The continuous random energy model (CREM) is a toy model of spin glasses on $\{0,1\}^N$ 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 $\beta_c$, which is the static phase transition point; moreover, for certain $A$, the algorithms work up to the known algorithmic threshold $\beta_G$ 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.

翻译：连续随机能量模型（CREM）是定义在$\{0,1\}^N$上的自旋玻璃玩具模型，其在极限情况下展现出无限层次的关联结构。我们基于马尔可夫链和顺序采样器，给出了两种在高温区近似采样CREM吉布斯分布的多项式时间算法。算法的运行时间与期望的全变差距离和失败概率呈代数关系，并与$(1/g')^{O(1)}$呈指数关系，其中$g'$是到某个逆温度阈值的间隙；这与先前仅能在KL散度上达到$o(N)$精度的结果形成对比。如果CREM的协方差函数$A$是凹函数，则算法可工作至临界阈值$\beta_c$，即静态相变点；此外，对于某些特定的$A$，算法可工作至Addario-Berry和Maillard (2020) 提出的已知算法阈值$\beta_G$，以获得非平凡的采样保证。我们的结果依赖于对配分函数波动的定量界限，以及从采样中获得的“倾斜”CREM的一个新的邻接性结果，后者具有独立的研究意义。我们还证明了谱隙以高概率呈指数级小，这表明代数依赖关系在马尔可夫链方法中可能是不可避免的。