Definition. Massive graph Green functions [fdip-000A]

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Let \(V\) be a finite vertex set, with positive edge weights, and let \(L\) be its weighted graph Laplacian. For \(m>0\), define the massive precision and its Green operator by \[Q_m=L+m^2I,\qquad G_m=Q_m^{-1}.\] The inverse exists even when the graph is disconnected. Indeed, the energy identity gives, for \(f\ne 0\), \[\begin {aligned} \langle f,Q_mf\rangle _V &=\sum _e w(e)(Df)(e)^2\\ &\quad +m^2\sum _v f(v)^2>0. \end {aligned}\] Thus \(Q_mf=0\) implies \(f=0\). Finite dimension and rank-nullity make this injective endomorphism surjective, so it is an isomorphism. The inverse-response lemma now gives a self-adjoint inverse and the unique solution \(f=G_mj\) to \(Q_mf=j\). The endomorphism \(Q_m\) is a positive precision parameter; its counting array realizes it as a positive symmetric matrix. For \(j\ne 0\), write \(x=G_mj\ne 0\); then \(\langle j,G_mj\rangle _V=\langle Q_mx,x\rangle _V>0\). On an empty vertex set the same map argument gives the unique isomorphism of zero modules; there are no point-source columns.

By the real spectral theorem, choose an orthonormal eigenbasis \(u_a\) of the self-adjoint operator \(L\), with eigenvalues \(\lambda _a\ge 0\). The energy identity proves this last inequality by evaluation on each unit eigenvector. Put \(q_a=\lambda _a+m^2>0\). Then \[\begin {aligned} Q_m&=\sum _a q_a u_au_a^T,\\ G_m&=\sum _a q_a^{-1}u_au_a^T. \end {aligned}\] Since \(u_a^Tu_b=\delta _{ab}\) and \(\sum _a u_au_a^T=I\), multiplication gives \(Q_mG_m=G_mQ_m=I\). Also \(G_m^T=G_m\), and \(f^TG_mf=\sum _a q_a^{-1}(u_a^Tf)^2>0\) for nonzero \(f\).

The Green kernel is defined from the inverse map by \(G_m(v,w)=(G_m\delta _w)(v)\). These are its matrix entries in the counting coordinate basis. Its column at \(w\) is the unique field solving \[Q_mG_m(\cdot ,w)=\delta _w,\] where \(\delta _w(v)\) is one at \(v=w\) and zero elsewhere. For a source field \(j\), the unique solution of \(Q_mf=j\) is \(f(v)=\sum _wG_m(v,w)j(w)\). No vertex-volume factor occurs because the vertex pairing and basis are counting.

This convention has to be matched when comparing lattice Green functions. Wingate's lattice action uses a positive operator built from minus the backward-forward difference and a mass term [wingate2025lattice, Section 4, Equations (4.2)-(4.5), p. 3]. For unit spacing, that difference term has the sign of our \(L\).

In [friedli2017statistical, Theorem 8.26, p. 399], the massive lattice precision is \[Q=-\frac {1}{2d}\Delta +m^2I.\] Here \(\Delta f(x)=\sum _{y\sim x}(f(y)-f(x))\). The factor \(1/(2d)\) and the lattice boundary conditions belong to that normalization. Identifying their covariance with an inverse requires that precision, not an unscaled adjacency Laplacian.

Mathlib's Matrix.PosDef.isUnit proves invertibility of a positive definite matrix, and Matrix.PosDef.inv proves positivity of its inverse. These apply to the matrix step after the graph-energy calculation supplies positivity. The kernel convention and its graph interpretation are specified here.