Approximating the partition function of the ferromagnetic Ising model with general external fields is known to be #BIS-hard in the worst case, even for bounded-degree graphs, and it is widely believed that no polynomial-time approximation scheme exists. This motivates an average-case question: are there classes of instances for which polynomial-time approximation schemes exist? We investigate this question for the random field Ising model on graphs with maximum degree $\Delta$. We establish the existence of fully polynomial-time approximation schemes and samplers with high probability over the random fields if the external fields are IID Gaussians with variance larger than a constant depending only on the inverse temperature and $\Delta$. The main challenge comes from the positive density of vertices at which the external field is small. These regions, which may have connected components of size $\Theta(\log n)$, are a barrier to algorithms based on establishing a zero-free region, and cause worst-case analyses of Glauber dynamics to fail. The analysis of our algorithm is based on percolation on a self-avoiding walk tree.
翻译:使用一般外部字段的铁磁性Ising 模型的分区功能已知为 #BIS-hard 在最坏的情况下是 #BIS-hard, 甚至对封闭度图形来说也是如此, 人们广泛认为不存在任何多边时近似方案。 这引发了一个平均案例问题: 是否存在多球时近似方案? 我们调查随机字段的随机模型是用最大度为$\Delta$的图形显示的。 如果外部字段是IID Gaussians, 差异大于常数, 仅取决于反向温度和$\Delta$, 我们确定是否存在完全多元时近似方案和抽样器, 随机字段的概率很高。 主要挑战来自外部字段小的脊椎正密度。 这些区域, 可能有大小为$\ Theta(\log n) 的相连接元的元件, 是在建立零无区域的基础上, 导致对Glabero动态进行最坏的大小分析失败。 我们的算法分析以正向的树本为基。