Definition. The Gaussian induced action [fdip-005R]

✍️sourceAGENTDRAFTED

Fix a finite nonempty vertex set \(V\), with \(n=|V|\), and a finite set \(E\) of labelled edges with source and head maps. Orientations are auxiliary. In the counting metric on \(\mathbb R^V\), put \(d_e=\delta _{h(e)}-\delta _{s(e)}\). For weights \(w_e\ge 0\) and mass \(m>0\), write \[L_w=\sum _{e\in E}w_e d_ed_e^T,\qquad Q=L_w+m^2I,\qquad G=Q^{-1}.\] A zero weight means the corresponding edge is deleted. Parallel labels remain distinct, and a loop has \(d_e=0\). Thus the boundary faces of this weight orthant describe subgraphs, extending the positive-weight background convention.

The Gaussian induced action, relative to counting-metric Lebesgue measure, is \[\mathcal W(w,m)=\frac 12\log \det Q.\] The Gaussian normalizer gives \[Z(w,m)=\int _{\mathbb R^V}e^{-\phi ^TQ\phi /2}\,d\phi =(2\pi )^{n/2}(\det Q)^{-1/2},\] so \(-\log Z=\mathcal W-(n/2)\log (2\pi )\). This scalar is invariant under graph relabeling and orthogonal conjugation of \(Q\). It does not by itself retain all graph or selector data.

For fixed \(m>0\), differentiation in the weights means differentiation of the smooth extension to \[\mathcal U_m=\{w\in \mathbb R^E:L_w+m^2I>0\}.\] This open convex set contains the nonnegative orthant. For \(w_e\ge 0\) and \(x\ne 0\), indeed, \[x^TQx=\sum _e w_e(d_e^Tx)^2+m^2\|x\|^2>0.\] In particular the formulas below remain meaningful at deleted edges, even though only nonnegative weights are interpreted as graph conductances.