Universality of Spectral Independence with Applications to Fast Mixing in Spin Glasses
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We study Glauber dynamics for sampling from discrete distributions μ on the hypercube {± 1}^n. Recently, techniques based on spectral independence have successfully yielded optimal O(n) relaxation times for a host of different distributions μ. We show that spectral independence is universal: a relaxation time of O(n) implies spectral independence. We then study a notion of tractability for μ, defined in terms of smoothness of the multilinear extension of its Hamiltonian – logμ – over [-1,+1]^n. We show that Glauber dynamics has relaxation time O(n) for such μ, 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 O(n) mixing time for random Hamiltonians, i.e. the classically studied mixed p-spin model at sufficiently high temperature. These results have significant downstream consequences for concentration of measure, statistical testing, and learning.
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