论文标题
具有镜型相互作用的单元圆上的点过程
A point process on the unit circle with mirror-type interactions
论文作者
论文摘要
我们考虑点过程\ begin {align*} \ frac {1} {z__ {n}} \ prod_ {1 \ leq j <k \ leq n} | e^{iθ_{j {j {J {J {J}}} - \qquadθ_{1}，\ ldots，θ_{n} \ in（-π，π]，\ quadβ> 0，\ end end {align*}其中$ z_ {n} $是归一化的常量。 $ e^{iθ_{1}}，\ ldots，e^{iθ_{n}} $与在真实行$ e^{ - iθ_{1}}中反射的镜像交互 我们研究$ \ sum_ {j = 1}^{n} g（θ_{j}）$的平滑线性统计量为$ n \ to \ infty $，其中$ g $ as $ g $ as $2π$ - periodic。我们证明可能会发生多种渐近情况：取决于$ g $，围绕平均值的领先订单波动（i）是$ n $ and $ n $ and Bernoulli，（ii）的订单$ 1 $，纯粹是高斯，（iii），（iii）是$ 1 $，纯粹是bernoulli and（pursy bernoulli of Bernoulli of Bernoulli of Bernoulli of Bernoulli and Iv）和1 $ 1 $。 $ bn_ {1}+（1-b）n_ {2} $，其中$ n_ {1}，n_ {2} $是两个独立的高斯人，$ b $是一个独立于$ n_ {1} $和$ n_ {2} $的bernoulli。上面的列表并不详尽：波动可以是订单$ n $，订单$ 1 $或$ o（1）$的订单，其他随机变量也可以在极限中出现。 我们还以$ z_ {n} $（和一些概括）获得大型$ n $渐近，直到和包括订单$ 1 $的期限。 我们的证明灵感来自McKay和Wormald [10]开发的方法，以估算相关的$ n $折叠积分。
We consider 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. The feature of this process is that the points $e^{iθ_{1}},\ldots,e^{iθ_{n}}$ interact with the mirror points reflected over the real line $e^{-iθ_{1}},\ldots,e^{-iθ_{n}}$. We study smooth linear statistics of the form $\sum_{j=1}^{n}g(θ_{j})$ as $n \to \infty$, where $g$ is $2π$-periodic. We prove that a wide range of asymptotic scenarios can occur: depending on $g$, the leading order fluctuations around the mean can (i) be of order $n$ and purely Bernoulli, (ii) be of order $1$ and purely Gaussian, (iii) be of order $1$ and purely Bernoulli, or (iv) be of order $1$ and of the form $BN_{1}+(1-B)N_{2}$, where $N_{1},N_{2}$ are two independent Gaussians and $B$ is a Bernoulli that is independent of $N_{1}$ and $N_{2}$. The above list is not exhaustive: the fluctuations can be of order $n$, of order $1$ or $o(1)$, and other random variables can also emerge in the limit. We also obtain large $n$ asymptotics for $Z_{n}$ (and some generalizations), up to and including the term of order $1$. Our proof is inspired by a method developed by McKay and Wormald [10] to estimate related $n$-fold integrals.
