The Place of the Church in the New Europe

Forrester, Duncan B.

doi:10.1163/157254392x00039

Cited by 1 publication

(3 citation statements)

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“…For our corresponding third order linear terms we can provide strict upper and lower bounds for their coefficients, as well as numerical evidence for their conjectured values. Our results are consistent with the exact results for the Ginibre ensemble at β = 2 given in [13] to the fourth order. The chiral ensembles which are our main focus depend explicitly on the number ν β 2 of exact zero-eigenvalues in the third order linear term.…”

Section: Introductionsupporting

confidence: 91%

“…although the coefficient of x is partly conjectured. It was derived rigorously in [13] including the coefficient of the next order term. In more detail, we proceed by splitting the sum over n in eq.…”

Section: Statement Of Large X Asymptotic Resultsmentioning

confidence: 99%

“…(2) 0 (r) in the large N limit keeping x = N r 2 fixed. Our method has the advantage that it can be easily extended to the chiral ensembles in the next section, although for this part it provides a weaker result than given in [13].…”

Section: 51)mentioning

confidence: 99%

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Gap probabilities in non-Hermitian random matrix theory

Akemann

Phillips

Shifrin

2009

Journal of Mathematical Physics

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We compute the gap probability that a circle of\ud radius r around the origin contains exactly k complex eigenvalues. Four different ensembles of random matrices are considered: the Ginibre ensembles and their chiral complex counterparts, with both complex (beta=2) or quaternion real (beta=4) matrix elements. For general non-Gaussian weights we give a Fredholm determinant or Pfaffian representation respectively, depending on the non-Hermiticity parameter. At maximal non-Hermiticity, that is for rotationally invariant weights, the product of Fredholm eigenvalues for beta=4 follows from beta=2 by skipping every second factor, in contrast to the known relation for Hermitian ensembles. On additionally choosing Gaussian weights we give new explicit expressions for the Fredholm eigenvalues in the chiral case, in terms of Bessel-K and incomplete Bessel-I functions. This compares to known results for the Ginibre ensembles in terms of incomplete exponentials. Furthermore we present an asymptotic expansion of the logarithm of the gap probability for large argument r at large N in all four ensembles, up to including the third order linear term. We can provide strict upper and lower bounds and present numerical evidence for its conjectured values, depending on the number of exact zero eigenvalues in the chiral ensembles. For the Ginibre ensemble at beta=2 exact results were previously derived by Forrester

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Section: Introductionsupporting

confidence: 91%

Section: Statement Of Large X Asymptotic Resultsmentioning

confidence: 99%