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. In the case $n=0$, the model reduces to statistical ensembles of self-avoiding loops, which can be described by Schramm–Loewner evolution (SLE) with $\kappa=8/3$. In this limit, the $O(n=0)$ model corresponds to a logarithmic conformal field theory (LCFT) with the central charge $c=0$. We study the LCFT correlation functions in the upper half-plane containing several twist operators in the bulk and a pair of the $\Phi_{1,2}$ boundary operators. By using a Coulomb gas representation for the correlation functions, we obtain explicit results for the probabilities of the SLE${}_{8/3}$ trace to pass in various ways about $N\geq 1$ marked points. When the points approach each other pairwise, the probabilities reduce to multipoint SLE Green's functions. We propose an explicit representation for the Green's functions in terms of the correlation functions of the bulk $\Phi_{3,1}$ and boundary $\Phi_{1,2}$ operators.