Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 212 / 2

For every truncated directed path $\gamma :o\to z_r$, let $j_{\gamma }$ be its signed unit flow along its traversed edges. Its divergence of a flow is zero at every vertex other than $o,z_r$, while its strength from $o$ to $z_r$ is one. The displayed expression in the question isIt is a nonnegative linear combination of such path flows. Hence it obeys flow conservation at every interior vertex, is antisymmetric on oppositely directed edges, and is supported on open edges. It is therefore a flow from $o$ to $z_r$ for every configuration $\omega$.

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The strength of the linear combination is the same linear combination of the unit strengths:Taking expectations cancels each denominator. Since the truncated paths partition the path space,

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Expanding the square and using the independence of the percolation edges givesThe ratio equals $p^{-N_r(\gamma ,{\gamma }^{\prime })}$, where $N_r$ is the number of common edges in the two truncated paths. It is at most $p^{-N}$ for $N=|\xi \cap {\xi }^{\prime }|$, under the convention in the hypothesis; in particular $N\ge 1$ because both paths contain $o$. For every integer $N\ge 1$,The tail-sum formula and the assumed exponential intersection tail therefore yield

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Let the energy of a flow be $\mathcal{E}(\theta )={\sum }_e\theta (e{)}^2$, with each unoriented edge counted once. Expanding as in part (i), dropping orientation signs, and summing over common edges givesuniformly in $r$.

Part (i), $\mathbb{E}{X}_r=1$, and the Paley-Zygmund inequality give a constant $a>0$ such that $\mathbb{P}(X_r>1/2)\ge a$ for every $r$. The preceding uniform expectation bound and Markov inequality allow a constant $L$ such thatOn this event, ${\theta }_{\omega }/X_r$ is a unit open flow from $o$ to $z_r$ with energy at most $4L$.

The events that there is such a bounded-energy unit flow from $o$ out of $B(o,r)$ are decreasing in $r$. Their intersection still has probability at least $a/2$. A diagonal compactness argument produces on this intersection a unit flow from $o$ to infinity, supported on its open cluster, with finite energy. The finite-energy flow criterion for transience makes that open cluster transient.

Thus a transient open cluster exists with positive probability. This existence event is a tail event: changing finitely many edges cannot destroy transience in every infinite component, because transience is invariant under finite graph modifications. The Kolmogorov zero-one law upgrades its probability to one.

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