math.PR — Probability

Aging in Spherical Spin Glass Dynamics

Contributed by P. M. Aronow, Patrick Lopatto

We study Langevin dynamics for the pure spherical $p$-spin glass with $p\ge3$, with covariance $Nq^p$ and generator $\Delta_S+\beta\nabla_SH_N\cdot\nabla_S$, started from a uniform point independent of the disorder. We let the dimension tend to infinity on fixed time horizons and study the long-time behavior of the resulting hard-spherical Cugliandolo–Kurchan equations for the correlation $C$ and the response $R$. At every positive temperature, their solution has a unique stationary limit at bounded time differences, which satisfies the fluctuation–dissipation theorem (FDT), and its energy converges. For $\beta\ge\beta_{\mathrm d}$, where $\beta_{\mathrm d}$ is the dynamical transition, we show that the plateau $q$ of the stationary correlation is the larger solution of the marginality relation $\beta^2p(p-1)q^{p-2}(1-q)^2=1$, and we compute the limiting energy. Correlations with every fixed earlier state tend to zero. The limits of large waiting time and large time difference therefore do not commute, which proves aging of the correlation, including at the transition point. We also prove that the response at a late time, as a function of the time difference, converges to its stationary profile in $L^2$ and uniformly over all earlier times; that FDT in the later time holds uniformly over all earlier times; and that the integral of the squared violation of FDT in the earlier time tends to zero, while for $\beta>\beta_{\mathrm d}$ the positive part $(\partial_sC-R)_+$ of this violation keeps a positive integrated mass. For $\beta\ge\beta_{\mathrm d}$, the stationary correlation approaches its plateau as a power law, with the exponent predicted by mode-coupling theory and a finite positive amplitude. For the cubic model and $\beta\ge\beta_{\mathrm d}$, including the transition point, we prove a lower bound on the total susceptibility and a product inequality for correlations at three times; for $\beta>\beta_{\mathrm d}$, we prove, for each fixed level below the plateau, an explicit asymptotic lower bound on the first passage time to that level multiplied by the entropy production during the passage. The aging proofs for the cubic and quartic models represent the correlation by vectors in a Hilbert space indexed by trees, and general pure interactions are treated with averages over finitely many independent futures and established algorithmic threshold bounds. The full fluctuation–dissipation relation of the aging regime and the time scale of aging remain open.