Multicritical swelling exponents for

When the space dimension d decreases from three to two, more and more interactions become relevant. Actually, for a chain of Brownian area S, let us consider a p-body interaction energy of the form 

This energy being a pure number, we immediately deduce the dimensional 

Therefore, the dimensionless parameter corresponding to this interaction is bpSp (p lkl/2 up to a numerical factor. The interaction is relevant if this parameter goes to infinity when S becomes infinite, i.e. if 

for d 3, the three-body interaction is relevant for d 8/3, the four-body interaction is also relevant, and so on. For d = 2, there is an infinite number of relevant interactions. 

Let us now consider a set of chains with two-body, three-body,. m-body interactions. We have a p-critical system (p m), if (p — 2) virial coefficients vanish 

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