  Weakly coupled oscillators are used throughout the physical sciences,
particularly in mathematical neuroscience to describe the interaction of
neurons in the brain. Systems of weakly coupled oscillators have a well-known
decomposition to a canonical phase model which forms the basis of our
investigation in this work. Particularly, our interest lies in examining the
stability of synchronous (phase-locked) solutions to this phase system:
solutions with phases having the same temporal frequency but differ through
time-independent phase-lags. The main stability result of this work comes from
adapting a series of investigations into random walks on infinite weighted
graphs. We provide an interesting link between the seemingly unrelated areas of
coupled oscillators and random walks to obtain algebraic decay rates of small
perturbations off the phase-locked solutions under some minor technical
assumptions. We also provide some interesting and motivating examples that
demonstrate the stability of phase-locked solutions, particularly that of a
rotating wave solution arising in a well studied paradigm in the theory of
coupled oscillators.
