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handout4 - 18.338J/16.394J The Mathematics of Innite Random...

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negationslash 18.338J/16.394J: The Mathematics of Infinite Random Matrices The Stieltjes transform based approach Raj Rao Handout #4, Thursday, September 16, 2004. 1 The eigenvalue distribution function For an N × N matrix A N , the eigenvalue distribution function 1 (e.d.f.) F A N ( x ) is defined as F A N ( x ) = Number of eigenvalues of A N x . (1) N As defined, the e.d.f. is right continuous and possibly atomic i.e. with step discontinuities at discrete points. In practical terms, the derivative of (1), referred to as the (eigenvalue) level density, is simply the appropriately normalized histogram of the eigenvalues of A N . The MATLAB code histn we distributed earlier approximates this density. A surprising result in infinite RMT is that for some matrix ensembles, the expectation E [ F A N ( x )] has a well defined i.e. not zero and not infinite limit. We drop the notational dependence on N in (1) by defining the limiting e.d.f. as F A ( x ) = lim E [ F A N ( x )] . (2) N →∞ This limiting e.d.f. 2 is also sometimes referred to in literature as the integrated density of states [2, 3]. Its derivative is referred to as the level density in physics literature [4]. The region of support associated with this limiting density is simply the region where dF A ( x ) = 0. When discussing the limiting e.d.f. we shall often distinguish between, its atomic and non-atomic components. 2 The Stieltjes transform representation One step removed from the e.d.f. is the Stieltjes transform which has proved to be an efficient tool for determining this limiting density. For all non-real z the Stieltjes (or Cauchy) transform of the probability measure F A ( x ) is given by 1 m A ( z ) = x z dF A ( x ) Im z negationslash = 0 . (3) The integral above is over the whole 3 or some subset of the real axis since for the matrices of interest, such as the Hermitian or real symmetric matrices, the eigenvalues are real. When we refer to the “Stieltjes transform of A ” in this paper, we are referring to m A ( z ) defined as in (3) expressed in terms of the limiting density dF A ( x ) of the random matrix ensemble A . 1 This is also referred to in literature as the empirical distribution function [1]. 2 Unless we state otherwise any reference to an e.d.f. or the level density. in this paper will refer to the corresponding limiting e.d.f. or density respectively. 3 While the Stieltjes integral is over the positive real axis, the Cauchy integral is more general [5] and can include complex contours as well. This distinction is irrelevant for several practical classes of matrices, such as the sample covariance matrices, where all of the eigenvalues are non-negative. Nonetheless, throughout this paper, (3) will be referred to as the Stieltjes transform with the implicit assumption that the integral is over the entire real axis.
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The Stieltjes transform in (3) can also be interpreted as an expectation with respect to the measure F A ( x ) such that 1 m A ( z ) = E X . (4) z x Since there is a one-to-one correspondence between the probability measure F A ( x ) and the Stieltjes transform, convergence of the Stieltjes
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