Sampling from the Continuous Random Energy Model in Total Variation Distance

The continuous random energy model (CREM) is a toy model of spin glasses on \0,1 that, in the limit, exhibits an infinitely hierarchical correlation structure. We give two polynomial-time algorithms to approximately sample from the Gibbs distribution of the CREM in the high-temperature regime, based on a Markov chain and a sequential sampler. The running time depends algebraically on the desired TV distance and failure probability and exponentially in (1/g')O(1), where $g'$ is the gap to a certain inverse temperature threshold; this contrasts with previous results which only attain $o(N)$ accuracy in KL divergence. If the covariance function A of the CREM is concave, the algorithms work up to the critical threshold βc, which is the static phase transition point; moreover, for certain A, the algorithms work up to the known algorithmic threshold βG proposed in Addario-Berry and Maillard (2020) for non-trivial sampling guarantees. Our result depends on quantitative bounds for the fluctuation of the partition function and a new contiguity result of the ``tilted" CREM obtained from sampling, which is of independent interest. We also show that the spectral gap is exponentially small with high probability, suggesting that the algebraic dependence is unavoidable with a Markov chain approach.