Rare Events Dominance in Non Equilibrium Critical Phenomena: Selected Examples

In the first chapter we will carefully investigate RG and scaling in Abelian Sandpiles. vVe will derive a real space RG rescaling procedure very similar in spirit to the one introduced by Pietronero et al. for SOC systems (5]. It can be used for both a classical uncorrelated Branching Process evolving on a lattice (which needs of a tuning condition in order to be critical) and for the Abelian Sandpile model, historically the first toy model introduced to explain SOC. We will see that in the case of the 2d Sand pile the treatment given in [5] can be better understood introducing a new critical exponent (. Such a novel exponent describes how the scaling properties of Avalanches in Abelian Sand piles and the Renormalization flow intrinsically depend on the SOC dynamics of the model. It is zero in the case of the classical BP while it signals SOC criticality in the 2d Abelian Sandpile taking a non trivial value. In view of these results a new scaling theory is introduced and numerically tested, solving an old controversy on the avalanche critical exponents in the 2d Abelian Sandpile model. The second chapter is devoted to the development of mean field for interface growth phenomena; striking result of this treatment is the description of the high dimensional behaviour of J( P Z when the number of neighbors becomes very high. Non Markovian long memory effects are shown to appear in the strong coupling phase. As we will pictorially show in the case of Sandpiles, statistics of rare events strongly affects analysis and interpretation of numerical data in most of non equilibrium critical phenomena. To understand how the scaling picture is modified by such rare events statistics, we will finally (third chapter) study the simplest case in which such tails appear: directed polymers in disordered media. We will extend the RG introduced by Derrida et al. [6] to the case of Levy distributed disorder; ?-.foltifractal scaling and Non Self Averaging properties are shown to appear in this description as unavoidable in order to characterize their critical properties