Sampling from the Gibbs measure of the continuous random energy model and the hardness threshold

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 $\beta\leq \beta_G$, the recursive sampling algorithm approximates the Gibbs measure with 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 own value.