Multi-point passage probabilities and Green's functions for SLE${}_{8/3}$

We consider a loop representation of the $O(n)$ model at the critical point. When $n=0$ the model represents ensembles of self-avoiding loops (i.e., it corresponds to SLE with $\kappa=8/3$), and can be described by the logarithmic conformal field theory (LCFT) with central charge $c=0$. We focus on the correlation functions in the upper-half plane containing the twist operators in the bulk, and a pair of the boundary one-leg operators. By using a Coulomb gas representation for the correlation functions, we obtain explicit results for probabilities of the SLE${}_{8/3}$ trace to wind in various ways about $N\geq 1$ marked points. When the points collapse pairwise the probabilities reduce to multi-point Green's functions. We propose an explicit representation for the Green's functions in terms of the correlation functions of the bulk 1/3-weight operators, and a pair of the boundary one-leg operators.