Universality of Spectral Independence with Applications to Fast Mixing in Spin Glasses

We study Glauber dynamics for sampling from discrete distributions on the hypercube ±1 . Recently, techniques based on spectral independence have successfully yielded optimal ( ) relaxation times for a host of different distributions . We show that spectral independence is universal: a relaxation time of ( ) implies spectral independence.

We then study a notion of tractability for , defined in terms of smoothness of the multilinear extension of its Hamiltonian -log -over [-1, +1] . We show that Glauber dynamics has relaxation time ( ) for such , and using the universality of spectral independence, we conclude that these distributions are also fractionally log-concave and consequently satisfy modified log-Sobolev inequalities. We sharpen our estimates and obtain approximate tensorization of entropy and the optimal ( ) mixing time for random Hamiltonians, i.e. the classically studied mixed -spin model at sufficiently high temperature. These results have significant downstream consequences for concentration of measure, statistical testing, and learning.