Boundary and bulk scaling-limit conjecture for the lambda-self-avoiding walk

Let $D\subset\mathbb C$ be bounded and let $z,w\in\overline D$ be distinct marked points, with the boundary regularity and lattice conventions required to define the corresponding measures $\nu_n$. For every ${\bf c}\leq1$, let $\beta$ be lattice dependent and let $b,\tilde b$ be lattice-independent scaling exponents. Define

$$b_\zeta=\begin{cases}b,&\text{if }\zeta\text{ is a boundary point},\tilde b,&\text{if }\zeta\text{ is an interior point}. \end{cases}$$

Boundary and bulk lambda-self-avoiding-walk scaling conjecture. The partition function and rescaled measures satisfy

$$Z_n\sim n^{-(b_z+b_w)}\Psi_D^*(z,w),\qquad n\to\infty,$$ $$\nu_D(z,w)=\lim_{n\to\infty}n^{b_z+b_w}\nu_n,$$

and

$$f\circ\nu_D(z,w)=|f'(z)|^{b_z}|f'(w)|^{b_w}\nu_{f(D)}(f(z),f(w)).$$

This is the rough boundary extension of the bulk conjecture; the paper points to a more precise formulation accounting for lattice issues and does not prove it.