×

zbMATH — the first resource for mathematics

Non-backtracking loop soups and statistical mechanics on spin networks. (English) Zbl 1366.82020
The authors define and study a Markov field of which the free energy density can be computed exactly in any dimension, which turns to be a problem related to a random-loop model referred to as loop soup. The problem is tackled by introducing a new ingredient which is a non-backtracking condition on the open loops. The striking result of the paper is that the distribution of the random network with prescribed boundary conditions is defined by a Gibbs distribution with a local Hamiltonian. The ansatz of the paper is to express the partition function of the model in terms of determinants of two different matrices which, for some values of the edge weights, involve transition matrices of some Markov processes.

MSC:
82B30 Statistical thermodynamics
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Aizenman, M, Proof of the triviality of \({{ϕ }_{d}}^{4}\) field theory and some Mean-field features of Ising models for \(d >4\), Phys. Rev. Lett., 47, 1-4, (1981)
[2] Aizenman, M, Geometric analysis of \({{ϕ }}^{4}\) fields and Ising models. parts I and II, Commun. Math. Phys., 86, 1-48, (1982) · Zbl 0533.58034
[3] Aizenman, M; Duminil-Copin, H; Sidoravicius, V, Random currents and continuity of Ising model’s spontaneous magnetization, Commun. Math. Phys., 334, 719-742, (2014) · Zbl 1315.82004
[4] Beneš, C., Viklund, F. J., Lawler, G. F.: Scaling limit of the loop-erased random walk Green’s function. (2014). arXiv:1402.7345v1 [math.PR] · Zbl 1049.60072
[5] Berlin, TH; Kac, M, The spherical model of a ferromagnet, Phys. Rev., 86, 821-835, (1952) · Zbl 0047.45703
[6] Biskup, M.: Reflection positivity and phase transitions in lattice spin models. In: Methods of Contemporary Mathematical Statistical Physics, vol. 1970. Lecture Notes in Mathematics, pp. 1-86. Springer-Verlag, Berlin (2009) · Zbl 1180.82041
[7] Brydges, D; Fröhlich, J; Spencer, T, The random walk representation of classical spin systems and correlation inequalities, Commun. Math. Phys., 83, 123-150, (1982)
[8] Brydges, D; Fröhlich, J; Sokal, AD, The random-walk representation of classical spin systems and correlation inequalities. II. the skeleton inequalities, Commun. Math. Phys., 91, 117-139, (1983)
[9] Camia, F; Gandolfi, A; Kleban, M, Conformal correlation functions in the Brownian loop soup, Nucl. Phys. B, 902, 483-507, (2016) · Zbl 1332.82083
[10] Crane, L, \(2\)-d physics and \(3\)-d topology, Commun. Math. Phys., 135, 615-640, (1991) · Zbl 0717.57007
[11] Crane, L, Conformal field theory, spin geometry, and quantum gravity, Phys. Lett. B, 259, 243-248, (1991)
[12] Fitzner, R; Hofstad, R, Non-backtracking random walk, J. Stat. Phys., 150, 264-284, (2013) · Zbl 1259.82043
[13] Fröhlich, J, On the triviality of \(λ ϕ _d^4\) theories and the approach to the critical point in \(d_{(-)} > 4\) dimensions, Nucl. Phys. B, 200, 281-296, (1982)
[14] Hashimoto, K.-I.: Zeta functions of finite graphs and representations of p-adic groups. Automorphic forms and geometry of arithmetic varieties, pp. 211-280 (1989)
[15] Horton, M.D., Stark, H.M., Terras, A.A.: What are zeta functions of graphs and what are they good for? Contemp. Math. 415, 173-190 (2006) · Zbl 1222.11109
[16] Ihara, Y, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math. Soc. Japan, 18, 219-235, (1966) · Zbl 0158.27702
[17] Kac, M; Clive Ward, J, A combinatorial solution of the two-dimensional Ising model, Phys. Rev., 88, 1332-1337, (1952) · Zbl 0048.45804
[18] Kager, W; Lis, M; Meester, R, The signed loop approach to the Ising model: foundations and critical point, J. Stat. Phys., 152, 353-387, (2013) · Zbl 1276.82009
[19] Lawler, G. F., Trujillo Ferreras, J. A.: Random walk loop soup. Trans. Am. Math. Soc. 359(2), 767-787 (2007) (electronic) · Zbl 1120.60037
[20] Lawler, G.F., Limic, V.: Random walk: a modern introduction. volume 123 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2010) · Zbl 1210.60002
[21] Lawler, GF; Werner, W, The Brownian loop soup, Probab. Theory Relat. Fields, 128, 565-588, (2004) · Zbl 1049.60072
[22] Jan, Y, Markov loops and renormalization, Ann. Probab., 38, 1280-1319, (2010) · Zbl 1197.60075
[23] Le Jan, Y.: Markov loops, coverings and fields. (2016). arXiv:1602.02708 · Zbl 1369.60075
[24] Le Jan, Y.: Markov loops, free field and Eulerian networks. J. Math. Soc. Japan 67(4), 1671-1680, 10 (2015) · Zbl 1337.60245
[25] Le Jan, Y.: Lectures from the 38th Probability Summer School held in Saint-Flour, 2008. In: Markov paths, loops and fields. Lecture Notes in Mathematics. École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School], vol. 2026. Springer, Heidelberg (2011). ISBN: 978-3-642-21215-4 · Zbl 1315.82004
[26] Lis, M, A short proof of the Kac-Ward formula., Ann. Inst. Henri Poincaré Comb. Phys. Interact, 3, 45-53, (2016) · Zbl 1331.05021
[27] Osterwalder, K; Schrader, R, Axioms for Euclidean green’s functions, Commun. Math. Phys., 31, 83-112, (1973) · Zbl 0274.46047
[28] Penrose, R.: Angular momentum: an approach to combinatorial space-time. In Quantum theory and beyond. Cambridge University Press, Cambridge (1971)
[29] Rovelli, C; Smolin, L, Spin networks and quantum gravity, Phys. Rev. D, 53, 5743-5759, (1995)
[30] Stark, HM; Terras, AA, Zeta functions of finite graphs and coverings, Adv. Math., 121, 124-165, (1996) · Zbl 0874.11064
[31] Symanzik, K, Euclidean quantum field theory, (1969), New York
[32] van de Brug, T., Camia, F., Lis, M.: Conformal fields from Brownian loops (2017) (In preparation) · Zbl 0717.57007
[33] Watanabe, Y., Fukumizu, K.: Graph zeta function in the bethe free energy and loopy belief propagation. In: Bengio, Y., Schuurmans, D., Lafferty, J.D., Williams, C.K.I., Culotta, A., (eds.) Advances in Neural Information Processing Systems 22, pp. 2017-2025. Curran Associates, Inc. (2009)
[34] Werner, W.: On the spatial Markov property of soups of unoriented and oriented loops. In: Sém. Probab. XLVIII. Springer (2016) (To appear) · Zbl 1370.60192
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.