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Marcenko-Pastur Distribution and McKay’s Law

In: Application of Integrable Systems to Phase Transitions

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  • C. B. Wang

    (Institute of Analysis)

Abstract

When the Lax pair method for the eigenvalue density is applied to the Laguerre and Jacobi polynomials, the Marcenko-Pastur distribution, McKay’s law and their generalizations can be obtained. The logarithmic divergences based on these density models will be discussed in this chapter by using the elliptic integrals and expansion method discussed before. Even though the associated matrix models and transitions have not been widely studied so far, the density models provide an alternative interpretation for the state changes and two-phase models of the random systems that are now important in applications typically in complexity subjects such as behavioral sciences and econophysics. The power-law distribution is also a very interesting research direction that has drawn the attention of many researchers to find the formulation of the power-law distributions. The Laplace transform is applied to discuss this problem in this chapter, showing that the power exponent in the power-law distribution can be obtained from the power behavior of the density model at the end point(s) of the interval where the eigenvalues are distributed.

Suggested Citation

  • C. B. Wang, 2013. "Marcenko-Pastur Distribution and McKay’s Law," Springer Books, in: Application of Integrable Systems to Phase Transitions, edition 127, chapter 0, pages 161-189, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-38565-0_7
    DOI: 10.1007/978-3-642-38565-0_7
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