Limiting Pair Correlation Research Articles

Statistical Regularity of Apollonian Gaskets

Abstract Apollonian gaskets are formed by repeatedly filling the gaps between three mutually tangent circles with further tangent circles. In this paper we give explicit formulas for the limiting pair correlation and the limiting nearest neighbor spacing of centers of circles from a fixed Apollonian gasket. These are corollaries of the convergence of moments that we prove. The input from ergodic theory is an extension of Mohammadi–Oh’s theorem on the equidisribution of expanding horospheres in the frame fundles of infinite volume hyperbolic spaces.

Leading corrections to the scaling function on the diagonal for the two-dimensional Ising model

In the neighbourhood of the critical point, the correlation length of the spin–spin correlation function of the two-dimensional Ising model diverges. The correlation function permits a scaling limit in which the separation N between spins goes to infinity, but the scaling variable remains fixed, where t is the coupling, and t = 1 the critical point. Previous work has specified these scaling functions (there is one for the critical point being approached from above, and another if approached from below) in terms of transcendents defined by a particular -form of the degenerate Painlevé V equation. For the diagonal–diagonal correlation, we characterise the first two leading large N correction terms to the scaling functions—these occur at orders N−1 and N−2—in terms of solutions of a second order linear differential equation with coefficients given in terms of these transcendents, and show how they can be computed. We show that the order N−1 is trivial and can be eliminated through appropriate variables so that the leading non-trivial correction is of order N−2. In this respect our solution is the first non-trivial correction to the scaling limit pair-correlation results given in the earlier literature, but restricted to the case with both spins on a diagonal.

Zeros of random Reinhardt polynomials

For a strictly pseudoconvex Reinhardt domain with smooth boundary in and a positive smooth measure on the boundary of , we consider the ensemble of polynomials of degree N with the Gaussian probability measure which is induced by . Our aim is to compute the scaling limit distribution function and scaling limit pair correlation function for zeros near a point . First, we apply the stationary phase method to the Boutet de Monvel–Sjöstrand Theorem to get the asymptotic for the scaling limit partial Szegö kernel around . Then by using the Kac–Rice formula, we compute the scaling limit distribution and pair correlation functions.

Pair correlation of angles between reciprocal geodesics on the modular surface

The existence of the limiting pair correlation for angles between reciprocal geodesics on the modular surface is established. An explicit formula is provided, which captures geometric information about the length of reciprocal geodesics, as well as arithmetic information about the associated reciprocal classes of binary quadratic forms. One striking feature is the absence of a gap beyond zero in the limiting distribution, contrasting with the analog Euclidean situation.

Pair correlation of roots of rational functions with rational generating functions and quadratic denominators

For any rational functions with complex coefficients $A(z), B(z)$ and $C(z)$, where $A(z)$, $C(z)$ are not identically zero, we consider the sequence of rational functions $H_{m}(z)$ with generating function $\sum H_{m}(z)t^{m}=1/(A(z)t^{2}+B(z)t+C(z))$. We provide an explicit formula for the limiting pair correlation function of the roots of $\prod_{m=0}^{n}H_{m}(z)$, as $n\rightarrow\infty$, counting multiplicities, on certain closed subarcs $J$ of a curve $\mathcal{C}$ where the roots lie. We give an example where the limiting pair correlation function does not exist if $J$ contains the endpoints of $\mathcal{C}$.

Correlation of fractions with divisibility constraints

AbstractLet \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ B=(B_{Q}\!)_{{Q \in {\mathbb N}}} $\end{document} be an increasing sequence of positive square free integers satisfying the condition that $ B_{{Q_1}}\vert B_{{Q_2}} $ whenever Q1 < Q2. For any subinterval I ⊂ [0, 1], let It is shown that if BQ ≪ Qlog log Q/4, then the limiting pair correlation function of the sequence \documentclass{article}\usepackage{amsmath,amssymb,mathrsfs,bm}\pagestyle{empty}\begin{document}$ ({\mathscr{F}_{{B}\!,_Q}(I)})_{Q \in {\mathbb N}} $\end{document} exists and is independent of the subinterval I. Moreover, the sequence is Poissonian if $ \lim_{Q \rightarrow \infty }{{\varphi (B_{Q}\!)}\over{B_{Q}\!}} = 0 $, and exhibits a very strong repulsion if $ \lim_{Q \rightarrow \infty }{{\varphi (B_{Q}\!)}\over{B_{Q}\!}} \ne 0 $, where φ is Euler's totient function. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Pair correlation of sums of rationals with bounded height

For each positive integer Q , let ℱ Q denote the Farey sequence of order Q . We prove the existence of the pair correlation measure associated to the sum ℱ Q + ℱ Q modulo 1, as Q tends to infinity, and compute the corresponding limiting pair correlation function.

Pair correlation of torsion points on elliptic curves

We prove the existence of the pair correlation measure associated to torsion points on the real locus E ( R ) of an elliptic curve E and provide an explicit formula for the limiting pair correlation function.

Pair Correlation Statistics for the Zeros of Lamé Polynomials

The joint eigenfunctions of a quantum completely integrable system can naturally be described in terms of products of Lame polynomials. In this paper, we compute the limiting pair correlation distribution for the zeros of Lame polynomials in various thermodynamic, asymptotic regimes. We give results both in the mean and pointwise, for an asymptotically full set of values of the parameters α0,. . .,αN.

Correlations Between Zeros and Supersymmetry

In our previous work [BSZ2], we proved that the correlation functions for simultaneous zeros of random generalized polynomials have universal scaling limits and we gave explicit formulas for pair correlations in codimensions 1 and 2. The purpose of this paper is to compute these universal limits in all dimensions and codimensions. First, we use a supersymmetry method to express the n-point correlations as Berezin integrals. Then we use the Wick method to give a closed formula for the limit pair correlation function for the point case in all dimensions.

Universality and scaling of correlations between zeros on complex manifolds

We study the limit as $N\to\infty$ of the correlations between simultaneous zeros of random sections of the powers $L^N$ of a positive holomorphic line bundle $L$ over a compact complex manifold $M$, when distances are rescaled so that the average density of zeros is independent of $N$. We show that the limit correlation is independent of the line bundle and depends only on the dimension of $M$ and the codimension of the zero sets. We also provide some explicit formulas for pair correlations. In particular, we provide an alternate derivation of Hannay's limit pair correlation function for SU(2) polynomials, and we show that this correlation function holds for all compact Riemann surfaces.

Level Spacings for Integrable Quantum Maps in Genus Zero

We consider the eigenvalue pair correlation problem for certain integrable quantum maps on the 2-sphere. The classical maps are time one maps of Hamiltonian flows of perfect Morse functions. The quantizations are unitary operators on spaces of homogeneous holomorphic polynomials of degree N in two complex variables. There are N eigenphases on the unit circle and the pair correlation problem is to determine the distribution of spacings between the eigenphases on the length scale of the mean level spacing. The Berry-Tabor conjecture says that for generic integrable systems, the spacings distribution should tend to the Poisson (uniform) limit as N tends to infinity. In this paper and in its addendum we prove a somewhat weak form of this conjecture: We define a large class of two parameter families of integrable Hamiltonians, and prove that for almost all elements of each family the limit pair correlation function is Poisson. In this paper, we take the limit along a slightly sparse subsequence of N's; in the addendum we take Cesaro means along the full seuqence.