Invariant sets of the period-two splitting map and the complete catalogue of periodic Gibbs measures for the q-state Potts model on the Cayley tree

The classification of periodic Gibbs measures is the main open part of the exact theory of the $q$-state Potts model on the Cayley tree $\Gamma_k$. By the Rozikov-Khakimov reduction, every periodic splitting Gibbs measure is either translation-invariant or $G_k^{(2)}$-periodic (period two, with respect to the index-two subgroup of even-length words), and the $G_k^{(2)}$-periodic measures correspond to the positive fixed points of an explicit self-map $W$ of $\mathbb{R}_+^{2q-2}$. Khakimov and Makhammadaliev (2019) proved that in the ferromagnetic regime ($k\ge2$, $q\ge3$) and in the antiferromagnetic $k=2$ regime all periodic Gibbs measures are translation-invariant, but their Remark 1 records that the explicit invariant sets $I_1$–$I_6$ do not completely describe all invariant sets of $W$; the exact solution counts obtained on the families $I_m$ and $I'_m$ and their coordinate permutations (Khakimov 2014; Khakimov-Khaydarov 2016; Makhammadaliev 2021) are likewise confined to explicitly exhibited invariant sets. The surviving open core is precise: decide whether the known invariant sets of $W$ already contain all of its positive fixed points, and otherwise exhibit an explicit invariant-set family containing the remaining fixed points; determine all fixed points off the known families — chiefly in the antiferromagnetic $k\ge3$ regime over the full temperature range and all $q$, including $q\ge k+1$ where neither existence nor exclusion of non-translation-invariant periodic measures is known; and thereby complete the catalogue of $G_k^{(2)}$-periodic Gibbs measures of the zero-field model.

Background

The Cayley tree $\Gamma_k$ of order $k\ge1$ is the infinite regular tree in which every vertex has exactly $k+1$ nearest neighbors; it is the Cayley graph of the group $G_k=\mathbb{Z}_2*\cdots*\mathbb{Z}_2$, the free product of $k+1$ copies of the two-element group. The $q$-state Potts model on $\Gamma_k$ is the spin system whose configurations assign to each vertex $x$ a value $\sigma(x)\in\{1,\dots,q\}$ with formal Hamiltonian $H(\sigma)=-J\sum_{\langle x,y\rangle}\delta_{\sigma(x)\sigma(y)}$, where the sum runs over nearest-neighbor pairs and $\delta$ is the Kronecker symbol; $J>0$ is the ferromagnetic case (neighbors prefer to agree) and $J<0$ the antiferromagnetic case (neighbors prefer to differ). Writing $\theta=e^{J/T}$ with $T>0$ the temperature, the antiferromagnetic regime corresponds to $0<\theta<1$. A Gibbs measure is an infinite-volume probability measure on configurations compatible with this interaction; the measures amenable to exact description on trees are the splitting Gibbs measures, encoded by boundary-law vectors $h_x\in\mathbb{R}^{q-1}$ attached to the vertices and satisfying the compatibility equation $h_x=\sum_{y\in S(x)}F(h_y,\theta)$, where $S(x)$ is the set of successors of $x$ directed away from a chosen root and $F$ is an explicit rational map (the compatibility criterion is reproduced as Theorem 1 of the source paper, Khakimov and Makhammadaliev 2019).

A splitting Gibbs measure is translation-invariant (TI) if $h_x$ is independent of $x$, and periodic if the family $\{h_x\}$ is constant on the cosets of a normal subgroup of finite index in $G_k$; periodic measures are the natural spatially modulated phases beyond the TI ones. The reduction theorem of Rozikov and Khakimov states that every periodic splitting Gibbs measure of this model is either TI or periodic with respect to the index-two subgroup $G_k^{(2)}\subset G_k$ of even-length words, so the whole periodic classification reduces to period two. For a $G_k^{(2)}$-periodic boundary law one has $h_x=h$ at even-depth and $h_x=l$ at odd-depth vertices; setting $x_i=e^{h_i}$, $y_i=e^{l_i}$, the compatibility equations become the fixed-point system

$$x_i=\left(\frac{(\theta-1)y_i+\sum_{j=1}^{q-1}y_j+1}{\sum_{j=1}^{q-1}y_j+\theta}\right)^{k},\qquad y_i=\left(\frac{(\theta-1)x_i+\sum_{j=1}^{q-1}x_j+1}{\sum_{j=1}^{q-1}x_j+\theta}\right)^{k},\qquad i=1,\dots,q-1,$$

