Abstract
Loop percolation, also known as the dense O(1) loop model, is a variant of critical bond percolation in the square lattice \({\mathbb{Z}^2}\) whose graph structure consists of a disjoint union of cycles. We study its connectivity pattern, which is a random noncrossing matching associated with a loop percolation configuration. These connectivity patterns exhibit a striking rationality property whereby probabilities of naturally-occurring events are dyadic rational numbers or rational functions of a size parameter n, but the reasons for this are not completely understood. We prove the rationality phenomenon in a few cases and prove an explicit formula expressing the probabilities in the “cylindrical geometry” as coefficients in certain multivariate polynomials. This reduces the rationality problem in the general case to that of proving a family of conjectural constant term identities generalizing an identity due to Di Francesco and Zinn-Justin. Our results make use of, and extend, algebraic techniques related to the quantum Knizhnik-Zamolodchikov equation.
Similar content being viewed by others
References
Andrews G.E.: Plane partitions V: the TSSCPP conjecture. J. Combin. Theory Ser. A 66, 28–39 (1994)
Andrews G.E., Burge W.H.: Determinant identities. Pac. J. Math. 158, 1–14 (1993)
Batchelor M., de Gier J., Nienhuis B.: The quantum symmetric XXZ chain at \({\Delta = -1/2}\), alternating-sign matrices and plane partitions. J. Phys. A 34, L265–L270 (2001)
Bressoud D.M.: Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge University Press, Cambridge (1999)
Cantini L., Sportiello A.: Proof of the Razumov-Stroganov conjecture. J. Comb. Theory Ser. A 118, 1549–1574 (2011)
Di Francesco, P., Zinn-Justin, P., Zuber, J.-B.: Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain. J. Stat. Mech. P08011 (2006)
Dyson F.: Statistical theory of the energy levels of complex systems. I. J. Math. Phys. 3, 140–156 (1962)
Feigin B., Jimbo M., Miwa T., Mukhin E.: Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials. Int. Math. Res. Not. 2003, 1015–1034 (2003)
de Gier J.: Loops, matchings and alternating-sign matrices. Discrete Math. 298, 365–388 (2005)
de Gier J., Lascoux A., Sorrell M.: Deformed Kazhdan-Lusztig elements and Macdonald polynomials. J. Combin. Theory Ser. A 119, 183–211 (2012)
Fonseca T., Zinn-Justin P.: On the doubly refined enumeration of alternating sign matrices and totally symmetric self-complementary plane partitions. Electron. J. Combin. 15, R81 (2008)
Fonseca, T., Zinn-Justin, P.: On some ground state components of the O(1) loop model. J. Stat. Mech. Theory Exp. P03025 (2009)
Good J.I.: Short proof of a conjecture of Dyson. J. Math. Phys. 11, 1884 (1970)
Grimmett G.: Percolation, 2nd edn. Springer, Berlin (1999)
Gunson J.: Proof of a conjecture of Dyson in the statistical theory of energy levels. J. Math. Phys. 3, 752–753 (1962)
Kasatani M.: Subrepresentations in the polynomial representation of the double affine Hecke algebra of type GL n at t k+1 q r-1 = 1. Int. Math. Res. Not. 2005, 1717–1742 (2005)
Krattenthaler C.: Determinant identities and a generalization of the number of totally symmetric self-complementary plane partitions. Electron. J. Combin. 4, R27 (1997)
Kuperberg G.: Another proof of the alternating sign matrix conjecture. Internat. Math. Res. Notes 1996, 139–150 (1996)
Lawler, G.F., Schramm, O., Werner, W.: One-arm exponent for critical 2D percolation. Electron. J. Probab. 7(2) (2002)
Mills W.H., Robbins D.P., Rumsey H.: Alternating sign matrices and descending plane partitions. J. Combin. Theory Ser. A 34, 340–359 (1983)
Mitra, S., Nienhuis, B., de Gier, J., Batchelor, M.T.: Exact expressions for correlations in the ground state of the dense O(1) loop model. J. Stat. Mech. Theory Exp. P09010 (2004)
Okada S.: Enumeration of symmetry classes of alternating sign matrices and characters of classical groups. J. Algebraic Combin. 23, 43–69 (2006)
Pasquier V.: Quantum incompressibility and Razumov-Stroganov type conjectures. Ann. Henri Poincaré 7, 397–421 (2006)
Pearce P.A., Rittenberg V., de Gier J., Nienhuis B.: Temperley-Lieb stochastic processes. J. Phys. A. 35, L661–L668 (2002)
Propp, J.: The many faces of alternating sign matrices. Discrete Mathematics and Theoretical Computer Science. In: Proceedings of DM-CCG, Conference Volume AA, pp. 43–58 (2001)
Razumov A.V., Stroganov Yu.G.: Combinatorial nature of ground state vector of O(1) loop model. Theor. Math. Phys. 138, 333–337 (2004)
Romik, D.: Connectivity patterns in loop percolation II: pipe percolation, in preparation
Sills A.V., Zeilberger D.: Disturbing the Dyson conjecture (in a GOOD way). Exp. Math. 15, 187–191 (2006)
Smirnov S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333, 239–244 (2001)
Stroganov Yu.: Izergin-Korepin determinant at a third root of unity. Theor. Math. Phys. 146, 53–62 (2006)
Temperley N., Lieb E.: Relations between the percolation and colouring problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the percolation problem. Proc. R. Soc. A 322, 251–280 (1971)
Wieland B.: A large dihedral symmetry of the set of alternating sign matrices. Electron. J. Combin. 7, R37 (2000)
Wilson K.: Proof of a conjecture by Dyson. J. Math. Phys. 3, 1040–1043 (1962)
Zeilberger D.: Proof of the alternating sign matrix conjecture. Electron. J. Combin. 3, R13 (1996)
Zeilberger, D.: Proof of a conjecture of Philippe Di Francesco and Paul Zinn-Justin related to the qKZ equations and to Dave Robbins’ two favorite combinatorial objects. Preprint (2007). http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/diFrancesco.pdf
Zinn-Justin, P.: Six-vertex, loop and tiling models: integrability and combinatorics. Habilitation thesis, arXiv:0901.0665
Zinn-Justin P., Di Francesco P.: Around the Razumov-Stroganov conjecture: proof of a multi-parameter sum rule. Electron. J. Combin. 12, R6 (2005)
Zinn-Justin P., Di Francesco P.: Quantum Knizhnik-Zamolodchikov equation, totally symmetric self-complementary plane partitions and alternating sign matrices. Theor. Math. Phys. 154, 331–348 (2008)
Zuber J.-B.: On the counting of fully packed loop configurations: some new conjectures. Electron. J. Combin. 11, R13 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Spohn
Rights and permissions
About this article
Cite this article
Romik, D. Connectivity Patterns in Loop Percolation I: the Rationality Phenomenon and Constant Term Identities. Commun. Math. Phys. 330, 499–538 (2014). https://doi.org/10.1007/s00220-014-2001-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-2001-5