Convergence of planar loop-erased random walk to SLE_2

Let $D$ be a bounded, simply connected domain containing the origin whose boundary is a Jordan curve, and let $U,V$ be disjoint, closed subarcs of $\partial D$. For the lattice approximation $A_n(D)$, boundary sets $U_n,V_n$, nearest-neighbor path set $\mathcal K_n$, self-avoiding paths $\mathcal W_n^*$ containing the unordered edge $\\{0,1\\}$, and the rescaled measure $\mu^{(n)}$ defined above, with $\mu_D(0;U,V)$ the corresponding $SLE_2$ path measure, Convergence conjecture. The two arbitrary constants in the definition of $\mu_D$ can be chosen so that, for every such $D,U,V$,

$$\lim_{n \rightarrow \infty} n^{3/4}\\,\mu^{(n)}=\mu_D(0;U,V).$$

In particular,

$$\lim_{n \rightarrow \infty}\sum_{\eta\in\mathcal W_n}\hat p_n(\eta)=\Psi_D(0;U,V).$$

This conjectures the convergence of loop-erased random walk to $SLE_2$ in simply connected Jordan domains; the source expects the analogous statement for finitely connected domains as well. The exponent $3/4$ reflects the boundary scaling exponent for loop-erased walk.