Path Properties The Scaling Limit

Scaling limits deal with the large N behavior of the whole polymer. This is the reason for giving here the following result that deals with the global quantity contact density, that is n(/ ) = f (/ ), cf. (1.21), which is in general defined only for / /3c- 

For what concerns (2.33), it follows directly (B.14) (and the lines that follow it for the case a = 1) and from the asymptotic behavior of f(-) close to pc (Theorem 2.1).  

Turning to scaling limits the results are in term of convergence in law of sequences of random sets, namely t/N) n [0, 1] at, insisting on the fact that (T/A )n[0,1] = T(7v) is a (random) closed subset of [0,1]. The topology on the space of closed subsets is the Hausdorff (or Matheron) one, we refer to Appendix A.5.4 for precise definitions. The first result we present is the following 

These are clear statements respectively of localization and of delocalization and they are proven in full generality. On the other hand the limit processes are trivial (and, due to that, the convergence is also in probability and not only in law). 

Let us estimate the right-hand side by observing that for 0 A m iV we have 

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