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The equilibrium behavior of vortices in the classical two-dimensional (2D) XYmodel with uncorrelated random phase shifts is investigated. The modeldescribes Josephson-Junction arrays with positional disorder, and hasramifications in a number of other bond-disordered 2D systems. The vortexHamiltonian is that of a Coulomb gas in a background of quenched randomdipoles, which is capable of forming either a dielectric insulator or a plasma.We confirm a recent suggestion by Nattermann, Scheidl, Korshunov, and Li [J.Phys. I (France) {\bf 5}, 565 (1995)], and by Cha and Fertig [Phys. Rev. Lett.{\bf 74}, 4867 (1995)] that, when the variance $\sigma$ of random phase shiftsis smaller than a critical value $\sigma_c$, the system is in a phase withquasi-long-range order at low temperatures, without a reentrance transition.This conclusion is reached through a nearly exact calculation of thesingle-vortex free energy, and a Kosterlitz-type renormalization group analysisof screening and random polarization effects from vortex-antivortex pairs. Thecritical strength of disorder $\sigma_c$ is found not to be universal, butgenerally lies in the range $0<\sigma_c<\pi/8$. Argument is presented tosuggest that the system at $\sigma>\sigma_c$ does not possess long-range glassyorder at any finite temperature. In the ordered phase, vortex pairs undergo aseries of spatial and angular localization processes as the temperature islowered. This behavior, which is common to many glass-forming systems, can bequantified through approximate mappings to the random energy model and to thedirected polymer on the Cayley tree. Various critical properties at theorder-disorder transition are calculated.Comment: 18 pages, 6 postscript figures, uses RevTex macro

Vortex statistics in a disordered two-dimensionalXYmodel