On an Average over the Gaussian Unitary Ensemble

Abstract

We study the asymptotic limit for large matrix dimension N of the partition function of the unitary ensemble (β = 2) with weightWe compute the leading order term of the partition function and of the coefficients of its Taylor 1 Introduction BackgroundIn Random Matrix Theory partition functions of ensembles whose probability measure is invariant under conjugation by unitary matrices (unitary or β = 2 ensembles) are integrals of the form 1where w(x) ≥ 0 is a weight function and usually J is either an interval, or the whole real line or the unit circle. The theory of orthogonal polynomials (see Szegő [15], pp. 23-28) implies that such integrals are determinants of Hankel or Toeplitz matrices, or a * The authors acknowledge financial support by the EPSRC grant EP/D505534/1.

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

“…[22,23,43]). The first attempt to study asymptotics of matrix models with an essential singularity was done by Mezzadri and Mo [31] and Brightmore et al [3] when they are considering asymptotic properties of the partition function (the normalization constant Z n (1.2), in our notation) associated with the following weight w(x; z, s) = exp − z 2 2x 2 + s x − x 2 2 , z ∈ R \ {0}, 0 ≤ s < ∞, x ∈ R.…”

Statistical correlations between locally randomized measurements: A toolbox for probing entanglement in many-body quantum states

“…9, the scaling behavior of the trace distance of the estimated density matrix (ρ A ) e for a pure product state ρ A is shown. For both protocols, we find that the numerical data, obtained with random unitaries sampled directly from the CUE [40], is well described by a scaling law…”

“…Graphical visualization of a pure ρ1 (green) and a mixed ρ1 (purple) singe qubit state with Bloch vectors (arrows) v1 and v2, respectively. Points correspond to 50 randomly rotated states, generated via the application of random unitaries sampled from the CUE [40] to ρ1 and ρ2, respectively. b) Histogram of the random variable ZU = Tr U ρU † σz for pure ρ1 (green) and mixed ρ2 (purple) state, the indicated standard deviation (multiplied with a factor √ 3) corresponds to the length of the Bloch vectors.…”

Universality in complex Wishart ensembles for general covariance matrices with 2 distinct eigenvalues

“…Although the case considered here is included in [8], the less direct Riemann-Hilbert approach used in this paper is useful in applications such as the computation of ensemble averages (see, e.g. [16]). …”