Self-avoiding surfaces

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Abstract
The set of self-avoiding random surfaces Sn(h) with n plaquettes and h boundary components is considered. The concatenation of the surfaces in Sn(h) and new constructions which either increase or decrease the number of boundary components of a surface are studied. These constructions make it possible to prove the existence of growth constants, beta h, for the cardinality Sn(h) of Sn(h) for each h>or=1 in two dimensions and h>or=0 in d>or=3 dimensions. The authors prove that beta h= beta 1 for all h>or=1 in d>or=2 dimensions. In addition, they prove that beta 0< beta 1 in d>or=3 dimensions and that in two dimensions beta 1< beta , where beta is the growth constant of the set Sn, the set of all self-avoiding surfaces in two dimensions. Finally, by postulating the existence of a critical exponent phi h for each set Sn(h), by assuming that sn(h) approximately n- phi h beta nh, they derive bounds on phi h from the constructions defined on the surface in Sn(h).