2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/107023We revisit the problem of a two-dimensional polymer ring subject to an inflating pressure differential. The ring is modeled as a freely jointed closed chain of N monomers. Using a Flory argument, mean-field calculation and Monte Carlo simulations, we show that at a critical pressure, $p_c \sim N^{-1}$, the ring undergoes a second-order phase transition from a crumpled, random-walk state, where its mean area scales as $<A> \sim N$, to a smooth state with $<A>\sim N^2$. The transition belongs to the mean-field universality class. At the critical point a new state of polymer statistics is found, in which $<A>\sim N^{3/2}$. For $p>>p_c$ we use a transfer-matrix calculation to derive exact expressions for the properties of the smooth state.9 pages, 8 figuresSoft Condensed MatterStatistical Mechanicstext