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

The continuum Gaussian free field $\Gamma$ on ${\mathbb{R}}^3$ is the centered Gaussian random distribution indexed by smooth compactly supported test functions, with covariancewhere $G$ is the Green kernel of $-\Delta$. Equivalently, its covariance operator is $(-\Delta {)}^{-1}$.

A finite measure $\mu$ can index the field when its Green energyis finite. On the two-dimensional square, the singularity is locally integrable because polar area contributes $rdr$ while $G$ contributes $1/r$. The near-diagonal integral is therefore proportional to ${\int }_0^{\epsilon }dr<\infty$, and the mean $\Gamma (\mu )$ is a well-defined centered Gaussian random variable.

On the segment the energy containswhich diverges logarithmically along the diagonal. The uniform segment measure therefore has infinite Green energy, so its mean cannot be defined as an $L^2$ Gaussian-field pairing.

The restriction and normalization make $\mu$ the unique conformal-restriction loop measure surrounding the origin, up to the fixed multiplicative normalization. By the classification of planar conformal restriction measures, it is the image of the rooted Brownian loop measure at the origin under the map taking a Brownian loop to its outer boundary. It is scale invariant and sigma-finite, with finite mass after imposing lower and upper diameter cutoffs.

For every bounded simply connected $D$, translation $z\mapsto z+x$ is a conformal map from $D$ to $D+x$. Conformal restriction says that translating ${\rho }_D$ gives ${\rho }_{D+x}$. Exhausting the plane by such domains proves that the whole measure $\rho$ is translation invariant. Consequentlywhere $T_x(\gamma )=\gamma +x$ and ${\mu }_x$ restricts $\rho$ to loops surrounding $x$.

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.

Restrict ${\mu }_x$ to outer boundaries that stay in $D$ and surround $y$. By part (B1), this is the image under outer-boundary formation of rooted Brownian loops through $x$ whose outer boundary stays in $D$ and surrounds $y$. But this restriction is exactly the restriction of ${\rho }_D$ to loops surrounding both $x$ and $y$. Interchanging $x$ and $y$ gives the same intrinsic restriction of ${\rho }_D$, so it can equally be represented by outer boundaries of Brownian loops rooted at $y$ that surround $x$.