Sharp Estimates on the Partition Function

As we have already stressed, the analogy with (2.18) is evident and it is probably not surprising for the reader that from such a formula one can extract the sharp behavior of the partition function. It should however be noted that, while in the positive recurrent set-up ( 1) the theory of Markov renewals is well developed, fewer results are available in the literature on the mass renewal function of null recurrent Markov renewals. Moreover the results, even only at the level of sharp asymptotic behavior of the partition function, are more involved. As we shall see, this complexity is not only of a technical nature, but it really reflects a substantially larger variety of phenomena that can be observed in weakly inhomogeneous models, with respect to homogeneous ones. 

We will not give a full proof of Theorem 3.4, but we will stress on the ideas that lead to results. And case by case we will analyze the dependence of the constants on 77 since it has a direct impact on the behavior of the polymer trajectories. 

Prom (3.23) and (3.20) we immediately obtain the statement in Theorem 3.4(1) with 

Of course if we are considering a polymer chain between n and N instead of 0 and N the asymptotic behavior of changes in the obvious 

In this case there is no exponential growth and one is left with estimating the mass renewal function the notation is the same as for 1, just keep in mind that now b = 0, so A° (n) = Ma, (n), Ta,f n) = Ma p n) fi/ and P( ) is the Markov transition matrix of J, with invariant measure V, Va = Ca a- We Will uot give the details of the proof, that can be found in [Caravenna et al. (2005)], but we stress that the proof is a matter of deahng with a return distribution that is a random superposition of return laws with 0=1/2 and trivial L( ), so the N dependence in Theorem 3.4(2) does not come as a surprise once we consider the corresponding result (2.15) 

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We will start by studying the order of the transition, thus completing what we have started in Section 1.2.2. But the main focus of this chapter is on showing how the tools of renewal theory yield sharp estimates on the partition function of the model and how from these sharp estimates one can get very precise information on the path properties of the polymer. 

Sharp estimates on the partition function of very general random walk based models (a = 1/2) can be found in [Caravenna et al. (2006a)]. For (p,q)-walks we mention [fsozaki and Yoshida (2001)]. In this section we generalize the approach in [Caravenna et al. (2006a) ] to general a and we heavily rely on renewal mass function estimates, treated in detail Appendix A.5. Of particular importance are the sharp results in [Doney (1997)] that allow an... 

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