Proposition. Projected Gaussian laws and the massless limit [fdip-0034]
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  • October 29, 2026

For the kernel decomposition, let \(\nu _m\) be the projection onto \(H\) of the law with precision \(L+m^2I\), \(m>0\). Then \(\nu _m\) converges to the constrained law \(\nu _H\) in total variation, meaning \(\sup _{A\text { Borel}}|\nu _m(A)-\nu _H(A)|\to 0\). For every fixed polynomial \(P\) on \(H\), its expectations converge too.

Proof. 

  • October 29, 2026

By kernel factorization, the densities on \(H\) are \(f_m=w_m/Z_{H,m}\), where \[\begin {aligned} w_m(x)&=e^{-h(x,(L_H+m^2I_H)x)/2},\\ w_0(x)&=e^{-h(x,L_Hx)/2}. \end {aligned}\] They are continuous, \(w_m\to w_0\) pointwise, and \(0\leq w_m\leq w_0\), with \(\int _H w_0=Z_H<\infty \). Dominated convergence gives \(Z_{H,m}\to Z_H>0\). For sufficiently small \(m\), \(Z_{H,m}\geq Z_H/2\); hence \[|f_m-f_0|\leq 3w_0/Z_H.\] Another application gives \(\int _H|f_m-f_0|\to 0\). For each Borel \(A\), \(|\nu _m(A)-\nu _H(A)|\leq \int _H|f_m-f_0|\), proving the uniform claim. For a fixed polynomial, the dominating function \(3|P|w_0/Z_H\) is integrable by finite polynomial moments, so \(\int P f_m\to \int P f_0\). Each use of dominated convergence applies along every positive sequence \(m_j\to 0\); the sequential criterion gives the real-parameter limits as \(m\downarrow 0\). When \(H=0\), these laws are all the same point mass.

For a fixed source \(j\in E\) and \(\ell _j(x)=h(j,x)\), the inverse-covariance formula and the kernel split give \[\begin {aligned} \operatorname {Var}_{Q_m}(\ell _j) &=m^{-2}\|P_Kj\|^2\\ &\quad +\sum _{\lambda _a>0} \frac {h(j,u_a)^2}{\lambda _a+m^2}. \end {aligned}\] Every term is nonnegative. The finite sum converges; the first term is finite in the limit precisely when \(P_Kj=0\). Thus the variance has a finite limit if and only if \(j\in H\), in which case it is \(h(j,L^+j)\).

For a connected nonempty graph with positive edge weights, put \(n=|V|\). The source \(u_0=\mathbf 1_V/\sqrt n\) has variance \(m^{-2}\) as a source. Its contribution to every Green-kernel entry is \(1/(nm^2)\), and \(G_m(v,v)\geq 1/(nm^2)\). A field difference has source \(\delta _v-\delta _w\in H\), so its variance converges to the constrained variance. On a disconnected graph this holds for vertices in the same component; a difference between components has a nonzero kernel projection. These observable limits do not supply a normalized full-space massless density.

The analytic ingredient is [tao2011measure, Theorem 1.4.49, pp. 111-112, preliminary version]. The three dominators above verify its hypotheses for these Gaussian laws and their fixed polynomial observables.