Coupling of $c=-2$ and $c=\frac{1}{2}$ and $c=0$ conformal field theories: The geometrical point of view

The coupling of the $c=-2$, $c=\frac{1}{2}$ and $c=0$ conformal field theories are numerically considered in this paper. As the prototypes of the couplings, $(c_1=-2)\oplus (c_2=0)$ and $(c_1=-2)\oplus (c_2=\frac{1}{2})$, we consider the BTW model on the 2D square critical site-percolation and the BTW model on Ising-correlated percolation lattices respectively. Some geometrical techniques are used to characterize the presumable conformal symmetry of the resultant systems. Using the Schramm-Loewrner evolution (SLE), we find that the algebra of the central charges of the coupled models is closed, namely the former case results to two-dimensional self-avoiding walk (SAW) fixed point corresponding to $c=0$ conformal field theory, whereas the latter one results to the two-dimensional critical Ising fixed point corresponding to the $c=\frac{1}{2}$ conformal field theory.