Index Distribution of Cauchy Random Matrices

Using a Coulomb gas technique, we compute analytically the probability $\mathcal{P}_β^{(C)}(N_+,N)$ 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 $\mathcal{P}_β^{(C)}(N_+,N)\approx \exp\left[-βN^2 ψ_C(N_+/N)\right]$, where $β$ is the Dyson index of the ensemble. The rate function $ψ_C(κ)$ 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 $\mathrm{Var}(N_+)\sim σ_C\ln N$, where $σ_C=2/(βπ^2)$ 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 $β=2$, can be derived using orthogonal polynomials.

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