Conformal Field Theories in Random Domains and Stochastic Loewner Evolutions

where L is the number of links of the lattice, NC the number of clusters in the configuration C and ‖C‖ the number of links inside the NC clusters, usually called FK-clusters. Criticality is then encoded in the fractal nature of these clusters. The stochastic Loewner evolutions (SLE) [2] are mathematically well-defined processes describing the growth of random sets, called the SLE hulls, and of random curves, called the SLE traces, embedded in the two-dimensional plane. The growths of these sets are encoded into families of random conformal maps satisfying specific evolution equations. Their distribution depends on a real parameter κ. The connexion between critical systems and SLE growths may intuitively be undertsood as follows: imagine considering the Q-state Potts models on a lattice covering the upper half plane with boundary conditions on the real line such that all spins on the negative real axis are frozen to the same identical value while spins on the right of the origin are free with non assigned values. Then, in each configuration there exists a FK-cluster growing from the negative half real axis into the upper half plane whose boundary starts at the origin. In the continuum limit, this boundary curve is conjectured to be statistically equivalent to a