Pinning model disordered

Very general return times are treated in [Alexander and Sidoravicius (2006)], that deals with disordered pinning models the homogeneous case is treated in detail, but only at the level of the contact fraction (namely, at the level of Large Deviations estimates). 

Fig. 4.1 A (possible) plot of the free energy for a disordered pinning model (thick line) and for three approximating periodic models, at f3 fixed. Even if lim7 ->oo =...

This chapter will be devoted to the analysis of the phase diagram of the disordered pinning model ... 

The smoothing effect of disorder in pinning models is a controversial issue in the physical literature, at least for a > 1. These are precisely the return probabilities considered to be of relevance for the DNA denat-uration modeling (see Section 1.4 and relative bibliographic complements). Relevant papers that attack the problem of regularity in disordered Poland-Scheraga models are in particular [Cule and Hwa (1997)], [Tang and Chate (2001)], [Blossey and Carlon (2003)], [Schafer (2005)], [Coluzzi (2005)],... 

This section is devoted to applying the homogeneous localization strategy, already exploited in Section 5.2 in the context of disordered pinning models. But let us first prove the soft part of Theorem 6.1, namely ... 

It is natural to ask what the optimal value for C2, in Theorem 7.7, is. Lemma 7.10 is rough (no effort is made to track the constants, but the method itself does not appear to be very sharp). Based on Theorem 7.5(2), one is possibly tempted to say that C2 can be chosen arbitrarily close to /x(/ , h), but this is not completely evident. In the restricted set-up of the disordered pinning model based on simple random walk with hard wall condition, with law denoted by however a precise result can be proven [Toninelli (2006)]. Precisely the result is that for every (/ , k) G the limit... 

We focus on the disordered pinning model. The changes that are needed to extend the results to the copolymer case are summed up in Section 8.4. 

Fig. 9.2 The disordered pinning model, based on simple random walk, with binary charges. In the two cases / = 0.1, h instead takes value 0.0045 and 0.0048. In both cases to suppress fluctuations we plot only a point every 10 . A slope is visible in both cases and it suggests a free energy of about 10 in the first case and of about 1.5 -10 in the second (note that the slope has to be divided by 2 since the real length of the system is 2N). Note also that in the second case the partition function is still well below 1 for a system of length 200 millions. For h = 0.0049 (data not plotted) the slope is less clear. However, since logcosh(O.l) = 0.004991..., these graphs suggest that the quenched critical point is very close to the annealed one (see Section 5.5). With reference to (9.3), we have chosen A = B = 6 and No = 1000.

If I try to characterize such a subset, keeping in mind the motivations (that come from physics, biology, chemistry, material science, etc.), I end up with a list (1 - - d)-dimensional pinning models, (1-1-1 )-dimensional wetting models, adsorption models, copolymer models, DNA denaturation models, etc. and to each model in this list one should probably add the adjective disordered, since this work is mostly focused on disordered systems, even if non-disordered systems do play a central role. But such a list may at first appear quite disordered in itself... 

We have seen in this section how quantum fluctuations can reconcile the predictions of 7r-electron models to the experimental observations on thin Aims. However, remembering that disorder is also an effective mechanism to pin excited states, it is possible that the parametrization of the 7r-electron models (derived with short oligomers) is simply not valid for long polymers. 

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