Exponents for Hamiltonian paths on random bicubic maps and KPZ. (English) Zbl 1520.81122
Summary: We evaluate the configuration exponents of various ensembles of Hamiltonian paths drawn on random planar bicubic maps. These exponents are estimated from the extrapolations of exact enumeration results for finite sizes and compared with their theoretical predictions based on the Knizhnik, Polyakov and Zamolodchikov (KPZ) relations, as applied to their regular counterpart on the honeycomb lattice. We show that a naive use of these relations does not reproduce the measured exponents but that a simple modification in their application may possibly correct the observed discrepancy. We show that a similar modification is required to reproduce via the KPZ formulas some exactly known exponents for the problem of unweighted fully packed loops on random planar bicubic maps.
MSC:
| 81S40 | Path integrals in quantum mechanics |
| 70G70 | Functional analytic methods for problems in mechanics |
| 46T25 | Holomorphic maps in nonlinear functional analysis |
| 81V25 | Other elementary particle theory in quantum theory |
| 05A15 | Exact enumeration problems, generating functions |
| 82D80 | Statistical mechanics of nanostructures and nanoparticles |
Software:
OEISOnline Encyclopedia of Integer Sequences:
a(n) is the number of words of length 2n in the language F, the language of ”parity-constrained-shuffle of well-parenthesized words”.References:
| [1] | Guitter, E.; Kristjansen, C.; Nielsen, J. L., Hamiltonian cycles on random Eulerian triangulations, Nucl. Phys. B, 546, 3, 731-750 (1999) · Zbl 0984.82023 |
| [2] | Knizhnik, V. G.; Polyakov, A. M.; Zamolodchikov, A. B., Fractal structure of 2D—quantum gravity, Mod. Phys. Lett. A, 03, 08, 819-826 (1988) |
| [3] | Di Francesco, P.; Golinelli, O.; Guitter, E., Meanders: exact asymptotics, Nucl. Phys. B, 570, 3, 699-712 (2000) · Zbl 0984.82024 |
| [4] | Di Francesco, P.; Guitter, E.; Jacobsen, J. L., Exact meander asymptotics: a numerical check, Nucl. Phys. B, 580, 3, 757-795 (2000) · Zbl 0984.82025 |
| [5] | Di Francesco, P.; Guitter, E., Geometrically constrained statistical systems on regular and random lattices: from folding to meanders, Phys. Rep., 415, 1, 1-88 (2005) |
| [6] | Reshetikhin, N. Y., A new exactly solvable case of an O(n)-model on a hexagonal lattice, J. Phys. A, Math. Gen., 24, 2387 (1991) · Zbl 0737.60093 |
| [7] | Blöte, H. W.J.; Nienhuis, B., Fully packed loop model on the honeycomb lattice, Phys. Rev. Lett., 72, 1372-1375 (Feb 1994) |
| [8] | Batchelor, M. T.; Suzuki, J.; Yung, C. M., Exact results for Hamiltonian walks from the solution of the fully packed loop model on the honeycomb lattice, Phys. Rev. Lett., 73, 2646-2649 (Nov 1994) · Zbl 1020.82543 |
| [9] | Kondev, Jane; de Gier, Jan; Nienhuis, Bernard, Operator spectrum and exact exponents of the fully packed loop model, J. Phys. A, Gen. Phys., 29, Article 03 pp. (1996) · Zbl 0905.60088 |
| [10] | Dupic, T.; Estienne, B.; Ikhlef, Y., Three-point functions in the fully packed loop model on the honeycomb lattice, J. Phys. A, Math. Theor., 52, 20, Article 205003 pp. (Apr 2019) · Zbl 1509.82025 |
| [11] | Nienhuis, Bernard, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, J. Stat. Phys., 34, 731-762 (1984) · Zbl 0595.76071 |
| [12] | David, F., Conformal field theories coupled to 2-D gravity in the conformal gauge, Mod. Phys. Lett. A, 03, 17, 1651-1656 (1988) |
| [13] | Distler, Jacques; Kawai, Hikaru, Conformal field theory and 2D quantum gravity, Nucl. Phys. B, 321, 2, 509-527 (1989) |
| [14] | Schramm, Oded, Scaling limits of loop-erased random walks and uniform spanning trees, Isr. J. Math., 118, 221-288 (2000) · Zbl 0968.60093 |
| [15] | Sheffield, Scott, Exploration trees and conformal loop ensembles, Duke Math. J., 147, 1, 79-129 (2009) · Zbl 1170.60008 |
| [16] | Duplantier, Bertrand, Higher conformal multifractality, J. Stat. Phys., 110, 3-6, 691-738 (2003) · Zbl 1041.60074 |
| [17] | Duplantier, B., Conformal fractal geometry & boundary quantum gravity, (Lapidus, M. L.; van Frankenhuysen, M., Fractal Geometry and Applications: a Jubilee of Benoît Mandelbrot, Part 2. Fractal Geometry and Applications: a Jubilee of Benoît Mandelbrot, Part 2, Proc. Sympos. Pure Math., vol. 72 (2004), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 365-482 · Zbl 1068.60019 |
| [18] | Kager, Wouter; Nienhuis, Bernard, A guide to stochastic Löwner evolution and its applications, J. Stat. Phys., 115, 5-6, 1149-1229 (2004) · Zbl 1157.82327 |
