Asymptotic properties of correlations between chain ends Fishers result

3 Asymptotic properties of correlations between chain ends Fisher s result 

Let us consider a d-dimensional hypercubic lattice and, on this lattice, the selfavoiding chains of origin O. Let PN (r) be the probability distribution of the end-to-end vector, r. This vector r has coordinates qv.qd which are integers. The parity of r is by definition the parity of qj. 

we see immediately that PN (r) differs from zero only if r and N have the same parity. In this case, and for large values of N, it is possible to postulate a scaling law of the form 

The asymptotic properties of fn (x) are determined by Fisher s theorem. Note that the values x S 1 correspond, for the chain, to stretched configurations. 

The function fD (x) which, for a self-avoiding chain, describes the asymptotic behaviour of PN(r) when N - oo, decreases when x-+ oo as exp [ — Da X / I V ] ( )d = constant, v = critical exponent) 

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