论文标题
复杂晶格点的对数的成对相关性
Pair correlations of logarithms of complex lattice points
论文作者
论文摘要
我们研究了各种尺度上复杂线中$ \ mathbb z $ lattice点的复杂对数对的相关性，证明了成对相关函数的存在。我们证明，在线性缩放下，这对相关性表现出水平排斥，因为它有时会发生在统计物理学中。我们证明，在超级线性尺度上，质量现象的总损失，以及在sublinear尺度上的泊松行为。 Euler权重的情况适用于从最大Margulis cusp邻里的常见垂直地理弧的长度相关性，该弧在Bianchi orbifold $ \ Mathrm {psl} _2（\ Mathbb z [i]）\ backsslash \ Mathb rh^33 cr^33 cr^33 c^33 c.2（\ MATHBB Z [i]）_2（\ MATHBB z [i]）
We study the correlations of pairs of complex logarithms of $\mathbb Z$-lattice points in the complex line at various scalings, proving the existence of pair correlation functions. We prove that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics. We prove total loss of mass phenomena at superlinear scalings, and Poissonian behaviour at sublinear scalings. The case of Euler weights has applications to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp neighborhood to itself in the Bianchi orbifold $\mathrm{PSL}_2(\mathbb Z[i]) \backslash\mathbb H^3_{\mathbb R}$.
