Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-22T20:58:43.460Z Has data issue: false hasContentIssue false

The Cluster Expansion in Combinatorics

Published online by Cambridge University Press:  23 May 2024

By
Matthew Jenssen
Edited by
Felix Fischer  and
Robert Johnson
Felix Fischer
Affiliation:
Queen Mary University of London
Robert Johnson
Affiliation:
Queen Mary University of London
Get access

Summary

The cluster expansion is a classical tool from statistical physics used to study the phase diagram of interacting particle systems. Recently, the cluster expansion has seen a number of applications in combinatorics and the field of approximate counting/sampling. In this article, we give an introduction to the cluster expansion and survey some of these recent developments.

Type
Chapter
Information
Surveys in Combinatorics 2024 , pp. 55 - 88
Publisher: Cambridge University Press
Print publication year: 2024

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Balogh, József, Garcia, Ramon I, and Li, Lina, Independent sets in the middle two layers of boolean lattice, Journal of Combinatorial Theory, Series A 178 (2021), 105341.CrossRefGoogle Scholar
Barvinok, Alexander, Computing the permanent of (some) complex matrices, Foundations of Computational Mathematics 16 (2016), no. 2, 329–342.CrossRefGoogle Scholar
Barvinok, Alexander, Combinatorics and complexity of partition functions, Algorithms and Combinatorics, vol. 30, Springer, 2017.Google Scholar
Björklund, Andreas, Husfeldt, Thore, Kaski, Petteri, and Koivisto, Mikko, Computing the Tutte polynomial in vertex-exponential time, Proceedings of the Forty-ninth Annual Symposium on Foundations of Computer Science, FOCS 2008, IEEE, 2008, pp. 677–686.Google Scholar
Borgs, Christian, Chayes, Jennifer, Helmuth, Tyler, Perkins, Will, and Tetali, Prasad, Efficient sampling and counting algorithms for the potts model on Fd at all temperatures, Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, 2020, pp. 738–751.CrossRefGoogle Scholar
Cai, Jin-Yi, Galanis, Andreas, Goldberg, Leslie Ann, Guo, Heng, Jerrum, Mark, Štefankovič, Daniel, and Vigoda, Eric, # BIS-hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness region, Journal of Computer and System Sciences 82 (2016), no. 5, 690–711.CrossRefGoogle Scholar
Cai, Jin-Yi and Liu, Tianyu, An fptas for the square lattice six-vertex and eight-vertex models at low temperatures, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2021, pp. 1520–1534.CrossRefGoogle Scholar
Cannon, Sarah and Perkins, Will, Counting independent sets in unbalanced bipartite graphs, Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2020, pp. 1456–1466.CrossRefGoogle Scholar
Carlson, Charles, Davies, Ewan, and Kolla, Alexandra, Efficient algorithms for the potts model on small-set expanders, arXiv preprint arXiv:2003.01154 (2020).Google Scholar
Carlson, Charlie, Davies, Ewan, Fraiman, Nicolas, Kolla, Alexandra, Potukuchi, Aditya, and Yap, Corrine, Algorithms for the ferromagnetic potts model on expanders, 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), IEEE, 2022, pp. 344–355.CrossRefGoogle Scholar
Chebolu, Prasad, Goldberg, Leslie Ann, and Martin, Russell, The complexity of approximately counting stable matchings, Theoretical Computer Science 437 (2012), 35–68.CrossRefGoogle Scholar
Chen, Zongchen, Galanis, Andreas, Leslie A Goldberg, Will Perkins, James Stewart, and Eric Vigoda, Fast algorithms at low temperatures via markov chains, Random Structures & Algorithms 58 (2021), no. 2, 294–321.CrossRefGoogle Scholar
Davies, Ewan, Jenssen, Matthew, and Perkins, Will, A proof of the upper matching conjecture for large graphs, Journal of Combinatorial Theory, Series B 151 (2021), 393–416.CrossRefGoogle Scholar
Boer, David de, Buys, Pjotr, Peters, Han, and Regts, Guus, On boundedness of zeros of the independence polynomial of tor, arXiv preprint arXiv:2306.12934 (2023).Google Scholar
Dobrushin, RL, Perturbation methods of the theory of gibbsian fields, Lectures on Probability Theory and Statistics: Ecole d’Eté de Probabilités de Saint-Flour XXIV—1994 (1996), 1–66.CrossRefGoogle Scholar
Dobrushin, Roland L, Estimates of semi-invariants for the ising model at low temperatures, Translations of the American Mathematical Society-Series 2 177 (1996), 59–82.CrossRefGoogle Scholar
Dyer, Martin, Goldberg, Leslie Ann, Greenhill, Catherine, and Jerrum, Mark, The relative complexity of approximate counting problems, Algorithmica 38 (2004), no. 3, 471–500.CrossRefGoogle Scholar
