Efficient Approximation of the CREM Gibbs Measure and the Hardness Threshold

Abstract

The continuous random energy model (CREM) is a toy model of disordered systems introduced by Bovier and Kurkova in 2004 based on previous work by Derrida and Spohn in the 80s. In a recent paper by Addario-Berry and Maillard, they raised the following question: what is the threshold \(\beta _G\), at which sampling approximately the Gibbs measure at any inverse temperature \(\beta >\beta _G\) becomes algorithmically hard? Here, sampling approximately means that the Kullback–Leibler divergence from the output law of the algorithm to the Gibbs measure is of order o(N) with probability approaching 1, as \(N\rightarrow \infty \), and algorithmically hard means that the running time, the numbers of vertices queries by the algorithms, is beyond of polynomial order. The present work shows that when the covariance function A of the CREM is concave, for all \(\beta >0\), a recursive sampling algorithm on a renormalized tree approximates the Gibbs measure with running time of order \(O(N^{1+\varepsilon })\). For A non-concave, the present work exhibits a threshold \(\beta _G<\infty \) such that the following hardness transition occurs: (a) For every \(\beta \le \beta _G\), the recursive sampling algorithm approximates the Gibbs measure with a running time of order \(O(N^{1+\varepsilon })\). (b) For every \(\beta >\beta _G\), a hardness result is established for a large class of algorithms. Namely, for any algorithm from this class that samples the Gibbs measure approximately, there exists \(z>0\) such that the running time of this algorithm is at least \(e^{zN}\) with probability approaching 1. In other words, it is impossible to sample approximately in polynomial-time the Gibbs measure in this regime. Additionally, we provide a lower bound of the free energy of the CREM that could hold its value.