Classification of weakly periodic Gibbs measures for the two-state hard-core model on Cayley trees: exact multiplicity, extremality, and the (k, i, lambda) phase diagram for normal divisors of index four

The two-state hard-core model (the random independent-set model with fugacity lambda) on the Cayley tree of order k admits, besides the unique translation-invariant Gibbs measure and the classical pair of two-periodic ('chessboard') measures, additional Gibbs measures that are weakly periodic: invariant under a normal divisor of index four in the automorphism group of the tree, but not under any subgroup defining the known periodic measures. Khakimov and Makhammadaliev (Theoretical and Mathematical Physics 204:1059-1078, 2020) proved uniqueness of such measures for a finite list of (k, i) cases and constructed new weakly periodic nonperiodic measures, but left the classification incomplete: outside the settled uniqueness cases, exact multiplicities are known only for k = 2 (i = 1, 2) and k = 3 (i = 1, above the critical value lambda_cr = 27/16, on one invariant set); for k >= 6, i = 1 only a lower bound of at least three measures (one of them translation-invariant) inside an explicit activity window is available; the extremality (purity) of every non-translation-invariant weakly periodic measure is undetermined; and no global (k, i, lambda) phase diagram exists. The problem posed is to complete this classification: for every tree order k >= 2 and subgroup parameter i, determine the exact number of weakly periodic Gibbs measures for every lambda > 0 with exact critical values, determine which of these measures are extreme points of the Gibbs simplex, and assemble the phase diagram.

Background

The Cayley tree $\Gamma^k$ of order $k$ is the infinite tree in which every vertex has $k+1$ nearest neighbours (also called the Bethe lattice in the physics literature). Its vertex set can be identified with the group $G_k$, the free product of $k+1$ cyclic groups of order two with generators $a_1,\dots,a_{k+1}$: each vertex is a word in the generators, and left multiplication realizes the natural translations of the tree. This group structure gives precise meaning to symmetry classes of measures (translation-invariant, periodic, weakly periodic).

The two-state hard-core (HC) model is the lattice gas of particles with excluded volume, equivalently the random independent-set model on the tree: configurations $\sigma\in\{0,1\}^V$ are admissible only if no two nearest neighbours are both occupied ($\sigma(x)\sigma(y)=0$ for every edge $\langle x,y\rangle$), with Hamiltonian $H(\sigma)=J\sum_{x\in V}\sigma(x)$ and activity (fugacity) $\lambda=e^{-J\beta}>0$ at inverse temperature $\beta$. A Gibbs measure is a probability measure on the admissible configurations satisfying the Dobrushin-Lanford-Ruelle (DLR) consistency condition between finite volumes and their boundary conditions. Each Gibbs measure represents an equilibrium phase, and non-uniqueness of Gibbs measures is the rigorous notion of a phase transition. The set of all Gibbs measures for a given Hamiltonian is a non-empty convex compact set, so every Gibbs measure decomposes uniquely over extreme points; describing all Gibbs measures and their extreme points is the central classification problem of the theory, and, as the source paper notes in its introduction, it has not been completed for any model.

On a tree, Gibbs measures of the HC model admit a boundary-law (splitting) representation: each Gibbs measure determines a collection of positive numbers $z_x>0$ assigned to the vertices and satisfying the functional equation $z_x=\prod_{y\in S(x)}(1+\lambda z_y)^{-1}$, where $S(x)$ is the set of direct successors of $x$ away from the root. Constant boundary laws give the translation-invariant measures, and the two-state HC model has exactly one translation-invariant Gibbs measure $\mu^*$ for every $\lambda$. A collection $z$ is called $\tilde G$-periodic for a subgroup $\tilde G\subseteq G_k$ of finite index if $z_{yx}=z_x$ for all $y\in\tilde G$. The periodic layer of the model is completely classified: every periodic Gibbs measure is either $\mu^*$ or one of the two-periodic ('chessboard') measures, which exist precisely for $\lambda>\lambda_{\rm cr}=k^k/(k-1)^{k+1}$ and form a pair $\mu_1,\mu_2$. Concerning purity, $\mu^*$ is extreme for small $\lambda$ and non-extreme for large $\lambda$ (a sufficient non-extremality condition is due to Suhov and Rozikov, and the extremality of $\mu^*$ is governed by reconstruction on trees as shown by Martin - reconstruction asks whether the value at the root of the tree can be recovered, with accuracy better than chance, from the values at vertices arbitrarily far away, and reconstructability of a measure is equivalent to its non-extremality), while $\mu_1$ and $\mu_2$ are extreme throughout their existence region (with an alternative proof via the reconstruction method for $k=2,3$).

Weakly periodic measures form the next symmetry layer. Fix a non-empty proper subset $A\subset\{1,\dots,k+1\}$ and put $i=|A|$; the set $H_A$ of words in which the generators $\{a_j: j\in A\}$ occur an even number of times is a subgroup, and $G_k^{(4)}=H_A\cap G_k^{(2)}$ (where $G_k^{(2)}$ is the subgroup of even-length words) is a normal divisor of index four. A collection $z$ is $G_k^{(4)}$-weakly periodic if $z_x=z_{ij}$ depends only on the pair of cosets $(i,j)$ of $x$ and its ancestor $x_\downarrow$ (the unique nearest neighbour of $x$ on the path to the root) in the factor group $G_k/G_k^{(4)}$, which has four cosets. In the parametrization of the source papers such a collection is described by four real values $(z_1,z_2,z_7,z_8)$, and solving the boundary-law equation under this ansatz reduces to finding fixed points of an explicit continuous map $W:\mathbb{R}^4\to\mathbb{R}^4$; the map leaves invariant the four sets $I_1=\{z_1=z_2=z_7=z_8\}$ (the diagonal, giving $\mu^*$), $I_2=\{z_1=z_7,\ z_2=z_8\}$, $I_3=\{z_1=z_2,\ z_7=z_8\}$ and $I_4=\{z_1=z_8,\ z_2=z_7\}$. A lemma of the theory states that on $I_2,I_3,I_4$ every weakly periodic Gibbs measure is either translation-invariant or genuinely weakly periodic, i.e. distinct from all periodic measures above. The principal counting tool is a lemma of Kesten on quadratic transformations: if a continuous self-map $f$ of an interval has a fixed point $\xi$ with $f'(\xi)<-1$, then $f$ has a two-cycle, which in the symmetric two-variable reductions produces additional fixed-point pairs.

