Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 5 / B3 / Solution

Part (B1) and the stated normalization determine $\mu$, and part (B2) then determines every ${\mu }_x$. Every self-avoiding loop surrounds some rational point. Since ${\mathbb{Q}}^2$ is countable, the restrictions of $\rho$ to loops surrounding rational points determine $\rho$ on the entire loop space, with overlaps already consistent because they arise from the same conformal-restriction law. Hence an existing normalized $\rho$ is unique.