| [19] | Rohde, Steffen; Schramm, Oded, Basic properties of SLE, Ann. Math. (2), 161, 2, 883-924 (2005) · Zbl 1081.60069 |
| [20] | Schramm, Oded; Sheffield, Scott, Contour lines of the two-dimensional discrete Gaussian free field, Acta Math., 202, 1, 21-137 (2009) · Zbl 1210.60051 |
| [21] | Smirnov, Stanislav, Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits, C. R. Acad. Sci., Ser. 1 Math., 333, 3, 239-244 (2001) · Zbl 0985.60090 |
| [22] | Smirnov, Stanislav, Conformal invariance in random cluster models. I: holomorphic fermions in the Ising model, Ann. Math. (2), 172, 2, 1435-1467 (2010) · Zbl 1200.82011 |
| [23] | Chelkak, Dmitry; Smirnov, Stanislav, Universality in the 2D Ising model and conformal invariance of fermionic observables, Invent. Math., 189, 3, 515-580 (2012) · Zbl 1257.82020 |
| [24] | Miller, Jason; Sheffield, Scott, Imaginary geometry. IV: interior rays, whole-plane reversibility, and space-filling trees, Probab. Theory Relat. Fields, 169, 3-4, 729-869 (2017) · Zbl 1378.60108 |
| [25] | Duplantier, B.; Sheffield, S., Liouville quantum gravity and KPZ, Invent. Math., 185, 333-393 (2011) · Zbl 1226.81241 |
| [26] | Duplantier, B.; Sheffield, S., Duality and KPZ in Liouville quantum gravity, Phys. Rev. Lett., 102, Article 150603 pp. (2009) |
| [27] | Rhodes, R.; Vargas, V., KPZ formula for log-infinitely divisible multifractal random measures, ESAIM Probab. Stat., 15, 358-371 (2011) · Zbl 1268.60070 |
| [28] | Duplantier, Bertrand; Rhodes, Rémi; Sheffield, Scott; Vargas, Vincent, Renormalization of critical Gaussian multiplicative chaos and KPZ relation, Commun. Math. Phys., 330, 1, 283-330 (2014) · Zbl 1297.60033 |
| [29] | Sheffield, Scott, Conformal weldings of random surfaces: SLE and the quantum gravity zipper, Ann. Probab., 44, 5, 3474-3545 (2016) · Zbl 1388.60144 |
| [30] | Duplantier, B.; Sheffield, S., Schramm-Loewner evolution and Liouville quantum gravity, Phys. Rev. Lett., 107, Article 131305 pp. (2011) |
| [31] | Duplantier, Bertrand; Miller, Jason; Sheffield, Scott, Liouville quantum gravity as a mating of trees, Astérisque, 427, 1-258 (2021) · Zbl 1503.60003 |
| [32] | Borga, Jacopo; Gwynne, Ewain; Sun, Xin, Permutons, meanders, and SLE-decorated Liouville quantum gravity (2022) |
| [33] | Di Francesco, P.; Guitter, E.; Kristjansen, C., Fully packed \(O(n = 1)\) model on random Eulerian triangulations, Nucl. Phys. B, 549, 3, 657-667 (1999) · Zbl 0944.82011 |
| [34] | Kazakov, Vladimir A.; Zinn-Justin, Paul, Two-matrix model with \(A B A B\) interaction, Nucl. Phys. B, 546, 3, 647-668 (1999) · Zbl 0958.81026 |
| [35] | Kostov, Ivan, Exact solution of the six-vertex model on a random lattice, Nucl. Phys., 575, 513-534 (2000) · Zbl 1037.82509 |
| [36] | The OEIS Foundation Inc. (2022), The on-line encyclopedia of integer sequences, published electronically at |
| [37] | Duplantier, Bertrand, Conformally invariant fractals and potential theory, Phys. Rev. Lett., 84, 7, 1363-1367 (2000) · Zbl 1042.82577 |
| [38] | Aizenman, Michael; Duplantier, Bertrand; Aharony, Amnon, Path-crossing exponents and the external perimeter in 2D percolation, Phys. Rev. Lett., 83, 1359-1362 (1999) |
| [39] | Zhan, Dapeng, Duality of chordal SLE, Invent. Math., 174, 2, 309-353 (2008) · Zbl 1158.60047 |
| [40] | Dubédat, Julien, Duality of Schramm-Loewner evolutions, Ann. Sci. Éc. Norm. Supér. (4), 42, 5, 697-724 (2009) · Zbl 1205.60147 |
| [41] | Duplantier, B., Harmonic measure exponents for two-dimensional percolation, Phys. Rev. Lett., 82, 3940-3943 (1999) · Zbl 1042.82560 |
| [42] | Duplantier, B., Two-dimensional copolymers and exact conformal multifractality, Phys. Rev. Lett., 82, 880-883 (1999) |
| [43] | Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin, The dimension of the planar Brownian frontier is 4/3, Math. Res. Lett., 8, 4, 401-411 (2001) · Zbl 1114.60316 |
| [44] | Lawler, G. F.; Schramm, O.; Werner, W., On the scaling limit of planar self-avoiding walk, (Lapidus, M. L.; van Frankenhuysen, M., Fractal Geometry and Applications: a Jubilee of Benoît Mandelbrot, Part 2. Fractal Geometry and Applications: a Jubilee of Benoît Mandelbrot, Part 2, Proc. Sympos. Pure Math., vol. 72 (2004), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 339-364 · Zbl 1069.60089 |
| [45] | Duplantier, Bertrand; Kostov, Ivan, Conformal spectra of polymers on a random surface, Phys. Rev. Lett., 61, 1433-1437 (1988) |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
