Canonical Equation K ₂₃ for Random Matrices with Independent Pairs of Entries with Different Variances and Equal Covariances

Abstract

In this chapter, we consider the problem of description of the limit spectral functions for random matrices ${\Xi _n} = [\xi _{ij}^{(n)}]_{i,j = 1}^n$ with independent pairs of entries $\left\{ {\left. {\xi _{ij}^{(n)},\;\xi _{ij}^{(n)}} \right\},\;i \ge j,} \right.\;i,j = 1, \ldots , n,$ which may have different variances and equal covariances. In this case, the spectral theory is much more complicated than the corresponding theory for random matrices whose entries have equal variances and the pairs of entries $\left\{ {\left. {\xi _{ij}^{(n)},\;\xi _{ij}^{(n)}} \right\},\;i \ge j,} \right.\;i,j = 1, \ldots , n,$ have equal covariances. As in the previous chapters, we consider the regularized V-transform

$$ V\{ t,s,z\} = {n^{ - 1}}Tr\;{[({\Xi _n} - \tau {I_n})({\Xi _n} - \tau {I_n})* - z{I_n}]^{ - 1}}, $$

where z = x + iy, y > 0, and τ = t + is. We prove limit theorems for V {t, s, z} for all τ and z, Imz > 0, and then find the limit n.s.f. of the matrix Ξ _n . We omit the technical details of our derivation and present only the main idea.