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 Incoming links: Electrical current from stopped random-walk traversals

Write

w_{x y} = w_{y x} > 0

for an edge conductance and

w_{x} = \sum_{y \sim x} w_{x y}

. Take

s \neq = t

, as implicit in a unit source-to-sink electrical network. An oriented current flow is antisymmetric,

i_{y x} = - i_{x y}

. Kirchhoff's first law is conservation of current:

\sum_{y} i_{x y} = 0

at every vertex other than the source and sink, with the divergences there equal to the prescribed total flow and its negative. Kirchhoff's second law says that the sum of oriented voltage drops around any closed cycle is zero. Ohm's law is

i_{x y} = w_{x y} (v_{x} - v_{y}), v_{x} - v_{y} = \frac{i _{x y}}{w _{x y}} .

Thus the resistance of an edge is

1/ w_{x y}

, and the cycle law reads

\sum_{j = 0}^{m - 1} i_{x_{j} x_{j + 1}} / w_{x_{j} x_{j + 1}} = 0

for

x_{m} = x_{0}

Let

τ = in f {n \geq 0 : X_{n} = t}

. Its expectation is finite. Indeed, from each state there is a path to

t

of length at most

∣ V ∣ - 1

with positive probability; finiteness gives a common lower bound

a > 0

on reaching

t

within that many steps. The Markov property then gives

P (τ > k (∣ V ∣ - 1)) \leq (1 - a)^{k}

G (s, x) = E_{s} \sum_{n = 0}^{τ - 1} 1_{X_{n} = x}, G (s, t) = 0.

Conditional expectation at every pre-stopping visit to

x

gives

E_{s} N_{x y} = G (s, x) p_{x y}

for the number

N_{x y}

of directed traversals. Hence the expected signed count is

u_{x y} = G (s, x) p_{x y} - G (s, y) p_{y x} = w_{x y} (\frac{G ( s , x )}{w _{x}} - \frac{G ( s , y )}{w _{y}}) .

This proves Ohm's law with voltage

v_{x} = G (s, x) / w_{x}

, grounded at

v_{t} = 0

. Summing its voltage drops along a cycle telescopes, proving Kirchhoff's second law.

For Kirchhoff's first law, count entrances and exits on each stopped path. Their net difference at

x

is the initial indicator minus the final indicator. Taking expectations,

\sum_{y} u_{x y} = E_{s} \sum_{n < τ} (1_{X_{n} = x} - 1_{X_{n + 1} = x}) = 1_{x = s} - 1_{x = t} .

Thus the source emits exactly one unit, the sink receives one unit, and all other vertices conserve current.

We also establish the uniqueness required for the conclusion. The cycle law implies that edge voltage drops are differences of vertex potentials: define a potential by summing drops along a path to

t

, and cycle sums make this independent of the path. If two unit flows obey both laws, their difference has the form

j_{x y} = w_{x y} (h_{x} - h_{y})

with zero divergence at every vertex. Therefore

0 = \sum_{x} h_{x} \sum_{y} j_{x y} = \sum_{{x, y} \in E} w_{x y} (h_{x} - h_{y})^{2} .

Strict positivity of the conductances and connectedness force

h

to be constant, so

j = 0

u_{x y} = i_{x y} for the unit source-to-sink current .