论文标题
具有反双向相互作用的单位圆上的点过程
A point process on the unit circle with antipodal interactions
论文作者
论文摘要
我们介绍了点过程\ begin {align*} \ frac {1} {z__ {n}} \ prod_ {1 \ leq j <k \ leq n} | e^{iθ_{iθ_{j {j}}}}} \qquad θ_{1},\ldots,θ_{n} \in (-π,π], \quad β> 0, \end{align*} where $Z_{n}$ is the normalization constant. This point process is attractive: it involves $n$ dependent, uniformly distributed random variables on the unit circle that attract each other. (For comparison, the well-studied c $β$ e涉及$ n $均匀分布的随机变量在彼此驱除的单位圆上。） 我们将表单的线性统计量$ \ sum_ {j = 1}^{n} g（θ_{j}）$作为$ n \ to \ infty $，其中$ g \ in c^{1，q} $和$2π$ - periodic。我们证明，平均值的领先顺序波动是$ n $的，由$ \ smash {\ big（g（u） - \ int _ { - π}^πg（θ）\ frac {dθ}} {dθ} {2π}}} {2π}}}} \ big）平均值的转向波动是$ \ sqrt {n} $和形式的$ \ nathcal {n} _ {\ mathbb {r}}（0,4G'（0,4G'（u）^{2}/β）/β）/β）\ sqrt \ sqrt {n} $ a g a g a g a g.差异。 我们的证明使用McKay和Isaev [8,6]开发的技术来获取相关$ n $折叠积分的渐近学。
We introduce the point process \begin{align*} \frac{1}{Z_{n}}\prod_{1 \leq j < k \leq n} |e^{iθ_{j}}+e^{iθ_{k}}|^β\prod_{j=1}^{n} dθ_{j}, \qquad θ_{1},\ldots,θ_{n} \in (-π,π], \quad β> 0, \end{align*} where $Z_{n}$ is the normalization constant. This point process is attractive: it involves $n$ dependent, uniformly distributed random variables on the unit circle that attract each other. (For comparison, the well-studied C$β$E involves $n$ uniformly distributed random variables on the unit circle that repel each other.) We consider linear statistics of the form $\sum_{j=1}^{n}g(θ_{j})$ as $n \to \infty$, where $g\in C^{1,q}$ and $2π$-periodic. We prove that the leading order fluctuations around the mean are of order $n$ and given by $\smash{\big(g(U)-\int_{-π}^πg(θ) \frac{dθ}{2π}}\big)n$, where $U \sim \mathrm{Uniform}(-π,π]$. We also prove that the subleading fluctuations around the mean are of order $\sqrt{n}$ and of the form $\mathcal{N}_{\mathbb{R}}(0,4g'(U)^{2}/β)\sqrt{n}$, i.e. that the subleading fluctuations are given by a Gaussian random variable that itself has a random variance. Our proof uses techniques developed by McKay and Isaev [8,6] to obtain asymptotics of related $n$-fold integrals.
