The six-vertex model on random planar maps revisited

被引:1
作者
Price, Andrew Elvey [1 ]
Zinn-Justin, Paul [2 ]
机构
[1] Univ Bordeaux, Lab Bordelais Rech Informat, UMR 5800, 351 Cours Liberat, F-33405 Talence, France
[2] Univ Melbourne, Sch Math & Stat, Melbourne, Vic 3010, Australia
基金
欧洲研究理事会;
关键词
Planar maps; Six vertex model; Eulerian orientations; Matrix integrals; Elliptic theta functions; Modular forms;
D O I
10.1016/j.jcta.2023.105739
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We address the six vertex model on a random lattice, which in combinatorial terms corresponds to the enumeration of weighted 4-valent planar maps equipped with an Eulerian orientation. This problem was exactly, albeit non-rigorously solved by Ivan Kostov in 2000 using matrix integral tech-niques. We convert Kostov's work to a combinatorial argu-ment involving functional equations coming from recursive decompositions of the maps, which we solve rigorously us-ing complex analysis. We then investigate modular properties of the solution, which lead to simplifications in certain special cases. In particular, in two special cases of combinatorial in-terest we rederive the formulae discovered by Bousquet-Melou and the first author.(c) 2023 Elsevier Inc. All rights reserved.
引用
收藏
页数:32
相关论文
共 50 条
[21]   Emptiness formation probability in the domain-wall six-vertex model [J].
Colomo, F. ;
Pronko, A. G. .
NUCLEAR PHYSICS B, 2008, 798 (03) :340-362
[22]   Domain Wall Six-Vertex Model with Half-Turn Symmetry [J].
Pavel Bleher ;
Karl Liechty .
Constructive Approximation, 2018, 47 :141-162
[23]   Asymmetric six-vertex model and the classical Ruijsenaars–Schneider system of particles [J].
A. V. Zabrodin ;
A. V. Zotov ;
A. N. Liashyk ;
D. S. Rudneva .
Theoretical and Mathematical Physics, 2017, 192 :1141-1153
[24]   Domain Wall Six-Vertex Model with Half-Turn Symmetry [J].
Bleher, Pavel ;
Liechty, Karl .
CONSTRUCTIVE APPROXIMATION, 2018, 47 (01) :141-162
[25]   Arctic Curves of the Six-Vertex Model on Generic Domains: The Tangent Method [J].
F. Colomo ;
A. Sportiello .
Journal of Statistical Physics, 2016, 164 :1488-1523
[26]   Finitary codings for gradient models and a new graphical representation for the six-vertex model [J].
Ray, Gourab ;
Spinka, Yinon .
RANDOM STRUCTURES & ALGORITHMS, 2022, 61 (01) :193-232
[27]   The Bethe ansatz for the six-vertex and XXZ models: An exposition [J].
Duminil-Copin, Hugo ;
Gagnebin, Maxime ;
Harel, Matan ;
Manolescu, Ioan ;
Tassion, Vincent .
PROBABILITY SURVEYS, 2018, 15 :102-130
[28]   The free energy singularity of the asymmetric six-vertex model and the excitations of the asymmetric XXZ chain [J].
Albertini, G ;
Dahmen, SR ;
Wehefritz, B .
NUCLEAR PHYSICS B, 1997, 493 (03) :541-570
[29]   ASYMMETRIC SIX-VERTEX MODEL AND THE CLASSICAL RUIJS']JSENAARS-SCHNEIDER SYSTEM OF PARTICLES [J].
Zabrodin, A. V. ;
Zotov, A. V. ;
Liashyk, A. N. ;
Rudneva, D. S. .
THEORETICAL AND MATHEMATICAL PHYSICS, 2017, 192 (02) :1141-1153
[30]   Exact results for the six-vertex model with domain wall boundary conditions and a partially reflecting end [J].
Hietala, Linnea .
LETTERS IN MATHEMATICAL PHYSICS, 2022, 112 (02)