For normal divisors of index two the weakly periodic classification is complete for all parameter values: every weakly periodic Gibbs measure coincides with $\mu^*$. Index four is therefore the first level at which genuinely new Gibbs measures appear, and there the classification is only partial: exact counts are known only for trees of order two and for one case on the order-three tree, a finite list of $(k,i)$ combinations is settled by uniqueness theorems, and for $k\ge 6$ only a lower bound is available. Whether the new measures are extreme, and how all branches fit together in a phase diagram, is unknown; and beyond all periodic and weakly periodic classes, the grand problem of describing all limiting Gibbs measures (including the non-periodic 'alternative' Gibbs measures discovered in 2023) remains open.

Problem Statement

For the two-state hard-core model with activity $\lambda>0$ on the Cayley tree $\Gamma^k$ of order $k\ge 2$, and for each normal divisor of index four of the form

$G_k^{(4)}=H_A\cap G_k^{(2)},\qquad H_A=\Big\{x\in G_k:\ \sum_{j\in A} w_x(a_j)\ \text{even}\Big\},\qquad i=|A|,$

where $G_k$ is the free product of $k+1$ copies of $\mathbb{Z}_2$ acting on the tree, $G_k^{(2)}$ is its subgroup of even-length words, $w_x(a_j)$ counts occurrences of the generator $a_j$ in the word $x$, and $A\subset\{1,\dots,k+1\}$ with $1\le i\le k$, determine:

(a) for every pair $(k,i)$ and every $\lambda>0$, the exact number of $G_k^{(4)}$-weakly periodic Gibbs measures - equivalently, the exact number of fixed points of the induced boundary-law map $W:\mathbb{R}^4\to\mathbb{R}^4$ on each of its invariant sets $I_1,\dots,I_4$, where $W$ acts on the four values $(z_1,z_2,z_7,z_8)$ of the weakly periodic collection and $I_1=\{z_1=z_2=z_7=z_8\}$, $I_2=\{z_1=z_7,\ z_2=z_8\}$, $I_3=\{z_1=z_2,\ z_7=z_8\}$, $I_4=\{z_1=z_8,\ z_2=z_7\}$ - together with the exact critical values of $\lambda$ at which this number changes (the full bifurcation structure in $\lambda$, including the counts at the critical values themselves);

(b) for each non-translation-invariant weakly periodic Gibbs measure identified in (a), whether it is an extreme point of the convex set of all Gibbs measures of the model;

(c) the assembled global $(k,i,\lambda)$ phase diagram of weakly periodic Gibbs measures. This is a presentation of the results of (a) only, adding no requirement beyond (a): it consolidates the exact counts and the exact critical $\lambda$ values obtained there into a single $(k,i,\lambda)$-indexed description.

Here a $G_k^{(4)}$-weakly periodic Gibbs measure is one corresponding to a collection $z_x=z_{ij}$ that depends only on the pair of cosets $(i,j)$ of $x$ and its ancestor in the factor group $G_k/G_k^{(4)}$; note that $A$ and its complement in $\{1,\dots,k+1\}$ define the same subgroup (since $\sum_{j\notin A}w_x(a_j)\equiv |x|-\sum_{j\in A}w_x(a_j)\pmod 2$), so the cases $(k,i)$ and $(k,k+1-i)$ concern the same subgroup up to relabelling of cosets and invariant sets; the parametrization by $i$ is retained as in the cited papers.

The following partial results from the cited literature calibrate the request but do not bound it: on $I_1$ (all $(k,i)$), on $I_3$ for $i=1$ and for $k=i$, and on $I_4$ for $(k,i)\in\{(2,1),(3,1),(3,2),(4,1),(4,2),(4,3),(5,1)\}\cup\{(k,k):k\ge 2\}$, the weakly periodic Gibbs measure is unique and coincides with the translation-invariant measure; on $I_2$ for $(k,i)=(2,1)$ and $(2,2)$ exactly two non-translation-invariant weakly periodic measures exist for $\lambda>4$; on $I_2$ for $(k,i)=(3,1)$ exactly two non-translation-invariant weakly periodic measures exist for $\lambda>27/16$; and on $I_4$ for $k\ge 6$, $i=1$, at least three weakly periodic Gibbs measures exist for $\lambda\in(\lambda_-(k),\lambda_+(k))$ (one of them translation-invariant, the others weakly periodic nonperiodic), with $\lambda_\pm(k)=(s_\pm+1)^k s_\pm$ and $s_\pm=(k-3\pm\sqrt{k^2-6k+1})/4$. Everything else - the exact counts for all remaining $(k,i)$ over the whole $\lambda$-axis, the critical values, the extremality determinations in (b), and the diagram in (c) - is the open content. Completing (a)-(c) settles the weakly periodic layer of the broader open problem of describing all limiting Gibbs measures of the hard-core model on Cayley trees, beyond which still wider classes (such as the alternative Gibbs measures discovered in 2023) remain to be mapped.

References

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