  We study the phase oscillator networks with distributed natural frequencies
and classical XY models both of which have a class of infinite-range
interactions in common. We find that the integral kernel of the self-consistent
equations (SCEs) for oscillator networks correspond to that of the saddle point
equations (SPEs) for XY models, and that the quenched randomness (distributed
natural frequencies) corresponds to thermal noise. We find a sufficient
condition that the probability density of natural frequency distributions is
one-humped in order that the kernel in the oscillator network is strictly
decreasing as that in the XY model. Furthermore, taking the uniform and
Mexican-hat type interactions, we prove the one to one correspondence between
the solutions of the SCEs and SPEs. As an application of the correspondence, we
study the associative memory type interaction. In the XY model with this
interaction, there exists a peculiar one-parameter family of solutions. For the
oscillator network, we find a non-trivial solution, i.e., a limit cycle
oscillation.
