We use SLE 6 paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.

The aim of this paper is to prove the following result. Consider the critical Ising model on the rescaled grid aZ 2 , then the renormalized magnetization fieldseen as a random distribution (i.e., generalized function) on the plane, has a unique scaling limit as the mesh size a ց 0. The limiting field is conformally covariant.

We compute analytically and in closed form the four-point correlation function in the plane, and the two-point correlation function in the upper half-plane, of layering vertex operators in the two dimensional conformally invariant system known as the Brownian Loop Soup. These correlation functions depend on multiple continuous parameters: the insertion points of the operators, the intensity of the soup, and the charges of the operators. In the case of the four-point function there is non-trivial dependence on five continuous parameters: the cross-ratio, the intensity, and three real charges. The four-point function is crossing symmetric. We analyze its conformal block expansion and discover a previously unknown set of new conformal primary operators.

It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE 6 . We provide here a detailed proof, which relies on Smirnov's theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy's formula). The version of convergence to SLE 6 that we prove suffices for the Smirnov-Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops.

We consider the dynamics of a model introduced recently by Bialas, Burda and Johnston. At equilibrium the model exhibits a transition between a fluid and a condensed phase. For long evolution times the dynamics of condensation possesses a scaling regime that we study by analytical and numerical means. We determine the scaling form of the occupation number probabilities. The behaviour of the two-time correlations of the energy demonstrates that aging takes place in the condensed phase, while it does not in the fluid phase. PACS: 02.50.Ey, 05.40.+j, 61.43.Fs, 75.50.Lk To be submitted for publication to Journal of Physics A 1 Introduction.