The gauge glass model offers an interesting example of a randomly frustrated system with a continuous O(2) symmetry. In two dimensions, the existence of a glass phase at low temperatures has long been disputed among numerical studies. To resolve this controversy, we examine the behavior of vortices whose movement generates phase slips that destroy phase rigidity at large distances. Detailed analytical and numerical studies of the corresponding Coulomb gas problem in a random potential establish that the ground state, with a finite density of vortices, is polarizable with a scale-dependent dielectric susceptibility. Screening by vortex/antivortex pairs of arbitrarily large size is present to eliminate the logarithmic divergence of the Coulomb energy of a single vortex. The observed power-law decay of the Coulomb interaction between vortices with distance in the ground state leads to a power-law divergence of the glass correlation length with temperature $T$. It is argued that free vortices possess a bound excitation energy and a nonzero diffusion constant at any $T>0$.
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