Langevin Dynamics of Non-reciprocally Coupled Spherical Spin Glasses

We analyze the limiting dynamics of two copies of a disordered mean-field spherical $p$-spin system coupled non-reciprocally to one another and initialized independently of the disorder. We prove that, on every finite time interval, as the system size tends to infinity, the empirical correlation and integrated response functions converge almost surely and in $L^q$ for $q \geq 1$, uniformly in time, to the unique non-random solution of a system of coupled integro-differential equations. For even $p$-spin models, we establish stationarity and exponential decay of the correlation and response functions in the high-temperature regime. In the special case of the spherical Sherrington–Kirkpatrick model, we derive explicit representations of the limiting correlation and integrated response functions. These representations allow us to characterize the long-time dynamics throughout the full range of temperatures. In particular, we prove the occurrence of aging in the low-temperature regime and fluctuation-dissipation behavior in the high-temperature regime, and provide a separate characterization of the dynamics at the critical temperature.