The phase diagram of the loop O(n) model on the hexagonal lattice, its hard-hexagon limit, and exponential decay of correlations in the planar spin O(n) model

The loop O(n) model — a random collection of mutually avoiding loops on the hexagonal lattice, with edge weight x and loop weight n — is the standard loop-level counterpart of the planar spin O(n) model. Duminil-Copin, Peled, Samotij and Spinka (2016) proved that for large n long loops are exponentially unlikely uniformly in x, exhibited a dilute disordered phase when n x^6 is small and a dense ordered phase, described by at least three Gibbs measures perturbing the three ground states, when n x^6 is large, and posed the open problems addressed here. The problem is to complete the rigorous phase picture of the two-dimensional O(n) system: establish or refute Nienhuis's predicted transition at x_c(n)=1/sqrt(2+sqrt(2-n)) for 0<n<=2, including its predicted Kosterlitz–Thouless nature; prove uniqueness of the Gibbs measure in the dilute large-n regime under arbitrary boundary conditions; classify the Gibbs measures in the dense regime; determine whether the large-n transition occurs at a single point x_c(n), what happens at criticality, and whether n x_c(n)^6 converges to Baxter's hard-hexagon critical fugacity; and prove Polyakov's conjecture that spin–spin correlations in the planar spin O(n) model decay exponentially for integer n>=3 at all positive temperatures.

Background

The loop O(n) model is a probability model introduced by Domany, Mukamel, Nienhuis and Schwimmer (1981) as a loop-level counterpart of the classical spin O(n) model of statistical mechanics. A configuration is a collection of mutually avoiding self-avoiding loops drawn on the edges of the hexagonal lattice (equivalently, a subgraph in which every vertex has degree 0 or 2); a configuration $\omega$ receives weight $x^{\lvert\omega\rvert}n^{\ell(\omega)}$, where $\lvert\omega\rvert$ is the number of occupied edges, $\ell(\omega)$ is the number of loops, $x>0$ is the edge weight and $n>0$ is the loop weight. Unlike the spin O(n) model — in which spins take values on the sphere $S^{n-1}$ and interact ferromagnetically, so that only integer $n$ are meaningful — the loop model is defined for every real $n>0$. The two are linked by a high-temperature expansion: the spin–spin correlation function of the planar spin O(n) model can be rewritten, heuristically, as a weighted sum over loop configurations together with two paths connecting the two spins, so that the typical loop length is a proxy for the decay of spin correlations. This expansion is not justified rigorously for all $x$, which is why loop-model results are analogues of, not substitutes for, spin-model statements.

Nienhuis (1982) predicted, from Coulomb-gas arguments, that for $0\le n\le 2$ the loop model on the hexagonal lattice has a critical edge weight

$x_c(n)=\frac{1}{\sqrt{2+\sqrt{2-n}}},$

separating a dilute phase ($x<x_c(n)$: loop lengths have exponential tails) from a dense phase ($x\ge x_c(n)$: loops occur on macroscopic scales), with the transition at $x_c(n)$ predicted to be of Kosterlitz–Thouless type for $0\le n\le 2$ — a transition whose correlation length diverges exponentially rather than polynomially, as in the two-dimensional XY model — in particular at the endpoint $n=2$, and conformal invariance in the dense phase. Rigorous confirmation was long limited to $n=1$ (the loops are then the domain walls of the triangular-lattice Ising model, with $x_c=1/\sqrt3$ known from the exact Ising solution) and $n=0$ (self-avoiding walks: Duminil-Copin and Smirnov proved that the connective constant — the exponential growth rate of the number of $k$-step self-avoiding walks on the hexagonal lattice — equals $\sqrt{2+\sqrt2}$, i.e. $x_c(0)=1/\sqrt{2+\sqrt2}$).

Duminil-Copin, Peled, Samotij and Spinka (2016) initiated the rigorous study of large $n$. They proved that for $n$ above an absolute constant $n_0$, uniformly in $x$, the probability that a loop of length at least $k$ surrounds a fixed vertex decays exponentially in $k$; that when $n x^6$ is sufficiently small (more precisely when $\max\{n,1\}x^6$ is small) the infinite-volume limits under vacant boundary conditions form a dilute, disordered phase in which a vertex is unlikely to be surrounded by any loop at all; and that when $n x^6$ is sufficiently large there exist at least three Gibbs measures — infinite-volume probability measures satisfying the DLR (Dobrushin–Lanford–Ruelle) consistency equations — each a small perturbation of one of the three ground states. The ground states are the three tilings of the plane by disjoint elementary hexagons (loops of length 6) centered on the three sublattices of the triangular lattice; this explains the combination $n x^6$, since the shortest possible loop has 6 edges. Passing formally to the limit $n\to\infty$ with $n x^6\to\lambda$ eliminates all non-trivial loops and leaves the hard-hexagon model: the hard-core lattice gas on the triangular lattice at fugacity $\lambda$, in which each site may be occupied only if its neighbors are not. Baxter solved that model exactly, showing that it has a critical fugacity

