Combinatorics and physics

Combinatorial physics or physical combinatorics is the area of interaction between physics and combinatorics.

## Contents

* 1 Overview
* 2 See also
* 3 References
* 4 Further reading
* 4.1 Combinatorics and statistical physics
* 4.2 Conference proceedings

## Overview

"Combinatorial Physics is an emerging area which unites combinatorial and discrete mathematical techniques applied to theoretical physics, especially Quantum Theory."[1]
"Physical combinatorics might be defined naively as combinatorics guided by ideas or insights from physics"[2]

Combinatorics has always played an important role in quantum field theory and statistical physics.[3] However, combinatorial physics only emerged as a specific field after a seminal work by Alain Connes and Dirk Kreimer,[4] showing that the renormalization of Feynman diagrams can be described by a Hopf algebra.

Combinatorial physics can be characterized by the use of algebraic concepts to interpret and solve physical problems involving combinatorics. It gives rise to a particularly harmonious collaboration between mathematicians and physicists.

Among the significant physical results of combinatorial physics, we may mention the reinterpretation of renormalization as a Riemann–Hilbert problem,[5] the fact that the Slavnov–Taylor identities of gauge theories generate a Hopf ideal,[6] the quantization of fields[7] and strings,[8] and a completely algebraic description of the combinatorics of quantum field theory.[9] An important example of applying combinatorics to physics is the enumeration of alternating sign matrix in the solution of ice-type models. The corresponding ice-type model is the six vertex model with domain wall boundary conditions.

## See also

* Mathematical physics
* Statistical physics
* Ising model
* Percolation theory
* Tutte polynomial
* Partition function
* Hopf algebra
* Combinatorics and dynamical systems
* Quantum mechanics

## References

1. ↑ 2007 International Conference on Combinatorial physics
2. ↑ Physical Combinatorics, Masaki Kashiwara, Tetsuji Miwa, Springer, 2000, ISBN 0-8176-4175-0
3. ↑ David Ruelle (1999). Statistical Mechanics, Rigorous Results. World Scientific. ISBN 978-981-02-3862-9.
4. ↑ A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem I, Commun. Math. Phys. 210 (2000), 249-273
5. ↑ A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem II, Commun. Math. Phys. 216 (2001), 215-241
6. ↑ W. D. van Suijlekom, Renormalization of gauge fields: A Hopf algebra approach, Commun. Math. Phys. 276 (2007), 773-798
7. ↑ C. Brouder, B. Fauser, A. Frabetti, R. Oeckl, Quantum field theory and Hopf algebra cohomology, J. Phys. A: Math. Gen. 37 (2004), 5895-5927
8. ↑ T. Asakawa, M. Mori, S. Watamura, Hopf Algebra Symmetry and String Theory, Prog. Theor. Phys. 120 (2008), 659-689
9. ↑ C. Brouder, Quantum field theory meets Hopf algebra, Mathematische Nachrichten 282 (2009), 1664-1690

## Further reading

* Some Open Problems in Combinatorial Physics, G. Duchamp, H. Cheballah
* One-parameter groups and combinatorial physics, G. Duchamp, K.A. Penson, A.I. Solomon, A.Horzela, P.Blasiak
* Combinatorial Physics, Normal Order and Model Feynman Graphs, A.I. Solomon, P. Blasiak, G. Duchamp, A. Horzela, K.A. Penson
* Hopf Algebras in General and in Combinatorial Physics: a practical introduction, G. Duchamp, P. Blasiak, A. Horzela, K.A. Penson, A.I. Solomon
* Discrete and Combinatorial Physics
* Bit-String Physics: a Novel "Theory of Everything", H. Pierre Noyes
* Combinatorial Physics, Ted Bastin, Clive W. Kilmister, World Scientific, 1995, ISBN 981-02-2212-2
* Physical Combinatorics and Quasiparticles, Giovanni Feverati, Paul A. Pearce, Nicholas S. Witte
* Fitzgerald, Hannah. "Physical Combinatorics of Non-Unitary Minimal Models". http://www.ms.unimelb.edu.au/publications/HonoursTheses-HannahFitzgerald.pdf. Retrieved 17 August 2014.
* Paths, Crystals and Fermionic Formulae, G.Hatayama, A.Kuniba, M.Okado, T.Takagi, Z.Tsuboi
* On powers of Stirling matrices, István Mező
* "On cluster expansions in graph theory and physics", N BIGGS — The Quarterly Journal of Mathematics, 1978 - Oxford Univ Press
* Enumeration Of Rational Curves Via Torus Actions, Maxim Kontsevich, 1995
* Non-commutative Calculus and Discrete Physics, Louis H. Kauffman, February 1, 2008
* Sequential cavity method for computing free energy and surface pressure, David Gamarnik, Dmitriy Katz, July 9, 2008

### Combinatorics and statistical physics

* "Graph Theory and Statistical Physics", J.W. Essam, Discrete Mathematics, 1, 83-112 (1971).
* Combinatorics In Statistical Physics
* Hard Constraints and the Bethe Lattice: Adventures at the Interface of Combinatorics and Statistical Physics, Graham Brightwell, Peter Winkler
* Graphs, Morphisms, and Statistical Physics: DIMACS Workshop Graphs, Morphisms and Statistical Physics, March 19-21, 2001, DIMACS Center, Jaroslav Nešetřil, Peter Winkler, AMS Bookstore, 2001, ISBN 0-8218-3551-3

### Conference proceedings

* Proc. of Combinatorics and Physics, Los Alamos, August 1998
* Physics and Combinatorics 1999: Proceedings of the Nagoya 1999 International Workshop, Anatol N. Kirillov, Akihiro Tsuchiya, Hiroshi Umemura, World Scientific, 2001, ISBN 981-02-4578-5
* Physics and combinatorics 2000: proceedings of the Nagoya 2000 International Workshop, Anatol N. Kirillov, Nadejda Liskova, World Scientific, 2001, ISBN 981-02-4642-0
* Asymptotic combinatorics with applications to mathematical physics: a European mathematical summer school held at the Euler Institute, St. Petersburg, Russia, July 9-20, 2001, Anatoliĭ, Moiseevich Vershik, Springer, 2002, ISBN 3-540-40312-4
* Counting Complexity: An International Workshop On Statistical Mechanics And Combinatorics, 10–15 July 2005, Dunk Island, Queensland, Australia
* Proceedings of the Conference on Combinatorics and Physics, MPIM Bonn, March 19–23, 2007

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