In this work, we study the annealed Potts model coupled to two-dimensional causal triangulations (CTs). Employing duality of graphs, we prove that in the thermodynamic limit, the Potts model coupled to causal triangulations with parameters β and μ is equivalent to a Potts model coupled to dual causal triangulations at the dual parameters β* = log(1 + q/(eβ − 1)) and μ* = μ − 3/2log(eβ − 1) + log q. This duality relation follows from the Fermi–Kurie representation for the Potts model. Employing our duality relation, we determine a region where the critical curve for the annealed model can be located. We also provide lower and upper bounds for the infinite-volume free energy.
REFERENCES
1.
J.
Ambjørn
, B.
Durhuus
, and T.
Jonsson
, Quantum Geometry: A Statistical Field Theory Approach
, No. 1 in Cambridge Monographs on Mathematical Physics (Cambridge University Press
, Cambridge, UK
, 1997
).2.
W. T.
Tutte
, “A census of planar triangulations
,” Can. J. Math.
14
, 21
–38
(1962
).3.
W. T.
Tutte
, “A census of planar maps
,” Can. J. Math.
15
, 249
–271
(1963
).4.
P.
Di Francesco
, P. H.
Ginsparg
, and J.
Zinn-Justin
, “2D gravity and random matrices
,” Phys. Rep.
254
, 1
–133
(1995
); e-print arXiv:hep-th/9306153.5.
G.
Schaeffer
, “Bijective census and random generation of Eulerian planar maps with prescribed vertex degrees
,” Elec. J. Comb.
4
(1997
), http://eudml.org/doc/119255.6.
J.
Bouttier
, P.
Di Francesco
, and E.
Guitter
, “Census of planar maps: From the one-matrix model solution to a combinatorial proof
,” Nucl. Phys. B
645
, 477
(2002
); e-print arXiv:cond-mat/0207682.7.
V. A.
Kazakov
, “Ising model on a dynamical planar random lattice: Exact solution
,” Phys. Lett. A
119
, 140
–144
(1986
).8.
D. V.
Boulatov
and V. A.
Kazakov
, “The Ising model on random planar lattice: The structure of phase transition and the exact critical exponents
,” Phys. Lett. B
186
, 379
(1987
).9.
J.
Ambjørn
and R.
Loll
, “Non-perturbative Lorentzian quantum gravity, causality and topology change
,” Nucl. Phys. B
536
, 407
–434
(1998
); e-print arXiv:hep-th/9805108.10.
B.
Durhuus
, T.
Jonsson
, and J. F.
Wheater
, “On the spectral dimension of causal triangulations
,” J. Stat. Phys.
139
, 859
–881
(2010
); e-print arXiv:0908.3643.11.
V.
Sisko
, A.
Yambartsev
, and S.
Zohren
, “A note on weak convergence results for uniform infinite causal triangulations
,” Brazilian J. Prob. Stat. (unpublished); e-print arXiv:1201.0264 (2013
).12.
V.
Sisko
, A.
Yambartsev
, and S.
Zohren
, “Growth of uniform infinite causal triangulations
,” J. Stat. Phys.
150
, 353
–374
(2013
).13.
J.
Ambjørn
, K. N.
Anagnostopoulos
, and R.
Loll
, “A new perspective on matter coupling in 2D quantum gravity
,” Phys. Rev. D
60
, 104035
(1999
); e-print arXiv:hep-th/9904012.14.
D.
Benedetti
and R.
Loll
, “Quantum gravity and matter: Counting graphs on causal dynamical triangulations
,” Gen. Relativ. Gravitation
39
, 863
–898
(2007
); e-print arXiv:gr-qc/0611075.15.
J.
Ambjørn
, K. N.
Anagnostopoulos
, R.
Loll
, and I.
Pushkina
, “Shaken, but not stirred –Potts model coupled to quantum gravity
,” Nucl. Phys. B
807
(1-2
), 251
–264
(2009
); e-print arXiv:0806.3506.16.
M.
Krikun
and A.
Yambartsev
, “Phase transition for the Ising model on the critical Lorentzian triangulation
,” J. Stat. Phys.
148
, 422
–439
(2012
); e-print arXiv:0810.2182.17.
V. A.
Malyshev
, A. A.
Yambartsev
, and A. A.
Zamyatin
, “Two-dimensional Lorentzian models
,” Moscow Math. J.
1
(3
), 439
–456
(2001
), http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mmj&paperid=30&option_lang=eng.18.
J. C.
Hernández
, A.
Yambartsev
, Y.
Suhov
, and S.
Zohren
, “Bounds on the critical line via transfer matrix methods for an Ising model coupled to causal dynamical triangulations
,” J. Math. Phys.
54
, 063301
(2013
); e-print arXiv:1301.1483.19.
C. M.
Fortuin
and R. W.
Kasteleyn
, “On the random-cluster model. I. Introduction and relation to other models
,” Physica
57
, 536
–564
(1972
).20.
J.
Ambjørn
and J.
Jurkiewics
, “The universe from scratch
,” Contemp. Phys.
47
, 103
–117
(2006
).21.
R. G.
Edwards
and A. D.
Sokal
, “Generalization of the Fortuin-Kasteleyn- Swendsen-Wang representation and Monte Carlo algorithm
,” Phys. Rev. D
38
(3
), 2009
–2012
(1988
).22.
V.
Beffara
and H.
Duminil-Copin
, “The self-dual point of the two- dimensional random-cluster model is critical for q ≥ 1
,” Probab. Theory Relat. Fields
153
, 511
–542
(2012
).23.
J.
Cerda-Hernández
, “Critical region for an Ising model coupled to causal dynamical triangulations
,” J. Stat. Mech.
2017
, 023209
.24.
G. R.
Grimmett
, “The stochastic random-cluster process and the uniqueness of random-cluster measures
,” Ann. Probab.
23
(4
), 1461
–1510
(1995
).25.
26.
G. M.
Napolitano
and T.
Turova
, “The Ising model on the random planar causal triangulation: Bounds on the critical line and magnetization properties
,” J. Stat. Phys.
162
(3
), 739
–760
(2016
); e-print arXiv:1504.03828.27.
C.
Domb
, “Configurational studies of the Potts models
,” J. Phys. A: Math., Nucl. Gen.
7
, 1335
(1974
).28.
F. Y.
Wu
, “The Potts models
,” Rev. Mod. Phys.
54
(1
), 235
–268
(1982
).29.
R. J.
Baxter
, “Exactly solved models in statistical mechanics
,” in Dover Books on Physics
(Dover Publications Inc.
, 2008
).30.
J. C.
Hernández
, A.
Yambartsev
, and S.
Zohren
, “On the critical probability of percolation on random causal triangulations
,” Braz. J. Probab. Stat.
31
(2
), 215
–228
(2017
).31.
© 2017 Author(s).
2017
Author(s)
You do not currently have access to this content.