Renormalization group flow of the two-dimensional hierarchical Coulomb gas

We consider a quasilinear parabolic differential equation associated with the renormalization group transformation of the two-dimensional hierarchical Coulomb system in the limit as the size of the block L down arrow 1. We show that the initial value problem is well defined in a suitable function space and the solution converges, as t --> infinity, to one of the countably infinite equilibrium solutions. The j(th) nontrivial equilibrium solution bifurcates from the trivial one at B-j = 87 pi / j(2), j = 1,2,....These solutions are fully described and we provide a complete analysis of their local and global stability for all values of inverse temperature B > 0. Gallavotti and Nicolo's conjecture on infinite sequence of "phases transitions" is also addressed. Our results rule out an intermediate phase between the plasma and the Kosterlitz-Thouless phases, at least in the hierarchical model we consider.