Spectral Gap Bounds for Langevin Dynamics in Mixed Spherical Spin Glasses
Abstract
We prove lower bounds on the relaxation time 1/γN,β of Langevin dynamics for mixed spherical spin glasses with even mixture ξ(x)=∑p≥4γp2xp, under a one-step-replica-symmetry-breaking-type standing assumption of strict threshold separation E0(ξ)>E1(ξ)>E2(ξ); the pure spherical p-spin glass with even p≥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 −NE1(ξ), combining the saddle complexity Θ1,ξ 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 1/2log(βe)−ΞEK1,ξ(−E1(ξ))−Cξ,bβ−1/2 with −ΞEK1,ξ(−E1(ξ))≥1/4logξ''(1)+1/4>0, so the refinement is a strict improvement for all sufficiently large fixed β; the onset temperature and constants b*(ξ) and Cξ,b produced by the proof are not numerically explicit. Two companion results -- a sequential Arrhenius bound with the explicit constant E0(ξ)−E1(ξ) 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.
Related
- Langevin Dynamics of Non-reciprocally Coupled Spherical Spin Glasses
- Rényi Tracking Bounds for Langevin Dynamics with Moving Targets
- Local Algebraic Stability of Wasserstein Gradient Descent-Ascent and Mean Field Langevin Dynamics without Spectral Gap
- An Autonomous Spectral Representation for Fully Asymmetric Spherical Sherrington-Kirkpatrick Dynamics
- Free energy of non-convex multi-species spherical spin glasses
BibTeX
@misc{khuzani2026spectral,
title = {{Spectral Gap Bounds for Langevin Dynamics in Mixed Spherical Spin Glasses}},
author = {Masoud Badiei Khuzani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.15157},
eprint = {2609.15157},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15157},
}
RIS
TY - UNPB
TI - Spectral Gap Bounds for Langevin Dynamics in Mixed Spherical Spin Glasses
AU - Masoud Badiei Khuzani
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.15157
UR - https://arxiv.org/abs/2609.15157
AB - We prove lower bounds on the relaxation time 1/γ_N,β of Langevin dynamics for mixed spherical spin glasses with even mixture ξ(x)=∑_p≥4γ_p^2x^p, under a one-step-replica-symmetry-breaking-type standing assumption of strict threshold separation E_0(ξ)>E_1(ξ)>E_2(ξ); the pure spherical p-spin glass with even p≥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 −NE_1(ξ), combining the saddle complexity Θ_1,ξ 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 1/2log(βe)−Ξ^EK_1,ξ(−E_1(ξ))−C_ξ,bβ^−1/2 with −Ξ^EK_1,ξ(−E_1(ξ))≥1/4logξ''(1)+1/4>0, so the refinement is a strict improvement for all sufficiently large fixed β; the onset temperature and constants b_*(ξ) and C_ξ,b produced by the proof are not numerically explicit. Two companion results -- a sequential Arrhenius bound with the explicit constant E_0(ξ)−E_1(ξ) 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.
ER -