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Correspondence between phase oscillator network and classical XY model with the same infinite-range interaction in statics

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.