Definition. The Gaussian partition function and probability law [fdip-0005]
AGENTDRAFTED
Let \(Q\) be a positive precision parameter on a
finite real inner-product space \((E,h)\), with canonical volume
\(\lambda _E\). Its Gaussian partition function is
\[Z_Q=\int _E w_Q\,d\lambda _E,
\qquad w_Q(x)=e^{-h(x,Qx)/2}.\]
The normalization theorem proves
\[0<Z_Q=\frac {(2\pi )^{n/2}}{\sqrt {\det Q}}<\infty ,
\qquad n=\dim E.\]
Its Gaussian probability law is the Borel measure
\[\mu _Q(A)=Z_Q^{-1}\int _A w_Q\,d\lambda _E.\]
In the counting field specialization this is exactly the density
\(Z_Q^{-1}\exp (-\phi ^TQ\phi /2)\) against the product reference measure.
In an orthonormal eigenchart the density is the product of the centered
scalar densities \(I(q_i)^{-1}e^{-q_i y_i^2/2}\). This identification
uses the scalar value and normalization, before computing covariance.
For \(n=0\), the action is zero at the unique point, \(Z_Q=1\), and
\(\mu _Q=\delta _0\). Replacing the reference volume by \(c\lambda _E\),
\(0<c<\infty \), multiplies \(Z_Q\) by \(c\) and leaves \(\mu _Q\)
unchanged, as in the reference-rescaling example.
A finite-sum partition function has the same
normalizing role. A finite vertex set gives a finite-dimensional configuration space.
Integrability of its Gaussian weight follows from the normalization theorem,
using positive precision.
The density convention is compared with
[friedli2017statistical, Lemma B.54 and Proposition B.55, pp. 504-505];
that source uses covariance rather than precision. Countable additivity
uses [tao2011measure, Theorem 1.4.44, pp. 107-108, preliminary version].
The normalization failures distinguish infinite normalizers,
zero weights and the zero-dimensional case.