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.

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 (unpublished); e-print arXiv:1201.0264 (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.
G. R.
Grimmett
,
The Random-Cluster Model
(
Springer
,
Berlin
,
2006
).
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.
J.
Gross
and
T.
Tucker
,
Topological Graph Theory
(
Wiley
,
Berlin
,
1987
).
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