Limit behavior of the spectrum in a class of large random matrices (Q1392339)

In [\textit{V. Marchenko} and \textit{L. Pastur}, Mat. Sb. 72(114), 507-536 (1967); English transl. Math. USSR, Sb. 1, 457-483 (1967; Zbl 0162.22501)] the limiting eigenvalue distribution \(\sigma(\lambda)\) of a certain ensemble \(H_N\) of random matrices is studied for \(N\to\infty\). It is shown that the Stieltjes transform \(f(z)\) of \(d\sigma(\lambda)\) is given by the relation \(f(z)= u(z,1)\), where \(u(z,t)\), \(t\in(0, 1)\) is the solution of the partial differential equation \[ u_t(z,t)+ c\tau(t)[1+ \tau(t)u(z, t)]^{-1} u_z(z, t)= 0,\quad u(z,0)= u_0(z) \] with \(u_0\), \(\tau\), and \(c\) determined by the ensemble \(\{M_N\}\). Also, the rule to determine the support of \(d\sigma\) via properties of \(u(z,1)\) is given. In the present paper, this equation is solved for all \(t\in(0,1)\) and a generalization of the rule mentioned is proved.