# Completing the picture for the smallest eigenvalue of real Wishart matrices

Akemann G, Guhr T, Kieburg M, Wegner R, Wirtz T (2014)
Physical Review Letters 113(5): 250201.

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Abstract
Rectangular real $N \times (N + \nu)$ matrices $W$ with a Gaussiandistribution appear very frequently in data analysis, condensed matter physicsand quantum field theory. A central question concerns the correlations encodedin the spectral statistics of $WW^T$. The extreme eigenvalues of $W W^T$ are ofparticular interest. We explicitly compute the distribution and the gapprobability of the smallest non-zero eigenvalue in this ensemble, both forarbitrary fixed $N$ and $\nu$, and in the universal large $N$ limit with $\nu$fixed. We uncover an integrable Pfaffian structure valid for all even values of$\nu\geq 0$. This extends previous results for odd $\nu$ at infinite $N$ andrecursive results for finite $N$ and for all $\nu$. Our mathematical resultsinclude the computation of expectation values of half integer powers ofcharacteristic polynomials.
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Akemann G, Guhr T, Kieburg M, Wegner R, Wirtz T. Completing the picture for the smallest eigenvalue of real Wishart matrices. Physical Review Letters. 2014;113(5):250201.
Akemann, G., Guhr, T., Kieburg, M., Wegner, R., & Wirtz, T. (2014). Completing the picture for the smallest eigenvalue of real Wishart matrices. Physical Review Letters, 113(5), 250201. doi:10.1103/PhysRevLett.113.250201
Akemann, G., Guhr, T., Kieburg, M., Wegner, R., and Wirtz, T. (2014). Completing the picture for the smallest eigenvalue of real Wishart matrices. Physical Review Letters 113, 250201.
Akemann, G., et al., 2014. Completing the picture for the smallest eigenvalue of real Wishart matrices. Physical Review Letters, 113(5), p 250201.
G. Akemann, et al., “Completing the picture for the smallest eigenvalue of real Wishart matrices”, Physical Review Letters, vol. 113, 2014, pp. 250201.
Akemann, G., Guhr, T., Kieburg, M., Wegner, R., Wirtz, T.: Completing the picture for the smallest eigenvalue of real Wishart matrices. Physical Review Letters. 113, 250201 (2014).
Akemann, Gernot, Guhr, T., Kieburg, Mario, Wegner, R., and Wirtz, T. “Completing the picture for the smallest eigenvalue of real Wishart matrices”. Physical Review Letters 113.5 (2014): 250201.
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PMID: 25554862
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arXiv 1409.0360

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