What's the current state of cluster expansions? A classic reference on cluster expansions in mathematical physics (specially statistical mechanics) is these lecture notes by professor Brydges for a les Houches course in 1984 on the mentioned topic. However, these lectures took place over 30 years ago and I wonder if the material is outdated by now. Is it still the best place to study cluster expansions in statistical mechanics? How did the results and/or techniques used in his lectures changed by now? Are there more modern references on this subject?
 A: An impressive result of the method of cluster expansion is to derive the van der Waals equation of state for gases. Indeed, two-body interactions give the corrections to the ideal gas law. So it can be used for practical calculations!
But as Abdelmalek has remarked, the method is often used in mathematical physics to prove results. Connections with combinatorics have also been pointed recently, including the related viral expansion.
Below are some recent references; the list is not exhaustive! Notice that the method is well explained in the recent book of Friedli and Velenik (Statistical mechanics of lattice systems. A concrete mathematical introduction. Cambridge University Press, Cambridge, 2018. xix+622 pp. ISBN: 978-1-107-18482-4).
Nguyen, Tong Xuan; Fernández, Roberto Convergence of cluster and virial expansions for repulsive classical gases. J. Stat. Phys. 179 (2020), no. 2, 448–484.
Jansen, Sabine; Tsagkarogiannis, Dimitrios Cluster expansions with renormalized activities and applications to colloids. Ann. Henri Poincaré 21 (2020), no. 1, 45–79.
Jansen, S. Cluster expansions for Gibbs point processes. Adv. in Appl. Probab. 51 (2019), no. 4, 1129–1178.
Pulvirenti, Elena; Tsagkarogiannis, Dimitrios Finite volume corrections and decay of correlations in the canonical ensemble. J. Stat. Phys. 159 (2015), no. 5, 1017–1039.
Jansen, Sabine; Tate, Stephen J.; Tsagkarogiannis, Dimitrios; Ueltschi, Daniel Multispecies virial expansions. Comm. Math. Phys. 330 (2014), no. 2, 801–817.
Morais, Thiago; Procacci, Aldo Continuous particles in the canonical ensemble as an abstract polymer gas. J. Stat. Phys. 151 (2013), no. 5, 830–849.
Faris, William G. Combinatorics and cluster expansions. Probab. Surv. 7 (2010), 157–206.
Poghosyan, Suren; Ueltschi, Daniel Abstract cluster expansion with applications to statistical mechanical systems. J. Math. Phys. 50 (2009), no. 5, 053509, 17 pp.
Faris, William G. A connected graph identity and convergence of cluster expansions. J. Math. Phys. 49 (2008), no. 11, 113302, 14 pp.
Fernández, Roberto; Procacci, Aldo Cluster expansion for abstract polymer models. New bounds from an old approach. Comm. Math. Phys. 274 (2007), no. 1, 123–140.
A: Let me add to Daniel's list a couple links to some hopefully pedagogical notes I wrote for a course I taught a while ago.

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* Notes on the cluster expansion
for the polymer gas, a.k.a., the Mayer expansion.


* Notes on cumulants and Moebius functions.


* Notes on the Brydges-Kennedy-Abdesselam-Rivasseau forest
interpolation formula.
For two hopefully instructive examples of applications to seemingly unrelated problems in many-body quantum mechanics and nonrelativistic QFT, see

*

*A. A., "The Ground State Energy of The Massless Spin-Boson Model".


*A. A. and D. Hasler, "Analyticity of The Ground State Energy For Massless Nelson Models".
