Motivated by the study of slow relaxations in glasses, we define a new class of random matrices, where the i'j'th element is exponential in
the distance between two points, and the points are randomly distributed in space. It turns out that the same mathematical model
describes various other physical problems, such as phonon localization and anomalous diffusion. By combining a renormalization group approach
with a direct calculation of the moments of the eigenvalue distribution, we are able to obtain a closed form for the spectrum, in
any dimension, and understand the localization properties of the eigenvectors. The eigenvectors are all localized, with a localization
length which diverges as one goes to vanishing eigenvalues (corresponding to slow relaxation modes). In the limit of very low
densities one obtains logarithmic relaxations, which can be viewed as the limit of anomalous diffusion when the power-law vanishes.