Functional Integral in Statistical Mechanics

Question

In this work, the author state that many problems in statistical mechanics center on the analysis of functional integrals of the form: \begin{equation} Z(\varphi') = \int d\mu(\varphi) e^{-V(\varphi+\varphi')} \equiv (\mu*e^{-V})(\varphi'), \tag{1}\label{1} \end{equation} where $\varphi = (\varphi_{x})_{x\in \Lambda}$ is a Gaussian process indexed by some finite set $\Lambda$ with joint probability measure $\mu$, mean zero and covariance $C_{xy}$.

I know that the Sine-Gordon transformation for systems interacting via two-body potential allows us to write the Grand Canonical partition function of the associate gas in the form of (\ref{1}). But I'd like to know where else does the functional integral (\ref{1}) take place in statistical mechanics. In what models we can express the partition function in terms of such an integral?

Answer 1

Many models of statistical mechaincs such as the Ising model are believed to be in the same universality class as models obtained as perturbations of Gaussian models (the search keyword is "Ginzburg-Landau"). The latter involve integrals of the form (1). Also the above integral is one of the main ways to define a renormalization group transformation: $\varphi'$ would the fixed low momentum field, while $\varphi$ is the high momentum field that gets integrated as part of a coarse-graining procedure.

You can find more information on the RG part in my two answers:

https://mathoverflow.net/questions/62770/what-mathematical-treatment-is-there-on-the-renormalization-group-flow-in-a-spac/63089#63089

and

What is the Wilsonian definition of renormalizability?