00-425 Leonardo F. Guidi, Domingos H. U. Marchetti

Renormalization Group Flow of the Two-Dimensional Hierarchical Coulomb Gas (162K, LaTeX 2e with 2 EPS Figures) Oct 31, 00

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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.

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