Abstract
The crossover between the random walk and the self-avoiding walk limits of a large ring polymer is described using a direct renormalisation theory. The methods of field theory renormalisation are applied directly to the ring polymer system to calculate the mean square radius of gyration (RG2)ring through first order in epsilon =4-(dimension of space). The expansion factor alpha 2 for (RG2)ring is not a universal function of the natural scaling variable x that characterises the strength of the self-repelling (excluded volume) interaction. The crossover exhibited by the radius of gyration ratio R(x)=(RG2)ring/(RG2)line is described by a universal scaling function of x and has a range space (1/2<or=R(x)<or=1/2(1+13/96 epsilon )) that is bounded below and above by finite universal numbers representing the random and self-avoiding limits, respectively.
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