We propose a systematic method to extract conformal loop models for rational conformal field theories (CFT). Method is based on defining an ADE model for boundary primary operators by using the fusion matrices of these operators as adjacency matrices
. These loop models respect the conformal boundary conditions. We discuss the loop models that can be extracted by this method for minimal CFTs and then we will give dilute O(n) loop models on the square lattice as examples for these loop models. We give also some proposals for WZW SU(2) models.
By interpreting the fusion matrix as an adjacency matrix we associate a loop model to every primary operator of a generic conformal field theory. The weight of these loop models is given by the quantum dimension of the corresponding primary operator.
Using the known results for the O(n) models we establish a relationship between these models and SLEs. The method is applied to WZW, $c<1$ minimal conformal field theories and other coset models.
We find the exact winding number distribution of Riemann-Liouville fractional Brownian motion for large times in two dimensions using the propagator of a free particle. The distribution is similar to the Brownian motion case and it is of Cauchy type.
In addition we find the winding number distribution of fractal time process, i.e., time fractional Fokker-Planck equation, in the presence of finite size winding center.
We have studied the iso-height lines on the $mathrm{WO_3}$ surface as a physical candidate for conformally invariant curves. We have shown that these lines are conformally invariant with the same statistics of domain walls in the critical Ising model
. They belong to the family of conformal invariant curves called Schramm-Loewner evolution (or $SLE_{kappa}$), with diffusivity of $kappa sim 3$. This can be regarded as the first experimental observation of SLE curves. We have also argued that Ballistic Deposition (BD) can serve as a growth model giving rise to contours with similar statistics at large scales.
