Canonical Equation K ₂₇ for Normalized Spectral Functions of Random Symmetric Block Matrices

Abstract

In this chapter, we consider random symmetric matrices ${\Xi _{n\;x\;n}} = (\xi _{ij}^{(n)})_{i,j = 1}^n$ with asymptotically independent entries. It is proved that, for almost all x and any ε > 0, under certain restrictions,

$$ \mathop {\lim }\limits_{n \to \infty } \;P\{ |{\mu _n}(x) - {F_n}(x)|\; > \;\varepsilon \} = 0, $$

where

$$ {\mu _n}(x) = {n^{ - 1}}\sum\limits_{k = 1}^n \chi ({\lambda _k}\;{\rm{ < }}x), $$

χ(λ _k < x) is the indicator function, λ _k are eigenvalues of the matrix ${\Xi _{n\;{\rm{x }}n}} = (\xi _{ij}^{(n)})_{i,j = 1}^n,$ F _n (x) is the distribution function whose Stieltjes transform is equal to

$$ \int_{ - \infty }^\infty {{{(x - z)}^{ - 1}}\;{\rm{d}}{F_n}(x) = {n^{ - 1}}\sum\limits_{k = 1}^p {Tr} \;{C_{kk}}(z),\;z = t + is, s \ne 0,} \; $$

and the block matrices C _kk (z), k = 1,..., p, of dimensionality q x q satisfy the system of canonical equations if K ₂₇

$$ {C_{kk}}(z) = {\{ {[{A_{pq\;{\rm{x }}pq}} - z{I_{pq\;{\rm{x }}pq}} - ({\delta _{lj}}\sum\limits_{s = 1}^p E \;H_{js}^{(n)}{C_{ss}}(z)H_{js}^{(n)T})_{l,j = 1}^p]^{ - 1}}\} _{kk}}, $$

where k = 1,..., p, A _{pq x pq} is a nonrandom matrix, I _{pq x pq} is the identity matrix, $H_{js}^{(n)}$ are random matrices of dimensionality q x q, p and q are some integers and notation {A} _kk means the kth diagonal block of size q x q of the matrix A.