Pinning model homogeneous

The random walk S defines the pinning model of Section 1.2.1, but the arguments really used only the distribution K -) of the return times to zero. In particular we could generalize the model to any homogeneous Markov chain S to avoid trivialities we should assume for example that P Sn = 0) > 0 for some n, but in general there is no reason to assume that 0 is recurrent for the chain. 

As it will be clear from the proof Theorem 1.7 is rather general (and it goes even well beyond the homogeneous set-up, c/. Section 1.10) one can for example state it for general pinning models, but one has to add the information on the entropic cost due to the last (incomplete) excursion, an information that is clearly not contained in the renewal process. 

While a quantitative analysis of the periodic pinning model of Section 1.6.1 is not immediate, on a qualitative level the mechanism of the transition is not new with respect to the corresponding homogeneous model. Now instead we are going to present a case in which the inhomogeneous character of the charge distribution is at the base of the localization mechanism. 

Notice in particular that there is no need to ask for k > 0, but of course one has to give up the interpretation of K -) as a sub-probability (it would rather be a combinatorial term, see in particular the way the Poland-Scheraga model has been introduced). Regardless of the value of K, we realize that, if c < —1, then exp( A )Z is the partition function of the homogeneous pinning model with polynomial decay of the return times (and critical point (3c)- Therefore there is a transition, between free energy equal to —k to larger than —k, at (3c- And it is... 

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). 

In this chapter we are going to treat in detail the general homogeneous pinning model introduced in Section 1.2.2 by giving its density with respect to the free process, the renewal r of law P, in equation (1.23) that we recall here ... 

In this chapter the aim is above all to stress the basic steps that allow to reduce the weakly inhomogeneous case to the homogeneous one and convey the idea that in this class of models one gets as far as in the homogeneous pinning model treated in Chapter 2. This reduction is not without a price, since in reality formulas become substantially more complex in the details. However it should be stressed from now that new phenomena appear in this set up and one of our purposes is to point out in an informal way the phenomenological richness of periodic models. 

We include in this section also another result that is in the spirit of Proposition 3.7. Consider the general model (3.1), but set = 0 for every n, that is there is no copolymer interaction (we denote by its free energy). The charges oj are instead periodic but not necessarily centered. The result we are going to state says that the periodic pinning model localizes more than the corresponding homogeneous model. 

Fig. 3.2 The free energy of the periodic pinning model compared to the one of the associated homogeneous model.

Before proving Theorem 5.6 let us discuss its relevance. For this it is useful to go back to Proposition 1.6 or to Theorem 2.1 these results, giving the behavior of the free energy of the homogeneous pinning model near the critical point, are summed up in Figure 1.6. If one rewrites (5.31) as... 

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 ... 

Another geometric difference between the reactor designs is the use of a 17 x 17 pin lattice in Sequoyah and North Anna as opposed to a 15 x 15 lattice in the other two core designs. The smaller diameter and pitch used in the 17 x 17 lattice result in a closer approximation to a homogeneous core than that represented by a 15 x 15 lattice, and self-shielding effects are reduced. Hence any bias in the numerical technique used in modeling self-shielding approaches could introduce discrepancies in the results. 

Quartercore symmetry was used flnoughout these calculations. For the square pitch lattices, the computer representation explicitly modeled each fuel rod, aluminum rod, intermediate band (Core IV), center grid plate (Core V), side sheet, moderator, base plate, and core tank in the X-Y direction. The fuel support structure above the water was not represented in the computer modeL The only difference in the modeling of the triangular pitch lattices (Cores I, II, and III) was that the fuel regions were homogenized except for the external mns in each fuel assembly. The outward half of each of these pins was explicitly modeled (see Table I). 

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