Solvable random-matrix ensemble with a logarithmic weakly confining potential

This work identifies a solvable (in the sense that spectral correlation functions can be expressed in terms of orthogonal polynomials), rotationally invariant random matrix ensemble with a logarithmic weakly confining potential. The ensemble, which can be interpreted as a transformed Jacobi ensemble, is in the thermodynamic limit characterized by a Lorentzian eigenvalue density. It is shown that spectral correlation functions can be expressed in terms of the nonclassical Gegenbauer polynomials $C_n^{(-1/2)}\left(x\right)$ with $n\ge 2$, which have been proven to form a complete orthogonal set with respect to the proper weight function. A procedure to sample matrices from the ensemble is outlined and used to provide a numerical verification for some of the analytical results. This ensemble is pointed out to potentially have applications in quantum many-body physics.

Figure 1

The difference between the normalized (to unity) eigenvalue density for $N$ finite and $N\to \infty$ for $N=10, N=100$, and $N=1000$. One observes that the difference scales as $1/N$.

Figure 2

A comparison between ${\rho }_N^{\left(2\right)}(0,x)$ (solid lines) and the result for Wigner-Dyson level statistics at $N\to \infty$ of Eq. (16) (dashed lines) for $N=25, N=100$, and $N=1000$. The finite-$N$ results have been scaled by $1/{[\rho \left(0\right)/N]}^2={\pi }^2$ and the linear transformation $x\to \pi x/N$ has been applied in order to set the mean level spacing at $x=0$ to unity for $N\to \infty$. The finite-$N$ curves approach the Wigner-Dyson result as $N$ increases.

Figure 3

A plot of the normalized (to unity) eigenvalue densities ${\rho }_N\left(x\right)/N$ (“analytical”) compared with a normalized [to unity on $x\in (-\infty ,\infty )]$ histogram of the eigenvalues for the random matrix ensemble proposed in this work (“numerical”) for $\beta =2$ at $N=10, N=100$, and $N=1000$. Perfect agreement can be observed.