quenched critical point differs from the annealed one, e.g., if the disorder distribution is Gaussian. In particular, for pinning models with loop exponent 0 < Î± < 1/2 this implies the existence of a transition from weak to strong disorder. For the copolymer model, under a (restrictive) condition on the law of the underlying renewal, we show that the critical point coincides with the one predicted via renormalization group arguments in the theoretical physics literature. A stronger result holds for a âreduced wetting model â introduced by T. Bodineau and G. Giacomin [5]: without restrictions on the law of the underlying renewal, the critical point coincides with the corresponding renormalization group prediction.
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 24 (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
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.
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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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.
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.
Comments on the influence of disorder for pinning model in correlated Gaussian environment
"... Abstract. We study the random pinning model, in the case of a Gaussian environment presenting powerlaw decaying correlations, of exponent decay a> 0. A similar study was done in a hierachical version of the model Berger and Toninelli (2013), and we extend here the results to the nonhierarchica ..."
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Abstract. We study the random pinning model, in the case of a Gaussian environment presenting powerlaw decaying correlations, of exponent decay a> 0. A similar study was done in a hierachical version of the model Berger and Toninelli (2013), and we extend here the results to the nonhierarchical (and more natural) case. We comment on the annealed (i.e. averaged over disorder) model, which is far from being trivial, and we discuss the influence of disorder on the critical properties of the system. We show that the annealed critical exponent νa is the same as the homogeneous one νpur, provided that correlations are decaying fast enough (a> 2). If correlations are summable (a> 1), we also show that the disordered phase transition is at least of order 2, showing disorder relevance if νpur < 2. If correlations are not summable (a < 1), we show that the phase transition disappears. 1.