Loop models on random maps via nested loops: the case of domain symmetry breaking and application to the Potts model

We use the nested loop approach to investigate loop models on random planar maps where the domains delimited by the loops are given two alternating colors, which can be assigned different local weights, hence allowing for an explicit Z2 domain symmetry breaking. Each loop receives a non-local weight n, as well as a local bending energy which controls loop turns. By a standard cluster construction that we review, the Q = n2 Potts model on general random maps is mapped to a particular instance of this problem with domain-non-symmetric weights. We derive in full generality a set of coupled functional relations for a pair of generating series which encode the enumeration of loop configurations on maps with a boundary of a given color, and solve it by extending well-known complex analytic techniques. In the case where loops are fully packed, we analyze in detail the phase diagram of the model and derive exact equations for the position of its non-generic critical points. In particular, we underline that the critical Potts model on general random maps is not self-dual whenever Q ≠ 1. In a model with domain-symmetric weights, we also show the possibility of a spontaneous domain symmetry breaking driven by the bending energy.This article is part of 'Lattice models and integrability', a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.

[1]  B. Nienhuis,et al.  Analytical calculation of two leading exponents of the dilute Potts model , 1982 .

[2]  M. Gaudin,et al.  O(n) model on a fluctuating planar lattice. Some exact results , 1989 .

[3]  P. Zinn-Justin.  The Dilute Potts Model on Random Surfaces , 1999 .

[4]  P. Di Francesco,et al.  2D gravity and random matrices , 1993 .

[5]  R. Baxter,et al.  Equivalence of the Potts model or Whitney polynomial with an ice-type model , 1976 .

[6]  T. T. Truong.  Structural properties of aZ(N2)-spin model and its equivalentZ(N)-vertex model , 1986 .

[7]  J. Bouttier,et al.  More on the O(n) model on random maps via nested loops: loops with bending energy , 2012, 1202.5521.

[8]  G. Parisi,et al.  Planar diagrams , 1978 .

[9]  P. Zinn-Justin.  The six-vertex model on random lattices , 2000 .

[10]  Ivan Kostov.  Exact solution of the six-vertex model on a random lattice , 2000 .

[11]  C. Fortuin,et al.  On the random-cluster model: I. Introduction and relation to other models , 1972 .

[12]  G. Bonnet,et al.  The Potts-q random matrix model: loop equations, critical exponents, and rational case , 1999 .

[13]  More on the exact solution of the O(n) model on a random lattice and an investigation of the case ∥n∥ > 2 , 1995, hep-th/9512052.

[14]  Henk W. J. Blöte,et al.  Critical behaviour of the fully packed loop model on the square lattice , 1996 .

[15]  Vladimir Kazakov,et al.  The ising model on a random planar lattice: The structure of the phase transition and the exact critical exponents , 1987 .

[16]  D. Shlyakhtenko,et al.  Loop Models, Random Matrices and Planar Algebras , 2010, 1012.0619.

[17]  J. Bouttier,et al.  Combinatorics of bicubic maps with hard particles , 2005 .

[18]  J. Bouttier,et al.  A recursive approach to the O(n) model on random maps via nested loops , 2011, 1106.0153.

[19]  Gilles Schaeffer,et al.  The degree distribution in bipartite planar maps: applications to the Ising model , 2002 .

[20]  B. Nienhuis.  Loop models , 2022 .

[21]  Exactly solvable Potts models, bond- and tree-like percolation on dynamical (random) planar lattice , 1988 .

[22]  Blöte,et al.  Fully packed loop model on the honeycomb lattice. , 1994, Physical review letters.

[23]  I. Kostov.  O($n$) Vector Model on a Planar Random Lattice: Spectrum of Anomalous Dimensions , 1989 .

[24]  Mireille Bousquet-Mélou,et al.  Counting colored planar maps: Algebraicity results , 2009, J. Comb. Theory, Ser. B.

[25]  Jean-Marc DAUL.  Q-states Potts model on a random planar lattice , 1995 .

[26]  F. Y. Wu,et al.  Nonintersecting string model and graphical approach: Equivalence with a Potts model , 1986 .

[27]  J. L. Gall,et al.  Scaling limits of random trees and planar maps , 2011, 1101.4856.

[28]  W. T. Tutte.  A Census of Planar Maps , 1963, Canadian Journal of Mathematics.

[29]  Exact solution of the O(n) model on a random lattice , 1995, hep-th/9506193.

[30]  Multicritical phases of the O(n) model on a random lattice , 1992, hep-th/9203030.

[31]  The O(n) model on a random surface: critical points and large-order behaviour , 1992, hep-th/9204082.

[32]  Grégory Miermont,et al.  Scaling limits of random planar maps with large faces , 2011 .