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 $\xi (x)={\sum }_{p\ge 4}{\gamma }_p^2{x}^p$, 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 $p\ge 4$, 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 $-N{E}_1(\xi )$, 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 $\frac{1}{2}\log (\beta e)-{\Xi }_{1,\xi }^{\mathrm{EK}}(-E_1(\xi ))-C_{\xi ,b}{\beta }^{-1/2}$ with $-{\Xi }_{1,\xi }^{\mathrm{EK}}(-E_1(\xi ))\ge \frac{1}{4}\log {\xi }^{\prime \prime }(1)+\frac{1}{4}>0$, so the refinement is a strict improvement for all sufficiently large fixed $\beta$; the onset temperature and constants $b_{\ast }(\xi )$ 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.

arXiv comment60 pages, 7 figures