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

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

Solved by gpt-5.6-sol high.

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.

Solved by gpt-5.6-sol high.