The 3-state Potts model on planar triangulations: explicit algebraic solution

TL;DR

This work settles the exact algebraic form of the 3-state Potts generating function on planar triangulations by proving that the family of near-triangulation generating functions $T_i(ν,w)$ is algebraic of degree 11, lying on a genus-1 curve in $(w,T_1)$ with a critical ν-value $ν_c=1+3/\sqrt{47}$ where the singular exponent is $6/5$; for other ν, the exponent is the standard map exponent $3/2$. Using a carefully constructed polynomial system and an elliptic parametrization, the authors derive the minimal polynomial for $T_1$ and show all $T_i$ (i≥1) belong to the same degree-11 extension, while establishing a duality to near-cubic maps and proving Salvy’s conjecture for properly 3-coloured near-cubic maps. They also analyze asymptotics and singularities, detailing the radius of convergence and exponent behavior, including KPZ-consistent exponents, across ν, and discuss negative ν regimes. The results provide a complete algebraic framework for the Potts model on planar maps and furnish new tools for related map-enumeration problems, with implications for dual models and potential extensions to general planar maps (BMN-26).

Abstract

We consider the $3$-state Potts generating function $T(ν,w)$ of planar triangulations; that is, the bivariate series that counts planar triangulations with vertices coloured in $3$ colours, weighted by their size (number of vertices, recorded by the variable $w$) and by the number of monochromatic edges (variable $ν$). This series was proved to be algebraic 15 years ago by Bernardi and the first author: this follows from its link with the solution of a discrete differential equation (DDE), and from general algebraicity results on such equations. However, despite recent progresses on the effective solution of DDEs, the exact value of $T(ν,w)$ has remained unknown so far -- except in the case $ν=0$, corresponding to proper colourings and solved by Tutte in the sixties. We determine here this exact value, proving that $T(ν,w)$ satisfies a polynomial equation of degree $11$ in $T$ and genus $1$ in $w$ and $T$. We prove that the critical value of $ν$ is $ν_c=1+3/\sqrt{47}$, with a critical exponent $6/5$ in the series $T(ν_c, \cdot)$, while the other values of $ν$ yield the usual map exponent $3/2$. By duality of the planar Potts model, our results also characterize the 3-state Potts generating function of planar cubic maps, in which all vertices have degree $3$. In particular, the annihilating polynomial, still of degree $11$, that we obtain for properly 3-coloured cubic maps proves a conjecture by Bruno Salvy from 2009.

Figures (13)

• Figure 1: A $3$-coloured rooted near-triangulation having $7$ vertices, $4$ monochromatic edges (in thick lines) and outer degree $4$. It contributes $w^7 \nu^4y^4$ in the series $T(\nu,w;y)$ counting such maps by vertices ($w$), monochromatic edges ($\nu$) and outer degree ($y$).
• Figure 2: Left: a rooted planar map with $3$ vertices and $4$ faces, having outer degree $4$. Right: the dual map, in dashed edges.
• Figure 3: The rooted near-triangulations of outer degree 1 with $v=2$ and $v=3$ vertices, and their Potts polynomials (divided by $q$).
• Figure 4: The branches of $\Delta_0$ (green/light), $\Delta_1$ (black) and $\Delta_2$ (red). The black dashed curve is the lower bound $\rho_1/\nu^3$ on the radius, for $\nu\ge 1$. The plot on the right zooms on the interval $[1,2]$. The radius $\rho_\nu$ first follows the top black branch, between $\nu=0$ and $\nu_c\simeq 1.44$, and then the top red branch.
• Figure 5: The value $c_\nu$ of $\dot{T}_1$ at its radius $\rho_\nu$ increases up to $\nu_c$ and decreases afterwards.

Theorems & Definitions (27)

• Theorem 1
• Corollary 2
• Proposition 3
• Proposition 4
• Remark
• proof
• Corollary 5
• Proposition 6
• Lemma 7: The case $\boldsymbol{\nu=0}$
• proof