A Statistical Test for Localization

As we have seen ElogZ AT is a super-additive sequence, cf. Theorem 4.2, and, by Proposition A.12, one has the following characterization 

What interests us in this section is that if we can find one value of N for which ElogZ w 0 then the system is localized. 

Let us give a simple application of such an idea for the copolymer model, based on the simple random walk, where one easily computes 

Elog Zfj for N = 10 or thereabout (one cannot go much beyond anyway). We will come back to this very interesting issue, but we anticipate that one needs rather N of the order of 10 (which is a remarkably small number, but 2i°°° is not). 

What we propose is to decide whether E log is positive by Monte Carlo sampling log Zpj Of course, by the law of large numbers, the empirical average of n independent copies of log that we denote by w (fV), is an asymptotic estimator of E log Z j,j but then one needs to decide how large n should be to decide that Un N) 0 really implies ElogZ 0. In order to tackle this point we resort to concentration inequalities. 

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