Fast mixing of Glauber dynamics for the Sherrington-Kirkpatrick model throughout the high-temperature phase, zero external field

Spectral-gap paper, Theorem 1.1: for fixed $0<\beta<1$, with probability tending to one over the disorder, $\mathrm{Var}_{\mu_0}(f)\le C_\beta\mathcal D_0(f)$ for all $f$ (zero field), so the uniform-site heat-bath chain has gap at least $1/(C_\beta n)$. Companions: worst-start total-variation cutoff at $\log n/(2\lambda(\beta))$ for $0\le\beta<1$; at $\beta=1$ mixing time $n^{2/3+o(1)}$ and slow mixing from typical starts before $n^{2/3}$; for $\beta>1$ stretched-exponential barriers from typical starts and an $\exp(n^{1-c})$ typical-start upper bound; at $\beta=1$ universal random limits of rescaled autocorrelations for Gaussian and Rademacher disorder. Not shown: nonzero external fields, or an $O(n\log n)$ bound from the gap paper alone.

Spectral-gap paper, Theorem 1.1: for fixed $0<\beta<1$, with probability tending to one over the disorder, $\mathrm{Var}_{\mu_0}(f)\le C_\beta\mathcal D_0(f)$ for all $f$ (zero field), so the uniform-site heat-bath chain has gap at least $1/(C_\beta n)$. Companions: worst-start total-variation cutoff at $\log n/(2\lambda(\beta))$ for $0\le\beta<1$; at $\beta=1$ mixing time $n^{2/3+o(1)}$ and slow mixing from typical starts before $n^{2/3}$; for $\beta>1$ stretched-exponential barriers from typical starts and an $\exp(n^{1-c})$ typical-start upper bound; at $\beta=1$ universal random limits of rescaled autocorrelations for Gaussian and Rademacher disorder. Not shown: nonzero external fields, or an $O(n\log n)$ bound from the gap paper alone.