  We consider a log-gas on a discretization of the unit circle. We prove that
if the gas is not too dense, or the number of particles in the gas is not too
large compared to the scale of the discretization, the absolute value of the
characteristic polynomial can be described in terms of a Gaussian
multiplicative chaos measure. This is done by analyzing discrete Toeplitz
determinants with Fisher-Hartwig singularities. In particular, we prove that if
the gas is not too dense, the classical Fisher-Hartwig conjecture holds for the
discrete Toeplitz determinant as well. Our analysis suggests that if the gas is
any denser than this, the formula needs to be modified.
