Index Distribution of Cauchy Random Matrices

Abstract

Using a Coulomb gas technique, we compute analytically the probability ${\mathcal{P}}_{\beta }^{(C)}(N_{+},N)$ that a large $N\times N$ 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 ${\mathcal{P}}_{\beta }^{(C)}(N_{+},N)\approx \exp \left[-\beta N^2{\psi }_C(N_{+}/N)\right]$, where $\beta$ is the Dyson index of the ensemble. The rate function ${\psi }_C(\kappa )$ 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 $\kappa =1/2$, has a quadratic behavior modulated by a logarithmic singularity. As a consequence, the variance of the index scales for large $N$ as $\mathrm{Var}(N_{+})\sim {\sigma }_C\ln N$, where ${\sigma }_C=2/(\beta {\pi }^2)$ 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 $\beta =2$, can be derived using orthogonal polynomials.

23 pages, 3 figures

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