AI 中文总结
该研究在随机几何图$\boldsymbol{\textit{G}}_{n,\boldsymbol{\textit{ε}}}$上的XY模型中，通过证明两点函数的非零下界，得出当逆温度$\boldsymbol{\textit{β}}$以合适速率趋于无穷时模型呈现长程序的结论。
AI 中文摘要
我们研究随机几何图$\boldsymbol{\textit{G}}_{n,\boldsymbol{\textit{ε}}}$上的经典XY模型，其中$\boldsymbol{\textit{G}}_{n,\boldsymbol{\textit{ε}}}$的构造方式为：在$d\boldsymbol{\textit{≥}}2$的有限域$\boldsymbol{\textit{Ω}}⊂\boldsymbol{\textit{R}}^d$中采样$n$个独立点，当两点间距为$\boldsymbol{\textit{ε}}>0$量级时便将二者连边。我们将$\boldsymbol{\textit{G}}_{n,\boldsymbol{\textit{ε}}}$称为随机环境。当$n→∞$时$\boldsymbol{\textit{ε}}→0$且速率足够慢，这类图可刻画$\boldsymbol{\textit{Ω}}$的几何结构。记逆温度为$\boldsymbol{\textit{β}}$，我们证明当$\boldsymbol{\textit{β}}$以依赖于$n$和$\boldsymbol{\textit{ε}}$的速率趋于无穷时，$\boldsymbol{\textit{G}}_{n,\boldsymbol{\textit{ε}}}$上的XY模型呈现长程序，具体而言，我们对两点函数证明了一个远离零的下界。我们的结果针对随机环境是淬火的：长程序以大概率成立，且当$n→∞$时该概率收敛到1。为证明该结论，我们还证明了环境以大概率足够正则，可应用凸性论证与Brascamp–Lieb不等式。
英文摘要
We study the classical $XY$-model on random geometric graphs $\mathcal{G}_{n, \varepsilon}$, which are obtained by sampling $n \in \mathbb{N}$ independent points in a finite domain $Ω\subset \mathbb{R}^d$, $d \geq 2$, and connecting two points by and edge if their distance is of order $\varepsilon > 0$. We refer to $\mathcal{G}_{n, \varepsilon}$ as the random environment. Letting $\varepsilon \to 0$ as $n \to \infty$ at a sufficiently slow rate, these graphs capture the geometry of $Ω$. Denoting the inverse temperature by $β$, we show that in the limit $β\to \infty$ at a rate depending on $n$ and $\varepsilon$, the $XY$-model on $\mathcal{G}_{n, \varepsilon}$ exhibits long range order in the sense that we prove a lower bound away from zero on the two-point function. Our result is quenched in the random environment: long-range order holds with large probability, converging to one as $n \to \infty$. To prove the statement, we show that with high probability the environment is sufficiently regular to apply a convexity argument and the Brascamp--Lieb inequality.
Comments34 pages, 4 figures
