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A method to calculate correlation functions for \(\beta =1\) random matrices of odd size. (English) Zbl 1172.82010
This paper deals with the \(\beta=1\) random matrix ensembles. For instance, in the case of the Wigner Gaussian ensembles the joint probability distribution of the eigenvalues of an \(N\times N\) matrix is
\[ P_{\beta,N}(x_1,\ldots,x_N)=\frac1{C_{\beta,N}}e^{-\beta \sum_{i=1}^N x_i^2/2} \prod_{i<j}|x_i-x_j|^\beta : \] the GOE corresponds to \(\beta=1\), the GUE to \(\beta=2\) and the GSE to \(\beta=4\). The authors are particularly interested in the \(\beta=1\) Ginibre ensemble. A particularity of \(\beta=1\) is that the computation of eigenvalue correlation functions follows a different route according to the parity of \(N\). The authors devise a way to get the odd-\(N\) eigenvalue correlation functions via a limiting procedure from the even-\(N\) ones; they apply this method first to the \(\beta=1\) Wigner ensemble (where the result is already well established via other methods) and then to the \(\beta=1\) Ginibre ensemble.

MSC:
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
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