The radial SLE scaling-limit conjecture for multiple self-avoiding walks

Suppose $D$ is a bounded, simply connected domain in $\mathbb{C}$ containing the origin, and let ${\bf z}=(z^1,\ldots,z^n)$ be an $n$-tuple of distinct, counterclockwise-oriented points on $\partial D$. Assume that near each $z^j$, the boundary is a straight line segment parallel to the coordinate axes. For lattice spacing $N^{-1}$, let $A_N$ approximate $ND$ in $\mathbb{Z}^2$, and let ${\bf z}_N=(z^1_N,\ldots,z^n_N)$ be lattice points corresponding to $N{\bf z}$. For the model with parameter $\cent\leq 1$, write $\nu_{A_N,\cent}$ for the measure on configurations and $\bar{\mathcal W}_{A_N}({\bf z}_N)$ for the relevant configuration space.

Radial SLE scaling-limit conjecture. There exist $b$, $\tilde b_n$, a critical value $\beta=\beta_{\cent}$, and a partition function $\Psi^*(D;{\bf z},0)$ such that, as $N\to\infty$,

$$\nu_{A_N,\cent}(\bar{\mathcal W}_{A_N}({\bf z}_N))\sim \Psi^*(D;{\bf z},0)N^{nb}N^{\tilde b_n}.$$

Moreover, the scaling limit $N^{-nb}N^{-\tilde b_n}\nu_{A_N,\cent}$ is $n$-radial $SLE_\kappa$, denoted $\mu_D({\bf z},0)$, with partition function $\Psi(D;{\bf z},0)$. If $f:D\to f(D)$ is conformal and $f(0)=0$, then

$$f\circ\mu_D({\bf z},0)=|f'({\bf z})|^b|f'(0)|^{\tilde b_n}\mu_{f(D)}(f({\bf z}),0),$$

where $f({\bf z})=(f(z^1),\ldots,f(z^n))$ and $f'({\bf z})=f'(z^1)\cdots f'(z^n)$. This conjectures the conformally covariant scaling limit of the lattice model to multiple radial Schramm–Loewner evolution; for other cases, the paper identifies the problem as open.