Engbers, John and Galvin, David, H-coloring tori, Journal of Combinatorial Theory, Series B 102 (2012), no. 5, 1110–1133.CrossRefGoogle Scholar
Erdős, Paul and Lovász, László, Problems and results on 3-chromatic hypergraphs and some related questions, Infinite and finite sets 10 (1975), no. 2, 609–627.Google Scholar
Erdös, P., Kleitman, D.J., and Rothschild, B., Asymptotic enumeration of Kn-free graphs, Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973) (1973), no. 17, 19–27.Google Scholar
Erdős, Paul and Rényi, Alfréd, On the evolution of random graphs, Publ. Math. Inst. Hung. Acad. Sci 5 (1960), no. 1, 17–60.Google Scholar
Galanis, Andreas, Ge, Qi, Štefankovič, Daniel, Vigoda, Eric, and Yang, Linji, Improved inapproximability results for counting independent sets in the hardcore model, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, Springer, 2011, pp. 567–578.CrossRefGoogle Scholar
Galanis, Andreas, Goldberg, Leslie Ann, and Stewart, James, Fast algorithms for general spin systems on bipartite expanders, ACM Transactions on Computation Theory (TOCT) 13 (2021), no. 4, 1–18.CrossRefGoogle Scholar
Galanis, Andreas, Stefankovic, Daniel, Vigoda, Eric, and Yang, Linji, Ferromagnetic Potts model: Refined #-BIS-hardness and related results, SIAM Journal on Computing 45 (2016), no. 6, 2004–2065.CrossRefGoogle Scholar
Galvin, David, On homomorphisms from the hamming cube to Z, Israel Journal of Mathematics 138 (2003), no. 1, 189–213.CrossRefGoogle Scholar
Galvin, David, A threshold phenomenon for random independent sets in the discrete hypercube, Combinatorics, Probability and Computing 20 (2011), no. 1, 27–51.CrossRefGoogle Scholar
Galvin, David, The independent set sequence of regular bipartite graphs, Discrete Mathematics 312 (2012), no. 19, 2881–2892.CrossRefGoogle Scholar
Galvin, David, Independent sets in the discrete hypercube, arXiv preprint arXiv:1901.01991 (2019).Google Scholar
Goldberg, Leslie Ann and Jerrum, Mark, Approximating the partition function of the ferromagnetic Potts model, Journal of the ACM (JACM) 59 (2012), no. 5, 25.CrossRefGoogle Scholar
Groeneveld, J, Two theorems on classical many-particle systems, Phys. Letters 3 (1962).CrossRefGoogle Scholar
Helmuth, Tyler, Jenssen, Matthew, and Perkins, Will, Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs, Annales de l’Institut Henri Poincare (B) Probabilites et statistiques 59 (2023), no. 2, 817–848.Google Scholar
Helmuth, Tyler, Perkins, Will, and Regts, Guus, Algorithmic Pirogov-Sinai theory, Probability Theory and Related Fields 176 (2020), 851–895.CrossRefGoogle Scholar
Janson, S., Luczak, T., and Rucinski, A., An exponential bound for the probability of nonexistence of a specified subgraph in a random graph, Random graphs’ 87, Proceedings, Poznán, 1987, eds. Karonn’ski, M., Jaworski, J. and Ruciński, A., 73–87.Google Scholar
Janson, Svante, Luczak, Tomasz, and Rucinski, Andrzej, An exponential bound for the probability of nonexistence of a specified subgraph in a random graph, Institute for Mathematics and its Applications (USA), 1988.Google Scholar
Jenssen, Matthew and Keevash, Peter, Homomorphisms from the torus, arXiv preprint arXiv:2009.08315 (2020).Google Scholar
Jenssen, Matthew, Keevash, Peter, and Perkins, Will, Algorithms for #BIS-hard problems on expander graphs, SIAM Journal on Computing 49 (2020), no. 4, 681–710.CrossRefGoogle Scholar
Jenssen, Matthew and Perkins, Will, Independent sets in the hypercube revisited, Journal of the London Mathematical Society 102 (2020), no. 2, 645–669.CrossRefGoogle Scholar
Jenssen, Matthew, Perkins, Will, and Potukuchi, Aditya, On the evolution of structure in triangle-free graphs, In preparation.Google Scholar
Jenssen, Matthew, Perkins, Will, and Potukuchi, Aditya, Independent sets of a given size and structure in the hypercube, Combinatorics, Probability and Computing 31 (2022), no. 4, 702–720.CrossRefGoogle Scholar
Jenssen, Matthew, Perkins, Will, and Potukuchi, Aditya, Approximately counting independent sets in bipartite graphs via graph containers, Random Structures & Algorithms (2023).CrossRefGoogle Scholar
Kahn, Jeff, An entropy approach to the hard-core model on bipartite graphs, Combinatorics, Probability and Computing 10 (2001), no. 3, 219–237.CrossRefGoogle Scholar
Kahn, Jeff and Park, Jinyoung, The number of 4-colorings of the hamming cube, Israel Journal of Mathematics 236 (2020), no. 2, 629–649.CrossRefGoogle Scholar
Korshunov, AD and Sapozhenko, AA, The number of binary codes with distance 2, Problemy Kibernet 40 (1983), 111–130.Google Scholar
