The fixed irreducible bridge ensemble for self-avoiding walks

We define a new ensemble for self-avoiding walks in the upper half-plane, the fixed irredicible bridge ensemble, by considering self-avoiding walks in the upper half-plane up to their $n$-th bridge height, $Y_n$, and scaling the walk by $1/Y_n$ to obtain a curve in the unit strip, and then taking $n\to\infty$. We then conjecture a relationship between this ensemble to $\SLE$ in the unit strip from $0$ to a fixed point along the upper boundary of the strip, integrated over the conjectured exit density of self-avoiding walk spanning a strip in the scaling limit. We conjecture that there exists a positive constant $\sigma$ such that $n^{-\sigma}Y_n$ converges in distribution to that of a stable random variable as $n\to\infty$. Then the conjectured relationship between the fixed irreducible bridge scaling limit and $\SLE$ can be described as follows: If one takes a SAW considered up to $Y_n$ and scales by $1/Y_n$ and then weights the walk by $Y_n$ to an appropriate power, then in the limit $n\to\infty$, one should obtain a curve from the scaling limit of the self-avoiding walk spanning the unit strip. In addition to a heuristic derivation, we provide numerical evidence to support the conjecture and give estimates for the boundary scaling exponent.