The spectrum of random matrices and integrable systems
نویسنده
چکیده
As the reader can find out from the excellent book by Mehta [], it is well known that, if the probability P (M ∈ dM) satisfies the following two requirements: (i) invariance under conjugation by unitary transformations M 7→ UMU, (ii) the random variables Mii, Re Mij , Im Mij , 1 ≤ i < j ≤ N are independent, then V (z) is quadratic (Gaussian ensemble). For this ensemble and for very large N , the probability P (an eigenvalue ∈ dz) tends to Wigner’s semi-circle distribution on the interval Appeared in: Group 21, Physical applications and Mathematical aspects of Geometry, Groups and Algebras, Vol.II, 835-852, Eds.:H.-D. Doebner, W. Scherer, C. Schulte, World scientific, Singapore, 1997.
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