Fabio Lucio Toninelli

Fractional Moment Bounds and Disorder Relevance for Pinning Models

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)xa0=xa0n−α-1L(n), with α ≥ 0 and 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 αxa0<xa01/2 disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents [3,28]. The same has been proven also for αxa0=xa01/2, but under the assumption that L(·) diverges sufficiently fast at infinity, a hypothesis that is not satisfied in the (1 + 1)-dimensional wetting model considered in [12,17], where L(·) is asymptotically constant. Here we prove that, if 1/2xa0< αxa0<xa01 or α >xa01, 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 so-called Harris criterion, disorder is therefore relevant in this case. In the marginal case αxa0=xa01/2, under the assumption that L(·) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed critical points, which is smaller than any power of the disorder strength, is positive: disorder is marginally relevant. Again, the case considered in [12,17] is out of our analysis and remains open.The results are achieved by setting the parameters of the model so that the annealed system is localized, but close to criticality, and by first considering a quenched system of size that does not exceed the correlation length of the annealed model. In such a regime we can show that the expectation of the partition function raised to a suitably chosen power

The High Temperature Region of the Viana-Bray Diluted Spin Glass Model*

{gamma in (0, 1)}

Disordered pinning models and copolymers: Beyond annealed bounds.

is small. We then exploit such an information to prove that the expectation of the same fractional power of the partition function goes to zero with the size of the system, a fact that immediately entails that the quenched system is delocalized.

Replica bounds for diluted non-Poissonian spin systems

We consider general disordered models of pinning of directed polymers on a defect line. This class contains in particular the (1+1)-dimensional interface wetting model, the disordered Poland–Scheraga model of DNA denaturation and other (1+d)-dimensional polymers in interaction with flat interfaces. We consider also the case of copolymers with adsorption at a selective interface. Under quite general conditions, these models are known to have a (de)localization transition at some critical line in the phase diagram. In this work we prove in particular that, as soon as disorder is present, the transition is at least of second order, in the sense that the free energy is differentiable at the critical line, so that the order parameter vanishes continuously at the transition. On the other hand, it is known that the corresponding non-disordered models can have a first order (de)localization transition, with a discontinuous first derivative. Our result shows therefore that the presence of the disorder has really a smoothing effect on the transition. The relation with the predictions based on the Harris criterion is discussed.

Disorder relevance at marginality and critical point shift

In this paper, we study the high temperature or low connectivity phase of the Viana–Bray model in the absence of magnetic field. This is a diluted version of the well known Sherrington–Kirkpatrick mean field spin glass. In the whole replica symmetric region, we obtain a complete control of the system, proving annealing for the infinite volume free energy and a central limit theorem for the suitably rescaled fluctuations of the multi-overlaps. Moreover, we show that free energy fluctuations, on the scale 1/N, converge in the infinite volume limit to a non-Gaussian random variable, whose variance diverges at the boundary of the replica-symmetric region. The connection with the fully connected Sherrington– Kirkpatrick model is discussed.

Estimates on path delocalization for copolymers at selective interfaces

The effect of disorder on pinning and wetting models has attracted much attention in theoretical physics. 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 alpha, 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 alpha < 1/2. Recent mathematical work has put these predictions on firm grounds. In renormalization group terms, the case alpha = 1/2 is a marginal case and there is no agreement in the literature as to whether one should expect disorder relevance or irrelevance at marginality. The question is particularly intriguing also because the case alpha = 1/2 includes the classical models of two-dimensional 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 three-dimensional space. Here we prove disorder relevance both for the general alpha = 1/2 pinning model and for the hierarchical version of the model proposed by B. Derrida, V. Hakim and J. Vannimenus (JSP, 1992), 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/beta^4) for beta small, if beta is the standard deviation of the disorder.

On the irrelevant disorder regime of pinning models.

We consider a general model of a disordered copolymer with ad-sorption. This includes, as particular cases, a generalization of thecopolymer at a selective interface introduced by Garel et al. [Euro-phys. Lett. 8 (1989) 9–13], pinning and wetting models in variousdimensions, and the Poland–Scheraga model of DNA denaturation.We prove a new variational upper bound for the free energy via anestimation of noninteger moments of the partition function. As anapplication, we show that for strong disorder the quenched criticalpoint differs from the annealed one, for example, if the disorder dis-tribution is Gaussian. In particular, for pinning models with loopexponent 0 < α< 1/2 this implies the existence of a transition fromweak to strong disorder. For the copolymer model, under a (restric-tive) condition on the law of the underlying renewal, we show thatthe critical point coincides with the one predicted via renormaliza-tion group arguments in the theoretical physics literature. A strongerresult holds for a “reduced wetting model” introduced by Bodineauand Giacomin [J. Statist. Phys. 117 (2004) 801–818]: without restric-tions on the law of the underlying renewal, the critical point coincideswith the corresponding renormalization group prediction.

Smoothing of depinning transitions for directed polymers with quenched disorder.

In this paper we extend replica bounds and free energy subadditivity arguments to diluted spin-glass models on graphs with arbitrary, non-Poissonian degree distribution. The new difficulties specific of this case are overcome introducing an interpolation procedure that stresses the relation between interpolation methods and the cavity method. As a byproduct we obtain self-averaging identities that generalize the Ghirlanda–Guerra ones to the multi-overlap case.