Where can one learn about the cluster expansion without a background in statistical mechanics?

Question

Apologies if this is better suited for mathSE. I would like to learn about the cluster expansion as is used in rigorous quantum field theory. I have read a lot on QFT, both from a physics and mathematical point of view, but I have not studied any statistical mechanics. Most of the resources I have tried either assume some familiarity with these expansion methods or with statistical mechanics.

Answer 1

The QFT cluster expansion is a bit different from the one used in statistical mechanics. The main use is to rigorously control the infinite volume limit $\Lambda\nearrow\mathbb{Z}^d$ for the pressure $\frac{\log Z(\Lambda)}{|\Lambda|}$ or correlation functions, and get convergent series expansions for these quantities. By working a bit harder one can also prove Borel summability of Feynman diagrammatic perturbation theory in infinite volume, yet with both UV (lattice) and IR (massive case) cutoffs. The key example to understand is say the massive $\phi^4$ model on the unit lattice $\mathbb{Z}^d$.

The procedure is in two steps.

1) Decoupling expansion: It is to write the finite volume partition function in the form $$ Z(\Lambda)=\sum_{\Pi\in{\rm Part}(\Lambda)} \prod_{A\in \Pi} c(A) $$ where ${\rm Part}(\Lambda)$ is the set of set partitions of the finite subset $\Lambda$ of $\mathbb{Z}^d$. Equivalently, one must write $$ Z(\Lambda)=\sum_{n\ge 0}\frac{1}{n!}\sum_{(A_1,\ldots,A_n)}c(A_1)\cdots c(A_n) $$ where one sums over ordered tuples of nonempty, pairwise disjoint subsets of $\Lambda$, with union equal to $\Lambda$. As a result we have a polymer gas representation $$ Z(\Lambda)=\prod_{x\in\Lambda} c\left(\{x\}\right) \times \sum_{n\ge 0}\frac{1}{n!}\sum_{(A_1,\ldots,A_n)}z(A_1)\cdots z(A_n) $$ where, in the sum over $(A_1,\ldots,A_n)$, we drop the requirement that $\cup A_i=\Lambda$ and add the one that the $A_i$ are not singletons. The new polymer activities $z$ are defined by $$ z(A)=\frac{c(A)}{\prod_{x\in A}c(\{x\})}\ . $$

2) The Mayer expansion for the polymer gas: We compute the logarithm of the partition function, thanks to the previous representation. Namely, $$ \log Z(\Lambda)=\sum_{x\in\Lambda}\log c(\{x\}) +\sum_{n\ge 1}\frac{1}{n!}\sum_{(A_1,\ldots,A_n)}\psi(A_1,\ldots,A_n)z(A_1)\cdots z(A_n) $$ where $\psi(A_1,\ldots,A_n)\in\mathbb{Z}$ is a suitable combinatorial coefficient. The $A_i$ now are not required to be disjoint. It is essential for convergence to prove that $\psi(A_1,\ldots,A_n)$ is bounded by the number of spanning trees with edges corresponding to overlaps $A_i\cap A_j\neq\varnothing$.

For both steps one needs a tree or forest interpolation formula. The simplest available is the so-called BKAR formula. Hence the first thing one can do to learn about cluster expansions in QFT would be to study the notes I wrote with a proof of this formula

https://mabdesselam.github.io/BKAR.pdf

You can then read

https://mabdesselam.github.io/polymergas.pdf

which explains how to use the BKAR formula in order to do the second step mentioned above, i.e., the Mayer expansion.

Finally, you can learn about the first step and how to put everything together in Section 4 of the article https://link.springer.com/article/10.1007/s10955-009-9789-y The article treats a complex Bosonic $\phi^4$ model, but it is easy to adapt it to the more common real $\phi^4$ model. The preprint version (probably with more typos) is available at https://arxiv.org/abs/0901.4756