  We consider an invariant random matrix model where the standard Gaussian
potential is distorted by an additional single pole of order $m$. We compute
the average or macroscopic spectral density in the limit of large matrix size,
solving the loop equation with the additional constraint of vanishing trace on
average. The density is generally supported on two disconnected intervals lying
on the two sides of the pole. In the limit of having no pole, we recover the
standard semicircle. Obtained in the planar limit, our results apply to
matrices with orthogonal, unitary or symplectic symmetry alike. The orthogonal
case with $m=2$ is motivated by an application to spin glass physics. In the
Sherrington-Kirkpatrick mean-field model, in the paramagnetic phase and for
sufficiently large systems the spin glass susceptibility is a random variable,
depending on the realization of disorder. It is essentially given by a linear
statistics on the eigenvalues of the coupling matrix. As such its large
deviation function can be computed using standard Coulomb fluid techniques. The
resulting free energy of the associated fluid precisely corresponds to the
partition function of our random matrix model. Numerical simulations provide an
excellent confirmation of our analytical results.
