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The conformation and scaling properties of self-avoiding fluid vesicles with zero extrinsic bending rigidity subject to an internal pressure increment $Delta p>0$ are studied using Monte Carlo methods and scaling arguments. With increasing pressure, there is a first-order transition from a collapsed branched polymer phase to an extended inflated phase. The scaling behavior of the radius of gyration, the asphericities, and several other quantities characterizing the average shape of a vesicle are studied in detail. In the inflated phase, continuously variable fractal shapes are found to be controlled by the scaling variable $x=Delta p N^{3 u/2}$ (or equivalently, $y = {<V>}/ N^{3 u/2}$), where $N$ is the number of monomers in the vesicle and $V$ the enclosed volume. The scaling behavior in the inflated phase is described by a new exponent $ u=0.787pm 0.02$.
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