固定磁化伊辛模型的计算阈值
Computational thresholds for the fixed-magnetization Ising model
摘要：
铁磁伊辛模型是一种磁性材料的模型，也是统计物理学的中心课题。它在近似计数的算法研究中也起着重要的作用：由于Jerrum和Sinclair的随机算法，在所有温度和所有图形上都可以近似具有均匀外场的铁磁伊辛模型的配分函数。在这里，我们展示了隐藏在模型内部的是难以计算的问题。对于类的有界度图，我们发现计算阈值的近似计数和采样问题的铁磁伊辛模型在固定的磁化强度（即，固定的数量$+1$和$-1$自旋）。特别地，令$β_c(Δ)$表示零场伊辛模型在无限$Δ$ -正则树上的临界逆温度，并且$η_{Δ,β,1}^+$表示零场$+$测度在逆温度$β$下在无限$Δ$ -正则树上的平均磁化强度，我们证明了，对于最大度$Δ$的图类：1。对于$β< β_c(Δ)$，存在用于所有磁化$η$的固定磁化伊辛模型的FPRAS和有效采样方案。2.对于$β> β_c(Δ)$，存在用于磁化$η$的固定磁化伊辛模型的FPRAS和有效采样方案，使得$|η| >η_{Δ,β,1}^+ $。3.对于$β> β_c(Δ)$，对于磁化$η$使得$|η| <η_{Δ,β,1}^+ $的固定磁化伊辛模型没有FPRAS，除非NP=RP。
英文摘要：
The ferromagnetic Ising model is a model of a magnetic material and a central topic in statistical physics. It also plays a starring role in the algorithmic study of approximate counting: approximating the partition function of the ferromagnetic Ising model with uniform external field is tractable at all temperatures and on all graphs, due to the randomized algorithm of Jerrum and Sinclair. Here we show that hidden inside the model are hard computational problems. For the class of bounded-degree graphs we find computational thresholds for the approximate counting and sampling problems for the ferromagnetic Ising model at fixed magnetization (that is, fixing the number of $+1$ and $-1$ spins). In particular, letting $β_c(Δ)$ denote the critical inverse temperature of the zero-field Ising model on the infinite $Δ$-regular tree, and $η_{Δ,β,1}^+$ denote the mean magnetization of the zero-field $+$ measure on the infinite $Δ$-regular tree at inverse temperature $β$, we prove, for the class of graphs of maximum degree $Δ$: 1. For $β< β_c(Δ)$ there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for all magnetizations $η$. 2. For $β> β_c(Δ)$, there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for magnetizations $η$ such that $|η| >η_{Δ,β,1}^+ $. 3. For $β> β_c(Δ)$, there is no FPRAS for the fixed-magnetization Ising model for magnetizations $η$ such that $|η| <η_{Δ,β,1}^+ $ unless NP=RP\@.
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