Index distribution of Cauchy random matrices

Using a Coulomb gas technique, we compute analytically the probability Pβ(C)(N+,N) that a large N × N Cauchy random matrix has N+ positive eigenvalues, where N+ is called the index of the ensemble. We show that this probability scales for large N as Pβ(C)(N+,N)≈exp [−βN2ψC(N+/N)], where β is the Dyson index of the ensemble. The rate function ψC(κ) is computed in terms of single integrals that are easily evaluated numerically and amenable to an asymptotic analysis. We find that the rate function, around its minimum at κ = 1/2, has a quadratic behavior modulated by a logarithmic singularity. As a consequence, the variance of the index scales for large N as Var(N+) ∼ σCln N, where σC = 2/(βπ2) is twice as large as the corresponding prefactor in the Gaussian and Wishart cases. The analytical results are checked by numerical simulations and against an exact finite N formula which, for β = 2, can be derived using orthogonal polynomials. (paper)