Kotecký, Roman and Preiss, David, Cluster expansion for abstract polymer models, Communications in Mathematical Physics 103 (1986), no. 3, 491–498.CrossRefGoogle Scholar
Li, Lina, McKinley, Gweneth, and Park, Jinyoung, The number of colorings of the middle layers of the hamming cube, arXiv preprint arXiv:2304.03203 (2023).Google Scholar
Liao, Chao, Lin, Jiabao, Lu, Pinyan, and Mao, Zhenyu, Counting independent sets and colorings on random regular bipartite graphs, arXiv preprint arXiv:1903.07531 (2019).Google Scholar
Luczak, Tomasz, On triangle-free random graphs, Random Structures & Algorithms 16 (2000), no. 3, 260–276.3.0.CO;2-Q>CrossRefGoogle Scholar
Mantel, Willem, Problem 28, Wiskundige Opgaven 10 (1907), no. 60-61, 320.Google Scholar
Mayer, Joseph E, The statistical mechanics of condensing systems. i, The Journal of Chemical Physics 5 (1937), no. 1, 67–73.CrossRefGoogle Scholar
Mayer, Joseph E and Montroll, Elliott, Molecular distribution, The Journal of Chemical Physics 9 (1941), no. 1, 2–16.CrossRefGoogle Scholar
Mayer, Joseph Edward and Mayer, Maria Goeppert, Statistical mechanics, vol. 28, John Wiley & Sons New York, 1940.Google Scholar
Mousset, Frank, Noever, Andreas, Panagiotou, Konstantinos, and Samotij, Wojciech, On the probability of nonexistence in binomial subsets, The Annals of Probability 48 (2020), no. 1, 493–525.CrossRefGoogle Scholar
Osthus, Deryk, Hans Jürgen Prömel, and Anusch Taraz, For which densities are random triangle-free graphs almost surely bipartite?, Combinatorica 23 (2003), no. 1, 105–150.CrossRefGoogle Scholar
Patel, Viresh and Regts, Guus, Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials, SIAM Journal on Computing 46 (2017), no. 6, 1893–1919.CrossRefGoogle Scholar
Patel, Viresh and Regts, Guus, Computing the number of induced copies of a fixed graph in a bounded degree graph, Algorithmica 81 (2019), 1844–1858.CrossRefGoogle Scholar
Patel, Viresh and Regts, Guus, Approximate counting using taylor’s theorem: a survey, arXiv preprint arXiv:2212.08143 (2022).Google Scholar
Penrose, Oliver, Convergence of fugacity expansions for fluids and lattice gases, Journal of Mathematical Physics 4 (1963), no. 10, 1312–1320.CrossRefGoogle Scholar
Peters, Han and Regts, Guus, On a conjecture of Sokal concerning roots of the independence polynomial, Michigan Mathematical Journal 68 (2019), no. 1, 33–55.CrossRefGoogle Scholar
Prömel, Hans Jürgen and Steger, Angelika, Counting H-free graphs, Discrete Mathematics 154 (1996), no. 1-3, 311–315.CrossRefGoogle Scholar
Ruelle, David, Correlation functions of classical gases, Annals of Physics 25 (1963), no. 1, 109–120.CrossRefGoogle Scholar
Sapozhenko, AA, On the number of connected subsets with given cardinality of the boundary in bipartite graphs, Metody Diskret Analiz 45 (1987), 42–70.Google Scholar
Scott, Alexander D and Sokal, Alan D, The repulsive lattice gas, the independentset polynomial, and the lovász local lemma, Journal of Statistical Physics 118 (2005), 1151–1261.CrossRefGoogle Scholar
Scott, Alexander D and Sokal, Alan D, On dependency graphs and the lattice gas, Combinatorics, Probability and Computing 15 (2006), no. 1-2, 253–279.CrossRefGoogle Scholar
Shearer, James B., On a problem of spencer, Combinatorica 5 (1985), 241–245.CrossRefGoogle Scholar
Sly, Allan, Computational transition at the uniqueness threshold, Proceedings of the Fifty-first Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, IEEE, 2010, pp. 287–296.Google Scholar
Sly, Allan and Sun, Nike, Counting in two-spin models on d-regular graphs, The Annals of Probability 42 (2014), no. 6, 2383–2416.CrossRefGoogle Scholar
Spitzer, Frank, Markov random fields on an infinite tree, The Annals of Probability 3 (1975), no. 3, 387–398.CrossRefGoogle Scholar
Stark, Dudley and Wormald, Nick, The probability of non-existence of a subgraph in a moderately sparse random graph, Combinatorics, Probability and Computing 27 (2018), no. 4, 672–715.CrossRefGoogle Scholar
Weitz, Dror, Counting independent sets up to the tree threshold, Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, STOC 2006, ACM, 2006, pp. 140–149.Google Scholar
Wormald, Nicholas C, The perturbation method and triangle-free random graphs, Random Structures & Algorithms 9 (1996), no. 1-2, 253–270.3.0.CO;2-O>CrossRefGoogle Scholar
Yang, Chen-Ning and Lee, Tsung-Dao, Statistical theory of equations of state and phase transitions. i. theory of condensation, Physical Review 87 (1952), no. 3, 404.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×