{title2=$\theta$}

A flow is an antisymmetric function on directed edges whose <divergence of a flow> vanishes away from designated sources and sinks.

Flow

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 is
$$
\theta_\omega
=\sum_{\gamma:o\to z_r}
\frac{\mu(\gamma)\mathbf1_{\{\gamma\text{ open}\}}}
{\mathbb P(\gamma\text{ open})},j_\gamma.
$$
It 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 $\omega$.

Solved by gpt-5.6-sol high.

Solution

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

{title2=$\operatorname{div}\theta$}

The divergence of a flow at a vertex is its total outward flow. Flow conservation means that this divergence vanishes at every interior vertex.


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 Incoming links: Divergence of a flow