$\lambda_c=\tfrac{1}{2}\bigl(11+5\sqrt5\bigr)\approx 11.09017$

above which a second-order fluid–solid transition favors occupation of one sublattice. Baxter's derivation, however, rests on assumptions that have not been mathematically justified.

The 2016 paper closes with the following open questions: the phase diagram for small $n\le 2$; uniqueness of the Gibbs measure in the dilute regime under arbitrary, adversarially chosen boundary conditions; whether the three constructed measures exhaust the extremal Gibbs measures in the dense regime — an analogue of the Aizenman–Higuchi theorem, which states that the only extremal Gibbs states of the two-dimensional ferromagnetic Ising model are its two pure phases (an extremal measure is one that cannot be written as a nontrivial convex combination of other Gibbs measures), and which was extended to Potts models by Coquille, Duminil-Copin, Ioffe and Velenik; the structure of the intermediate regime $c<n x^6<C$, including whether there is a single transition point $x_c(n)$, what happens at it, and whether $n x_c(n)^6\to\lambda_c$ as $n\to\infty$; and the motivating spin-model conjecture of Polyakov (1975) that spin–spin correlations in the planar spin O(n) model decay exponentially for integer $n\ge 3$ at every positive temperature — the Mermin–Wagner theorem forbids long-range order in two dimensions for continuous symmetries, quasi-long-range order is expected at $n=2$, and exponential decay (mass generation) is expected for $n\ge 3$. A further open endpoint, not part of the components posed below, is extending the uniform-in-$x$ exponential decay of loop lengths from $n\ge n_0$ to all $n>2$.

Problem Statement

Complete the rigorous determination of the phase and correlation structure of the two-dimensional O(n) system, as posed in Section 4 of Duminil-Copin, Peled, Samotij and Spinka (2016). The problem has five linked components, which jointly determine the phase diagram of the loop O(n) model on the hexagonal lattice, its $n\to\infty$ asymptotics, and the spin-model statement these loop results approximate:

1. Small-n regime. For $0<n\le2$, establish or refute the predicted phase diagram: a transition at $x_c(n)=1/\sqrt{2+\sqrt{2-n}}$ between exponential decay of loop lengths when $x<x_c(n)$ and macroscopic loops when $x\ge x_c(n)$, including the Kosterlitz–Thouless nature of the transition predicted for $0<n\le2$ (in particular at the endpoint $n=2$) and the predicted conformal invariance for $x\ge x_c(n)$. If the prediction fails at some $n$, determine the correct phase diagram (location and order of the transition, and the behavior of loop lengths on both sides).

2. Dilute regime, large n. In the regime where $\max\{n,1\}x^6$ is sufficiently small, prove or disprove that the infinite-volume Gibbs measure is unique — that limits taken under arbitrary boundary conditions coincide with the vacant-boundary limit.

3. Dense regime, large n. In the regime where $n x^6$ is sufficiently large, prove or disprove that the three constructed sublattice-pure measures are the only extremal Gibbs measures — equivalently, that every Gibbs measure (translation-invariant or not) is a convex combination of the three.

4. Transition structure and the hard-hexagon limit. For large $n$, determine whether there is a single transition point $x_c(n)$ separating the dilute disordered phase from the dense ordered phase; establish the behavior of the model at $x=x_c(n)$ (uniqueness of the Gibbs state, order of the transition); and prove or disprove that $n x_c(n)^6\to\lambda_c=\tfrac12(11+5\sqrt5)$ as $n\to\infty$, where $\lambda_c$ is Baxter's critical fugacity for the hard-hexagon model.

5. Spin counterpart (Polyakov's conjecture). Prove or refute that spin–spin correlations in the planar spin O(n) model with integer $n\ge3$ decay exponentially in the distance at every positive temperature — the conjecture that the loop-model results of the source approximate.

An acceptable answer may confirm the conjectured picture or prove a corrected one where the prediction fails, but it must resolve the stated components rather than a surrogate or narrowed target.

References

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