Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 35 / 2

At criticality, percolation clusters occur on every scale. Count their filled outer perimeters, discard microscopic loops through a diameter cutoff, and then pass to a scaling limit. The cutoff gives a locally finite measure on loops of macroscopic size inside a bounded region; summing over scales gives a sigma-finite measure rather than a probability distribution.

The conformal invariance of planar percolation makes these limiting perimeter statistics covariant under conformal maps. Moreover, deciding whether a filled outer perimeter lies in a simply connected subdomain uses only the configuration it encloses: changing the configuration outside that subdomain does not change that perimeter. This is the restriction part of the argument. Conformal covariance together with this domain consistency motivates a nonzero conformal restriction measure on simple loops.

This is an informal construction principle, as requested; it does not replace the tightness, boundary-simplification and uniqueness theorems required to construct the continuum measure rigorously.

Use finite cutoff windows to avoid subtracting two infinite masses. Write $D$ for the unit disc, $D_U={\mathbf{L}}_D\setminus {\mathbf{L}}_U$ on loops surrounding $0$, and $A(U)=\nu (D_U)$. For $0<r<1$, let $H_r=D_{rD}$. These increase to all loops in ${\mathbf{L}}_D$ surrounding $0$ as $r\downarrow 0$.

On the finite-mass windows, the domain-containment events form a pi-system. Indeed, if $U,V$ are simply connected and contain $0$, a simple loop surrounding $0$ that lies in both has its filled interior in both. It therefore lies in the component $W$ of $U\cap V$ containing $0$, which is simply connected. Thus, on the support of $\nu$,Inclusion-exclusion for the finite deficits givesTaking $V=rD$ and subtracting from $\nu (H_r)$ gives the useful recovery formulaThe assumed generation of the loop sigma-field by domain-containment events and the uniqueness theorem for measures now determine the finite restriction to $H_r$. Increasing these windows recovers ${\nu }_D$; conformal transport recovers its restriction to any simply connected domain containing the marked point. If a simply connected domain omits $0$, no loop in it can surround $0$.

The argument uses finite annular deficits. This is the usual local-finiteness convention for these loop measures and is explicitly ensured by the finite logarithmic formula assumed in the continuation. Bare sigma-finiteness must not be used as permission for formal $\infty -\infty$ subtraction. This is the recovery of a loop measure from conformal deficits.

Let ${\varphi }_i:D\to U_i$ be the normalized maps, and put $V={\varphi }_1(U_2)$. A simple loop in $U_1$ has its interior there, so applying ${\varphi }_1^{-1}$ preserves the property of surrounding $0$. There is a disjoint decompositionThe conformal restriction property identifies the measure of the second set with $\nu ({\mathbf{L}}_D\setminus {\mathbf{L}}_{U_2})$. ThereforeThe composition fixes $0$ and has positive derivative there, so it is exactly the normalized map onto $V$. No interchange of infinite differences is needed: the argument is additivity on disjoint sets. The later logarithmic expression is consistent with this identity because $({\varphi }_1\circ {\varphi }_2{)}^{\prime }(0)={\varphi }_1^{\prime }(0){\varphi }_2^{\prime }(0)$.

Let $\Gamma (B)$ be the outer boundary of a planar compact set applied to the Brownian loop. For the pinned Brownian measure here, its outer boundary is almost surely simple and surrounds its root. If $U$ is simply connected, thenThe forward implication uses the fact that filling a compact subset of a simply connected domain cannot leave that domain. For the converse, the Brownian trace is contained in the closed filled interior of its outer boundary, and that interior also lies in $U$. This is precisely why simply connected domains are used.

Let $\eta ={\Gamma }_{\ast }\rho$, a pushforward measure on simple loops surrounding $0$. Its deficits areThey agree with those of $\lambda \eta$ and ${\nu }_D$. The recovery argument from part (i), now with explicitly finite deficits, givesIf $\lambda =0$, the measure is zero on every cutoff window and hence zero. The rooted Brownian loop itself is not a simple loop; taking its outer boundary is essential.

Restrict $\mu$ first to loops surrounding $0$. This satisfies the pointed restriction property, so for one constant $C$,A single rooted pushforward cannot equal the whole ${\mu }_D$: it only sees loops surrounding its root. The rest is recovered by moving that root.

For $z\in D$, choose a disc automorphism ${\psi }_z$ with ${\psi }_z(0)=z$. Conformal restriction yieldswith the same constant. Let $(z_j)$ be a countable dense set in $D$. Every simple loop contained in $D$ surrounds one of these points. Make this cover disjoint by $P_j=E_{z_j}\setminus {\bigcup }_{i<j}{E}_{z_i}$. For a measurable event $A$ of loops in $D$,This explicitly relates the entire loop measure to the single-root measure without overcounting a loop once for every point in its interior. It also proves that knowledge of $\rho$, up to normalization, determines ${\mu }_D$. Nontriviality gives $C>0$: otherwise all countably many pointed patches would vanish. Using the usual unrooted Brownian loop measure, the same construction is commonly expressed as its outer-boundary pushforward, up to a constant.

Put $a=1/4$ and $E-a=\{\gamma -a:\gamma \in E\}$. Both events consist of loops surrounding $0$; $E-a$ lies in the disc of radius $3/4$, so its loops remain in $D$. Translation invariance and the pointed representation implySince $C>0$, cancel it. Outer boundaries commute with translation, and $\Gamma (B)\in E-a$ is exactly $\Gamma (B+a)\in E$. HenceThe half-disc condition ensures that translating a relevant filled Brownian trace still lies inside the original unit-disc cutoff. This proves the asserted event identity; it does not claim translation invariance of the entire measure rooted at $0$.