  We study persistent currents in a Josephson junction array wrapped around a
cylinder. The $T=0$ quantum statistical mechanics of the array is equivalent to
the statistical mechanics of a classical $xy$ spin system in 2+1 dimensions at
the effective temperature $T^{*}=\sqrt{2JU}$, with $J$ being the Josephson
energy of the junction and $U$ being the charging energy of the superconducting
island. It is investigated analytically and numerically on lattices containing
over one million sites. For weak disorder and $T^{*}\ll J$ the dependence of
the persistent current on disorder and $T^{*}$ computed numerically agrees
quantitatively with the analytical result derived within the spin-wave
approximation. The high-$T^{*}$ and/or strong-disorder behavior is dominated by
instantons corresponding to the vortex loops in 2+1 dimensions. The current
becomes destroyed completely at the quantum phase transition into the
Cooper-pair insulating phase.
