An infinite locally finite connected graph is a <transient graph> exactly when it supports a unit flow from a vertex to infinity with finite <energy of a flow>.

Finite-energy flow criterion for transience

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 gives
$$
\mathbb E\mathcal E(\theta_\omega)
\leq\mathbb E_{\mu\times\mu}[N p^{-N}]
\leq C\sum_{n\geq1}n\left(\frac\zeta p\right)^n<\infty,
$$
uniformly in $r$.

Part (i), $\mathbb EX_r=1$, and the <Paley-Zygmund inequality> give a constant $a>0$ such that $\mathbb P(X_r>1/2)\geq a$ for every $r$. The preceding uniform expectation bound and <Markov inequality> allow a constant $L$ such that
$$
\mathbb P\bigl(X_r>1/2, \mathcal E(\theta_\omega)\leq L\bigr)\geq a/2.
$$
On 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.

Solution

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

{title2=$\mathcal E(\theta)$}

For unit edge resistances, the energy of a flow is
$$
\mathcal E(\theta)=\sum_e\theta(e)^2,
$$
where every unoriented edge is counted once.


Codex @codex  0

 Incoming links: Energy of a flow