that is, $z=W(z)$ for the self-map $W$ of $\mathbb{R}_+^{2q-2}$ with $z=(x_1,\dots,x_{q-1},y_1,\dots,y_{q-1})$; positive fixed points of $W$ correspond one-to-one to $G_k^{(2)}$-periodic splitting Gibbs measures. A subset $I\subseteq\mathbb{R}_+^{2q-2}$ with $W(I)\subseteq I$ is called an invariant set of $W$; solving the system separately on each invariant set is the standard reduction, because the symmetry restrictions collapse the $2(q-1)$ coupled equations to low-dimensional ones. The invariant sets identified in the literature are the sets $I_1$–$I_6$ of the source paper (for example, $I_1$: all $2q-2$ coordinates equal; $I_2$: the $x_i$ equal among themselves and the $y_i$ equal among themselves; $I_3$: $x_i=y_i$ for all $i$; $I_5$: $x_1=y_1=1$), the sets $I_m=\{(u,v): u=(x,\dots,x,1,\dots,1),\ v=(y,\dots,y,1,\dots,1)\}$ and $I'_m=\{(u,v): u=(x,\dots,x,1,\dots,1,y,\dots,y),\ v=(y,\dots,y,1,\dots,1,x,\dots,x)\}$ with blocks of length $m$ (for $I'_m$ one needs $2m\le q-1$), and all images of these under coordinate permutations. In the antiferromagnetic regime with $k\ge3$ the relevant critical value is $\theta_{\mathrm{cr}}=(k-q+1)/(k+1)$, positive only when $q<k+1$.

Three regimes are already settled. For $J>0$, $k\ge2$, $q\ge3$, every periodic Gibbs measure is TI (source paper, Theorem 5, extending the $q=3$ case of Rozikov-Khakimov); for $J<0$, $k=2$, $q\ge2$, the $G_k^{(2)}$-periodic Gibbs measure is unique and coincides with the unique TI measure (source paper, Theorem 3); both proofs solve the full fixed-point system without restricting to any named invariant set. For $J<0$, $k\ge3$, $3\le q<k+1$ and $0<\theta<\theta_{\mathrm{cr}}$, non-TI $G_k^{(2)}$-periodic Gibbs measures do exist: exactly $2\left(2^q-1+\sum_{m=1}^{\lfloor q/2\rfloor}\binom{q}{m}\binom{q-m}{m}\right)$ of them are known on the sets $\bigcup_m I_m\cup\bigcup_m I'_m$ and their coordinate permutations. Nothing is known about fixed points off these invariant sets, and in the antiferromagnetic regime with $q\ge k+1$ (where $\theta_{\mathrm{cr}}\le0$) neither existence nor exclusion of non-TI periodic measures is known. A nonzero external field, adding a term $-\alpha\sum_x\delta_{1\sigma(x)}$ to the Hamiltonian, changes the splitting equations to a symmetry-broken system that the survey literature treats as a separate, harder direction. The case $q=2$ is the Ising model, which has its own separate literature and is not covered by the Potts classification theorems; the problem below is posed, as in those theorems, for $q\ge3$. The case $k=1$ is the doubly infinite chain, an amenable graph outside the nonamenable regime this classification program studies; the problem is likewise posed, as in the classification theorems, for $k\ge2$. The source paper's Remark 1 records that the sets $I_1$–$I_6$ do not completely describe all invariant sets of $W$. The general theory of Gibbs measures on Cayley trees is systematized in Rozikov's monograph.

Problem Statement

For the $q$-state Potts model with $q\ge3$ on the Cayley tree $\Gamma_k$ of order $k\ge2$ with zero external field, let $W:\mathbb{R}_+^{2q-2}\to\mathbb{R}_+^{2q-2}$ be the period-two splitting map displayed in the Background above, whose positive fixed points $z=W(z)$ correspond one-to-one to the $G_k^{(2)}$-periodic splitting Gibbs measures. Determine, for every coupling $J$ (equivalently every $\theta=e^{J/T}>0$):

(1) the invariant sets of $W$ that carry positive fixed points: decide whether the known families — $I_1$–$I_6$, the $I_m$ with $m\le q-1$, the $I'_m$ with $2m\le q-1$, and their coordinate permutations — are exhaustive in the sense that every positive fixed point of $W$ lies in one of them, and if not, exhibit an explicit invariant-set family containing the remaining fixed points;

(2) all positive fixed points of $W$ lying outside the known invariant sets, together with the parameter values $(k,q,\theta)$ at which they exist — in particular, settle the antiferromagnetic case $k\ge3$ over the full range $0<\theta<1$ and every $q\ge3$, including $q\ge k+1$ where no non-translation-invariant periodic Gibbs measure is currently known and none has been excluded;

(3) the resulting complete catalogue of $G_k^{(2)}$-periodic Gibbs measures: for each $(k,q,\theta)$, the full list of such measures, their number, and which of them are translation-invariant.

The problem is complete when every $G_k^{(2)}$-periodic Gibbs measure of the zero-field model is accounted for — either by a proof that every positive fixed point of $W$ lies in the known invariant-set families, together with the determination of all fixed points on each family for every $(k,q,\theta)$ (extending the known solution counts beyond the antiferromagnetic range $3\le q<k+1$ in which they are currently established), or by an enlarged explicit family with the same two properties. The already-settled regimes (ferromagnetic $J>0$ with $k\ge2$, $q\ge3$; antiferromagnetic $k=2$) are part of the requested classification but require no new work; the scientific content lies in the remaining, chiefly antiferromagnetic $k\ge3$, regimes. The nonzero-external-field analogue, in which the permutation symmetry of the splitting equations is lost, is a recognized further direction and is not required for completion.

References

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