Moments of a single entry of circular orthogonal ensembles and Weingarten calculus

Consider a symmetric unitary random matrix $V=(v_{ij})_{1 \le i,j \le N}$ from a circular orthogonal ensemble. In this paper, we study moments of a single entry $v_{ij}$. For a diagonal entry $v_{ii}$ we give the explicit values of the moments, and for an off-diagonal entry $v_{ij}$ we give leading and subleading terms in the asymptotic expansion with respect to a large matrix size $N$. Our technique is to apply the Weingarten calculus for a Haar-distributed unitary matrix.Comment: 17 page

The ratio of e±pe^{\pm}p scattering cross sections predicted from the global fit of elastic epep data

We present predictions for the value of the cross section ratio $\sigma(e^+p \to e^+p)/\sigma(e^-p \to e^-p)$, determined from our fit of the elastic $ep$ cross section and polarization data. In this fit we took into account the phenomenological two-photon exchange dispersive correction. The cross section ratios which are expected to be measured by the VEPP-3 experiment are computed. The kinematical region which will be covered by the E04-116 JLab experiment is also considered. It is shown that for both experiments the predicted cross section ratios deviate from unity within more than $3\sigma$.Comment: 7 pages, 4 figure

General moments of the inverse real Wishart distribution and orthogonal Weingarten functions

Let $W$ be a random positive definite symmetric matrix distributed according to a real Wishart distribution and let $W^{-1}=(W^{ij})_{i,j}$ be its inverse matrix. We compute general moments $\mathbb{E} [W^{k_1 k_2} W^{k_3 k_4} ... W^{k_{2n-1}k_{2n}}]$ explicitly. To do so, we employ the orthogonal Weingarten function, which was recently introduced in the study for Haar-distributed orthogonal matrices. As applications, we give formulas for moments of traces of a Wishart matrix and its inverse.Comment: 29 pages. The last version differs from the published version, but it includes Appendi

Singular measures in circle dynamics

Critical circle homeomorphisms have an invariant measure totally singular with respect to the Lebesgue measure. We prove that singularities of the invariant measure are of Holder type. The Hausdorff dimension of the invariant measure is less than 1 but greater than 0

General moments of matrix elements from circular orthogonal ensembles

The aim of this paper is to present a systematic method for computing moments of matrix elements taken from circular orthogonal ensembles (COE). The formula is given as a sum of Weingarten functions for orthogonal groups but the technique for its proof involves Weingarten calculus for unitary groups. As an application, explicit expressions for the moments of a single matrix element of a COE matrix are given.Comment: 19 page