有界度图上的Lee-Yang零点与铁磁Ising模型的复杂性
Lee-Yang zeros and the complexity of the ferromagnetic Ising model on bounded-degree graphs
摘要：
研究了用复平面单位圆上的外场参数$λ$近似铁磁伊辛模型配分函数的计算复杂性。伊辛模型的复值参数与统计物理中的量子电路计算和相变有关，但也是Liu，Sinclair和Srivastava最近对所有$|λ|\neq 1$的确定性近似方案的关键。在这里，我们专注于未解决的复杂性图片上的单位圆，并在诱人的问题发生了什么$λ=1$，其中一方面的经典算法的Jerrum和辛克莱给出了一个随机的近似计划的实轴上建议易处理性，另一方面，李杨零的存在暗指计算的硬度。我们的主要结果建立了一个尖锐的计算过渡点$λ=1$，更一般的整个单位圆。对于一个整数$Δ\geq 3$和边缘相互作用参数$b\in (0,1)$，我们显示#P-硬度近似图上的最大程度$Δ$上的单位圆的弧的李杨零点是密集的分区函数。该结果与当$|λ|\neq 1$或当$λ$在单位圆的$1$周围的互补弧中时的已知近似算法形成对比。因此，我们的工作给出了一个直接的联系之间的存在/不存在的李-杨零和有界度图上的分区函数的有效逼近的易处理性。
英文摘要：
We study the computational complexity of approximating the partition function of the ferromagnetic Ising model with the external field parameter $λ$ on the unit circle in the complex plane. Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics, but have also been key in the recent deterministic approximation scheme for all $|λ|\neq 1$ by Liu, Sinclair, and Srivastava. Here, we focus on the unresolved complexity picture on the unit circle, and on the tantalising question of what happens around $λ=1$, where on one hand the classical algorithm of Jerrum and Sinclair gives a randomised approximation scheme on the real axis suggesting tractability, and on the other hand the presence of Lee-Yang zeros alludes to computational hardness. Our main result establishes a sharp computational transition at the point $λ=1$, and more generally on the entire unit circle. For an integer $Δ\geq 3$ and edge interaction parameter $b\in (0,1)$ we show #P-hardness for approximating the partition function on graphs of maximum degree $Δ$ on the arc of the unit circle where the Lee-Yang zeros are dense. This result contrasts with known approximation algorithms when $|λ|\neq 1$ or when $λ$ is in the complementary arc around $1$ of the unit circle. Our work thus gives a direct connection between the presence/absence of Lee-Yang zeros and the tractability of efficiently approximating the partition function on bounded-degree graphs.
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