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Introduction to Random Matrices
Here I = S j (a2j 1,a2j) andI(y) is the characteristic function of the set I. In the Gaussian Unitary Ensemble (GUE) the probability that no eigenvalues lie in I is equal to �(a). Also �(a) is a…
On orthogonal and symplectic matrix ensembles
The focus of this paper is on the probability,Eβ(O;J), that a setJ consisting of a finite union of intervals contains no eigenvalues for the finiteN Gaussian Orthogonal (β=1) and Gaussian Symplectic…
Level spacing distributions and the Bessel kernel
AbstractScaling models of randomN×N hermitian matrices and passing to the limitN→∞ leads to integral operators whose Fredholm determinants describe the statistics of the spacing of the eigenvalues of…
Extremal polynomials associated with a system of curves in the complex plane
- H. Widom
- Mathematics
- 1 April 1969
Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices
The usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants. From this representation one can deduce formulas for spacing…
Asymptotics in ASEP with Step Initial Condition
In previous work the authors considered the asymmetric simple exclusion process on the integer lattice in the case of step initial condition, particles beginning at the positive integers. There it…
Fredholm determinants, differential equations and matrix models
AbstractOrthogonal polynomial random matrix models ofN×N hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (ϕ(x)ψ(y)−ψ(x)ϕ(y))/x−y. This paper is…
Integral Formulas for the Asymmetric Simple Exclusion Process
In this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice $${\mathbb{Z}}$$ with nearest neighbor hopping rates p to…
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