Spectral Gap Bounds for Langevin Dynamics in Mixed Spherical Spin Glasses

Abstract

We prove lower bounds on the relaxation time $1/\gamma_{N,\beta}$ of Langevin dynamics for mixed spherical spin glasses with even mixture $\xi(x)=\sum_{p\ge4}\gamma_p^2x^p$, under a one-step-replica-symmetry-breaking-type standing assumption of strict threshold separation $E_0(\xi)>E_1(\xi)>E_2(\xi)$; the pure spherical $p$-spin glass with even $p\ge4$, for which the assumption is a theorem, is recovered as a corollary. The main result is an aggregate Eyring-Kramers bound whose exponent sums conductances over the exponentially many index-one saddles above the lowest saddle level $-NE_1(\xi)$, combining the saddle complexity $\Theta_{1,\xi}$ with a half-determinant Hessian statistic -- an entropic contribution that the classical single-saddle picture misses. The single new random-matrix ingredient of the mixed model is that the conditional Hessian at a critical point is a randomly shifted GOE matrix: the radial derivative is no longer determined by the energy (Euler's identity degenerates exactly in the pure case), and every landscape rate becomes a one-dimensional supremum over the scalar shift with a Gaussian penalty. At low temperature the aggregate exponent exceeds the unconditional free-energy bound by $\frac12\log(\beta e)-\Xi^{\mathrm{EK}}_{1,\xi}(-E_1(\xi))-C_{\xi,b}\beta^{-1/2}$ with $-\Xi^{\mathrm{EK}}_{1,\xi}(-E_1(\xi))\ge\frac14\log\xi''(1)+\frac14>0$, so the refinement is a strict improvement for all sufficiently large fixed $\beta$; the onset temperature and constants $b_*(\xi)$ and $C_{\xi,b}$ produced by the proof are not numerically explicit. Two companion results -- a sequential Arrhenius bound with the explicit constant $E_0(\xi)-E_1(\xi)$ and a fixed-temperature free-energy bound -- come with complete, self-contained proofs. All bounds are one-sided; identifying the mechanism the dynamics actually realizes would require a matching upper bound, which remains open.