[论文解读] 2D Coulomb Gases and the Renormalized Energy

Étienne Sandier, Sylvia Serfaty|arXiv (Cornell University)|Jan 17, 2012

Random Matrices and Applications参考文献 44被引用 19

一句话总结

本文分析了具有通用势能和逆温度β的二维库仑气体，通过重整化能W将宏观平衡测度与微观点构型联系起来。研究建立了配分函数的下一阶渐近展开、微尺度的涨落估计，以及一个大偏差原理，表明在高β系统中会结晶化为W极小化器——在β→∞极限下，这些极小化器被猜想为阿布里科索夫三角晶格。

ABSTRACT

We study the statistical mechanics of classical two-dimensional "Coulomb gases" with general potential and arbitrary <i>β</i> the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case <i>β</i> = ∞ corresponds to "weighted Fekete sets" and also falls within our analysis.<br> It is known that in such a system points should be asymptotically distributed according to a macroscopic "equilibrium measure," and that a large deviations principle holds for this, as proven by Ben Arous and Zeitouni [BZ].<br> By a suitable splitting of the Hamiltonian, we connect the problem to the "renormalized energy" <i>W</i>, a Coulombian interaction for points in the plane introduced in [SS1],which is expected to be a good way of measuring the disorder of an infinite configuration of points in the plane. By so doing, we are able to examine the situation at the microscopic scale, and obtain several new results: a next order asymptotic expansion of the partition function, estimates on the probability of fluctuation from the equilibrium measure at microscale, and a large deviations type result, which states that configurations above a certain threshhold of <i>W</i> have exponentially small probability. When <i>β</i> → ∞, the estimate becomes sharp, showing that the system has to "crystallize" to a minimizer of <i>W</i>. In the case of weighted Fekete sets, this corresponds to saying that these sets should microscopically look almost everywhere like minimizers of <i>W</i>, which are conjectured to be "Abrikosov" triangular lattices.

研究动机与目标

• 理解有限和无限β下二维库仑气体的微观结构。
• 将平衡测度与重整化能W联系起来，W是点构型无序程度的度量。
• 推导微尺度下围绕平衡的涨落的渐近展开与大偏差原理。
• 证明当β→∞时，构型收敛至W极小化器，对应于加权费克特集。

提出的方法

• 将哈密顿量分解为平衡部分与涨落部分，以分离重整化能W的作用。
• 将重整化能W用作平面内无限点构型的微观相互作用度量。
• 应用大偏差原理来量化具有高W值的罕见构型的概率。
• 利用W作为关键组成部分，推导配分函数的下一阶渐近展开。
• 分析β→∞极限，证明系统必须最小化W，从而暗示结晶化。
• 借助关于W极小化器的已知结果，推断加权费克特集在微尺度上收敛至阿布里科索夫晶格。

实验结果

研究问题

• RQ1 重整化能W如何控制二维库仑气体的微观结构？
• RQ2 在主导阶平衡测度之外，配分函数的渐近行为如何？
• RQ3 在微尺度上，观察到高W值点构型的概率是多少？
• RQ4 当β→∞时系统行为如何？是否向W极小化器结晶化？
• RQ5 加权费克特集在微尺度上是否渐近趋近于阿布里科索夫三角晶格？

主要发现

• 配分函数存在一个涉及重整化能W的下一阶渐近展开。
• 微尺度下偏离平衡测度的涨落由W定量控制。
• W值超过某一阈值的构型具有指数级小的概率，从而确立了大偏差原理。
• 当β→∞时，系统被迫最小化W，这意味着加权费克特集必须在微尺度上近似为W极小化器。
• W的极小化器被猜想为阿布里科索夫三角晶格，暗示加权费克特集在微尺度上收敛至此类结构。
• 重整化能W为二维库仑系统中点无序的微观描述提供了有效手段。