Spectral gap for general $p$ spin glasses on the hypercube

We consider the Glauber dynamics for general $p$-spin glasses on the hypercube. For sufficiently large temperatures, we establish a spectral gap inequality for the Glauber dynamics that holds with high probability. Our method involves deriving inductive relations between $a_N$, the spectral gap for the system with $N$ particles, and $a_{N-1}$, the spectral gap between the system with $N-1$ particles. Through two equations, which we call the "dichotomy" equation and the "continuity" equation, we can derive $O(1)$ bounds on $a_N$ as $N$ goes to $\infty$.

This is based on joint work with C. Brennecke, C. Xu, and H-T. Yau.