The orthogonal circular ensemble

Robinson, Merle Michael

doi:10.1103/physrevd.45.2872

Cited by 3 publications

(1 citation statement)

References 16 publications

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“…A highlight of this line of investigation, which includes [37][38][39][40]42], was the work of Hisakado [28][29][30]. In the latter, for the potential V (θ) = t cos θ, both the Toda lattice equation and Virasoro constraints were used to characterise the corresponding partition function in terms of a solution of the Painlevé III equation.…”

Section: Jhep02(2015)173mentioning

confidence: 99%

Loop equation analysis of the circular β ensembles

Witte

Forrester

2015

J. High Energ. Phys.

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We construct a hierarchy of loop equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures Re β > 0 and number of eigenvalues N . Using matching arguments for the resolvent functions of linear statistics f (ζ) = (ζ + z)/(ζ − z) in a particular asymptotic regime, the global regime, we systematically develop the corresponding large N expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its general Fourier coefficient or moment m k to an equivalent length. The leading large N , large k, k/N fixed form of the moments can be related to the small wave-number expansion of the structure function in the bulk, scaled Dyson circular ensemble, known from earlier work. From the moment expansion we conjecture some exact partial fraction forms for the low k moments. For all of the forgoing results we have made a comparison with the exactly soluble cases of β = 1, 2, 4, general N and even, positive β, N = 2, 3.

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