Leonardo F. Guidi, Domingos H. U. Marchetti
Renormalization Group Flow of the Two-Dimensional Hierarchical Coulomb Gas
(162K, LaTeX 2e with 2 EPS Figures)

ABSTRACT.  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\downarrow 1$. We show that the initial value problem is well defined in a 
suitable function space and the solution converges, as $t\rightarrow \infty$, 
to one of the countably infinite equilibrium solutions. The $j$--th 
nontrivial equilibrium solution bifurcates from the trivial one at $\beta 
_{j}=8\pi /j^{2}$, $j=1,2,\ldots $. These solutions are fully described and 
we provide a complete analysis of their local and global stability for all 
values of inverse temperature $\beta >0$. Gallavotti and Nicol\'{o}'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.