Example. Pinning and mean zero on two vertices [fdip-0036]
Example. Pinning and mean zero on two vertices [fdip-0036]
Join vertices \(0,1\) by one edge of weight \(w>0\). In counting coordinates its Laplacian is \[L=w\begin {pmatrix}1&-1\\-1&1\end {pmatrix}.\] The unit mean-zero vector \(u=(1,-1)/\sqrt 2\) has eigenvalue \(2w\). Thus the mean-zero law has \[\begin {aligned} Z_H&=\sqrt {2\pi }/\sqrt {2w},\\ L^+&=\frac 1{4w}\begin {pmatrix}1&-1\\-1&1\end {pmatrix}. \end {aligned}\] The factor \(1/\sqrt 2\) is required because the reference measure is the induced orthonormal volume on \(H\).
Pinning \(\phi _0=0\) leaves the unit vector \(\delta _1\); its action is \(w\phi _1^2/2\). The pinned law therefore has \[\begin {aligned} Z_F&=\sqrt {2\pi }/\sqrt w,\\ C_F&=\begin {pmatrix}0&0\\0&1/w\end {pmatrix}. \end {aligned}\] These ambient coordinate covariances differ. But for the difference \(\phi _0-\phi _1\), applying the two covariance forms to \((1,-1)\) gives \(1/w\) in both cases. This agreement of one observable does not identify the laws. For a single vertex the zero-pinned and mean-zero spaces are both zero, so that special case has no such distinction.
These are direct one-dimensional Gaussian calculations using the positive density in [friedli2017statistical, Proposition B.55, p. 505] and the two explicitly specified induced volumes.