David Munton

Quenched disorder in a hierarchical Coulomb gas model

The effects of quenched disorder on the two-dimensional Coulomb gas are studied in the hierarchical approximation. The quenched random variables interact with the charges via a potential that decays as an inverse power (α) of the distance. Recursion relations for the single block charge activities are derived in which the quenched variables explicitly appear. In a linear approximation, for allα⩾1, with some restrictions on the variance of the normally distributed random variables, it is shown that the charge activities converge to the Kosterlitz-Thouless fixed point for all sufficiently low temperatures and sufficiently large blocks. The annealed system is also examined. This model is shown to have a Kosterlitz-Thouless phase only for an intermediate range of temperatures. At low temperatures the activities can diverge, and large charges can exist on all length scales.