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 ↓ 1. We show that the initial value problem is well defined in a suitable function space and the solution converges, as t → ∞, to one of the countably infinite equilibrium solutions. The j–th nontrivial equilibrium solution bifurcates from the trivial one at βj = 8π/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 β > 0. Gallavotti and Nicoló’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.