The Fixman-Freire Algorithm

We present now an algorithm that allows to compute, in an approximate fashion, the partition function in not much more than 0(N) computations. We give it for conciseness in the case of pinning. We make the choice K(n) = Ck, Ck is a positive constant, for every n N (the algorithm 

This result follows directly from [Beylkin and Monzon (2005), Theorem A.l], where an explicit procedure for determining the approximation coefficients is given. But beyond the quantitative aspects of Proposition 9.1, let us replace K n) with Kpiin) = X)I=i exp(—6jn), and denote by the partition function of the model with this new (AT-dependent) return distribution, and let us go back to (9.5) to observe that we can write 

The problem of giving quantitative bounds on N,u - Zn u, Starting for example from a statement like Proposition 9.1, will not be considered here. All the same we remark that if K]g(n) K(n) for n = then Zj Zig u and such an observation makes clear the interest of the Fixman-Preire scheme in view of establishing localization. 

The recursion scheme for copolymer models (with or without adsorption) is conceptually analogous, even if it is technically slightly more involved (see [Garel and Monthus (2005c)]). 

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