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 Incoming links: Transfer current

Transfer-current theorem Created 2026-09-28 Updated 2026-11-05

In a finite connected graph with unit electrical resistance on each edge, choose distinct undirected edges

e_{1}, \dots, e_{k}

and orient each once. Define the transfer current

K (e, f)

as the signed electric current through

f

when one unit of electric current enters the tail of

e

and leaves its head. For a uniform spanning tree

T

Pr (e_{1}, \dots, e_{k} \in T) = det [K (e_{i}, e_{j})]_{i, j = 1}^{k} .

(1)

Reversing one orientation negates its row and column, leaving the determinant unchanged. Distinctness is necessary: repeating an edge repeats a row and gives zero, although its inclusion can have positive probability.

Electrical reciprocity makes

K

symmetric. The two-edge case therefore gives

Cov (1_{e \in T}, 1_{f \in T}) = - K (e, f)^{2} \leq 0 (e \neq = f) .

(2)

Thus edge inclusions are negatively correlated.^[ref] The positive-weight version uses the corresponding conductance-normalized transfer current. Every extendable finite forest has positive inclusion probability, since at least one positively weighted spanning tree contains it.

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