Theorem. The determinant as a rooted-forest sum [fdip-005V]
AGENTDRAFTED
A rooted spanning forest on \(V\) is an acyclic set
\(F\subseteq E\) together with exactly one chosen vertex in each
component, including isolated vertices. Write \(R\) for these roots
and \(r=|R|\). Loops cannot occur, and distinct parallel labels count
as distinct choices. With the data of Definition [fdip-005R],
\[\det (L_w+m^2I)
=\sum _{(F,R)}\left (\prod _{e\in F}w_e\right )m^{2|R|}.\]
Normalize these nonnegative terms to obtain a probability law on
rooted spanning forests. It satisfies
\[\begin {aligned}
\mathbb P(e\in F)&=w_ed_e^TGd_e,\\
\mathbb E[|R|]&=m^2\operatorname {tr}G,\\
\mathbb E[|F|]&=n-m^2\operatorname {tr}G.
\end {aligned}\]
If the positive-weight support is connected, with weighted
spanning-tree sum \(\tau _w\), then
\[\det (L_w+m^2I)=n\tau _wm^2+O(m^4)
\qquad (m\downarrow 0).\]
Thus the spanning-tree coefficient appears after division by
\(m^2\); the unscaled determinant tends to zero.
Equivalently, add a new boundary vertex \(*\) with an edge of
weight \(m^2\) to each old vertex. Attach \(*\) to the chosen root
of each forest component; this is a bijection to spanning trees
of the augmented graph. Its Dirichlet Laplacian at \(*\) is
\(L_w+m^2I\). This is the trivial-connection case of Kenyon's
boundary theorem: cycle factors vanish, leaving trees rooted at
the boundary [kenyon2011spanning, Theorem 7, Section 8, p. 15].
The conductance extension is given in
[kenyon2011spanning, Section 7.1, Theorem 5, p. 13].
The proof above includes the needed weights and zero-weight faces.
Chaiken's all-minors matrix-tree theorem enumerates directed
forests by the minors of a Laplacian whose columns sum to zero
[chaiken1982minors, pp. 319-329]. The expansion proved
above is the undirected weighted case, obtained from
Cauchy-Binet. It does not reproduce Chaiken's directed sign rule.
TauCeti's
Matrix.det_mul_transpose_eq_sum_plucker_sq
gives the Cauchy-Binet identity for a matrix over a commutative
ring with rows indexed by \(\mathrm {Fin}(n)\) and a finite ordered
column set. Reindexing the present real matrix supplies those
hypotheses. Its Plücker coordinates are the maximal minors.
This supports the sum-of-squares step; the classification of
nonzero forest minors and the probability consequences are
proved here.