A Fully Inhomogeneous Non-Disordered Model

Here instead we give a general result on the following case consider 

Theorem 5.9 Choose K -) as in Definition 1.4 and assume that T,k = 1 and mx = oo, so that the free process is null recurrent. Moreover assume that a 0. Let to be a numeric sequence taking values in 0,1, A 0 and a = c or i. The following three statements are equivalent  

We therefore have a criterion for localization for general pinning models with arbitrary (deterministic ) dilution. For the case a = 0 see Section 5.7. Note moreover that if the underlying renewal is positive recurrent, that is mx oo, then the model (A/jv( r)] E q [Wjv(v)] and the latter 

Lemma 5.10 Under the hypotheses of Theorem 5.9, if there exists (5 0 and No such that 

But since Em SN, then n/N 6/ 2M) for N sufficiently large and the term between parentheses can be made larger than 1 by choosing M sufficiently large (recall that C ) This completes the proof.  

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