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 Incoming links: Effective resistance to infinity

Edge deletion preserves recurrence 2026-10-14

Deleting an edge of a locally finite recurrent graph leaves recurrent Markov chain connected components. On an infinite component choose a root away from the deleted edge's endpoints, so its vertex degree does not change. The Rayleigh monotonicity principle increases resistance to wired outer boundaries, and effective resistance to infinity therefore remains infinite. Finite components are recurrent Markov chain too. This is a recurrence-type comparison; at an endpoint, changes in the vertex degree factor prevent a blanket monotonicity claim about first-return probabilities.

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Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 30 / 1 / b / Solution Created 2026-10-03 Updated 2026-10-14

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Use

T_{y} = in f {n \geq 0 : X_{n} = y}

and the positive return time

T_{x}^{+} = in f {n \geq 1 : X_{n} = x}

. The walk has transitions

p_{uv} = c_{uv} / d (u)

d (u) = \sum_{v} c_{uv}

; its transition matrix defines a reversible Markov chain, although the transition probabilities need not be symmetric.

The voltage

h (u) = P_{u} (T_{x} < T_{y})

equals one at

x

, zero at

y

, and is a harmonic function on a graph away from the terminals:

h (u) = \sum_{v} p_{uv} h (v)

. For this unit electric potential difference the electric current is

I = \sum_{v} c_{xv} (1 - h (v)) = \frac{1}{R ( x , y )} .

(1)

The first-step analysis therefore proves the return probability from effective resistance identity:

P_{x} (T_{x}^{+} < T_{y}) = \sum_{v} p_{xv} h (v) = 1 - \frac{I}{d ( x )} = 1 - \frac{1}{d ( x ) R ( x , y )} .

(2)

Finite irreducibility ensures eventual hitting of one of the terminals, so no missing escape event appears in this finite-network calculation.

For an infinite locally finite connected graph, exhaust it by finite sets and wire their exteriors to one sink. The event of exiting before the first return decreases to the event of never returning. Thus

P_{o} (T_{o}^{+} = \infty) = \frac{1}{d ( o ) R ( o , \infty )},

(3)

with

1/\infty = 0

. In particular recurrence is equivalent to infinite effective resistance to infinity.

After deleting an edge, consider any infinite component and choose a root

o

away from that edge's endpoints. Its vertex degree

d (o)

is unchanged. On every finite wired exhaustion the Rayleigh monotonicity principle makes the resistance no smaller; the limit of a sequence does too. A recurrent original network consequently leaves a recurrent component. Finite components give recurrent walks as well, with an isolated graph vertex viewed as an absorbing state. This proves that edge deletion preserves recurrence. The phrase “no less recurrent” refers to this recurrence comparison: at a deleted-edge endpoint the vertex degree factor also changes, so monotonicity of its first-return probability alone should not be inferred.

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