This paper analyzes the size and shape of a closed, two-dimensional random walk with a pressure difference p between the inside and outside, which couples to an algebraic (signed) area. This pressurized-random-walk (PRW) model is, in some respects, closely related to a computer model studied by Leibler, Singh, and Fisher [Phys. Rev. Lett. 59, 1989 (1987)]. Since all terms in the Hamiltonian are quadratic in the position-vector field r, the partition function and its derivatives can be evaluated exactly. The most notable feature of the PRW model is an instability, which occurs at ‖p‖=${\mathit{p}}_{\mathit{c}}$. For ‖p‖<${\mathit{p}}_{\mathit{c}}$, the system has a finite algebraic area and an anisotropic shape; for ‖p‖≥${\mathit{p}}_{\mathit{c}}$, the algebraic area diverges and the shape is circular. The asphericity is also calculated. A form of bending rigidity, also quadratic in r, is introduced into the model; however, the resulting macroscopic properties are quite different from those one might ordinarily expect. This difference can be traced to the absence of a fixed link size in the model.

• Received 26 September 1991

DOI:https://doi.org/10.1103/PhysRevA.45.3629