Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 226 / 1 / b / ii

Table of contents
- Solution ii

Solution

Assume

A \subset B

and use the corrected strict identity from part (i). If a walk starts at

y \in A

and never returns to

A

, then transience makes

L_{B} < \infty = H_{A}

. Thus

cap (A) = y \in A \sum μ_{y} P_{y} (H_{A} = \infty) \leq y \in A \sum μ_{y} P_{y} (L_{B} < H_{A}, L_{B} \geq 0) = P_{e_{B}} (H_{A} < \infty) \leq z \in B \sum e_{B} (z) = cap (B) .

This proves monotonicity of the capacity of a finite set for a transient random walk.

Solved by gpt-5.6-sol high.

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