This question has been cross-posted on Mathoverflow : https://mathoverflow.net/questions/451652/locality-and-restriction-properties-for-self-avoiding-and-loop-erasing-random-wa
I have been trying to understand how scaling-limits of suitable families of random curves on a lattice, for instance self-avoiding walks with a uniform measure or percolation-cluster boundaries, turn out to be related to a suitable Schramm-Loewner evolution $\operatorname{SLE}_{\kappa}$ for givne $\kappa$.
In the case of self-avoiding walks it seems that pinning down the constant $\kappa$ is essentially done by establishing that self-avoiding walks have a suitable restriction-property, which will then survive in the limit, and the only Schramm-Loewner evolution which will have this restriciton property will be $\operatorname{SLE}_{8/3}$.
It is however unclear to me exactly how the restriction property will look at the level of random self-avoiding walks with uniform measure. The only candidate definition I have seen states that if we are given a graph $G=(V,E)$ then a family of random curves on $G$ will have the restriction property if for any subgraph $G'$ the law of our family on $G$, conditioned to stay in $G'$, will be the same as the law of our family of curves in $G'$. For self-avoiding random walks with a uniform measure this is clearly satisfied, but it appears to me that it would be equally satisfied for for instance loop-erasing random walks with a uniform measure, a family of curves which in the scaling-limit behaves like $\operatorname{SLE}_2$. Hence I am not sure this is the correct definition of the restriction property on the level of self-avoding random graphs.
A similar question can be asked in the case of percolation cluster boundaries. In this case the crucial property pinning down $\kappa$ appears to be locality, but I have been unable to find a clear definition and a proof that percolation cluster boundaries meet this condition, but self-avoiding random walks or loop-erasing random walks do not.
I would be very interested in seeing explicitly definitions of these two concepts in a way which would make it possible to prove directly that self-avoding walks satisfy one but not the other, whereas percolation cluster boundaries satisfy the other but not the first (and ideally of course in a way which would survive passing to the scaling limit, and would reproduce the usual definition of these two concepts in the context of $\operatorname{SLE}_k$.