Phase diagram of the loop $O(n)$ model with large $n$ and its hard-hexagon limit

The loop $O(n)$ model on the hexagonal lattice is a probability measure on collections of non-intersecting loops in which each edge receives weight $x$ and each loop weight $n$. Duminil-Copin, Peled, Samotij and Spinka (2016) proved that for large $n$ loop lengths decay exponentially at every $x$, identified a dilute disordered phase when $n x^6$ is small and a dense ordered phase with three distinct sublattice-ordered Gibbs measures when $n$ and $n x^6$ are large, and posed the following open questions: whether a single critical point $x_c(n)$ separates the two phases and what its structure is; whether $n x_c(n)^6$ converges, as $n\to\infty$, to Baxter's hard-hexagon critical fugacity $\lambda_c=(2\cos(\pi/5))^5$, as predicted by the formal convergence of the model to the hard-hexagon model; and whether the three ordered Gibbs measures are the only extremal ones in the ordered regime while the Gibbs measure is unique in the dilute regime under arbitrary boundary conditions. The large-$n$ phase-diagram questions, including the hard-hexagon limit, remain open; progress to date has been limited to the $n\le 2$ critical regime and to adjacent integrable or monotone models.

Background

The loop $O(n)$ model is a model of random non-intersecting loops on the hexagonal lattice $\mathbb{H}$. A loop configuration is a subgraph of $\mathbb{H}$ in which every vertex has degree either 0 or 2, so that its edges decompose into disjoint closed loops; the probability of a configuration $\omega$ in a finite domain is proportional to $x^{|\omega|} n^{\ell(\omega)}$, where $|\omega|$ is the number of edges, $\ell(\omega)$ the number of loops, $x\ge 0$ is the edge weight and $n\ge 0$ is the loop weight. The model was introduced in the physics literature as an approximation of the spin $O(n)$ model — spins on $\mathbb{H}$ taking values on a sphere in $\mathbb{R}^n$ with ferromagnetic nearest-neighbor interaction — whose high-temperature expansion produces loop configurations weighted this way; the two models are believed to lie in the same universality class, though the expansion relating them is heuristic. Special cases include the self-avoiding walk ($n=0$), Ising domain walls ($n=1$) and critical percolation ($n=x=1$). Nienhuis predicted that for $0\le n\le 2$ the model is critical on the curve $x_c(n)=1/\sqrt{2+\sqrt{2-n}}$, separating exponentially decaying loop lengths ($x<x_c$) from macroscopic loops ($x\ge x_c$), while for $n>2$ loop lengths should decay exponentially at every $x$.

Infinite-volume behavior is encoded in Gibbs measures (DLR measures): probability measures on loop configurations of $\mathbb{H}$ whose conditional distributions on finite domains coincide with the finite-volume measures defined by the weight above. Duminil-Copin, Peled, Samotij and Spinka (2016) proved that for all sufficiently large $n$ the length of a loop surrounding a given vertex has an exponential tail uniformly in $x$, and described two phases: when $n x^6$ is small (and $x$ small), configurations under vacant boundary conditions (boundary conditions forcing no loop edges on the boundary) converge to a dilute, disordered Gibbs measure supported on configurations with no infinite paths, whereas when $n$ and $n x^6$ are large there are at least three distinct Gibbs measures, each a small perturbation of one of the three periodic ground states obtained by filling one sublattice of the dual triangular lattice $\mathbb{T}$ with trivial loops of length 6 (a single such loop carries weight $n x^6$). This suggests that, as $n\to\infty$ with $\lambda = n x^6$ held fixed, the model converges formally to the hard-hexagon model — the hard-core lattice gas on $\mathbb{T}$ at fugacity $\lambda$, in which occupied sites may not be adjacent — solved exactly by Baxter, who computed its critical fugacity $\lambda_c=(2\cos(\pi/5))^5=(11+5\sqrt5)/2\approx 11.09017$ and found a second-order fluid–solid transition there; Baxter's derivation, however, rests on assumptions that have not been mathematically justified. What remains open is the structure between the two regimes already controlled: whether for each large $n$ there is a single critical point $x_c(n)$ and what happens there, whether $n x_c(n)^6\to\lambda_c$, and how many extremal Gibbs measures exist in each phase. The main methodological obstacles are that the model has no known positive-association (FKG) property in this regime (an FKG property is known only for $n\ge 1$ and $n x^2\le 1$), and that the Aizenman–Higuchi-type classification of Gibbs states available for the planar Ising and Potts models — which identifies all extremal Gibbs measures as the pure phases — relies on random-cluster representations (the FK-percolation model, an edge-percolation representation of Potts spin systems from which their Gibbs states can be classified) which the loop model lacks; at $x=\infty$ (and at $x=0$) many further Gibbs measures can be constructed, so the classification questions concern finite $x$.

Problem Statement

For the loop $O(n)$ model on the hexagonal lattice with loop weight $n$ and edge weight $x$, determine the phase structure for all sufficiently large $n$ and finite $x>0$. Specifically:

(1) Transition structure: prove or disprove that there is a single critical value $x_c(n)\in(0,\infty)$ such that for $x<x_c(n)$ the model is in a dilute, disordered phase while for $x>x_c(n)$ it is in a dense, ordered phase approximating one of the three sublattice ground states; and determine the structure at $x=x_c(n)$, namely the number of extremal Gibbs measures at criticality and the order of the transition in $x$ (predicted to be second order, with a unique Gibbs state at $x_c(n)$, by analogy with the hard-hexagon model).

(2) Hard-hexagon limit: prove or disprove that $n\,x_c(n)^6\to\lambda_c=(2\cos(\pi/5))^5=(11+5\sqrt5)/2$ as $n\to\infty$, where $\lambda_c$ is Baxter's critical fugacity of the hard-hexagon model on the triangular lattice, consistent with the formal convergence of the loop model to the hard-hexagon model as $n\to\infty$ with $\lambda = n x^6$ fixed.

(3) Gibbs-state classification: (a) in the ordered regime ($n\ge C$ and $n x^6\ge C$ for some constant $C$, with $x<\infty$), determine whether every Gibbs measure is a convex combination of the three sublattice-ordered Gibbs measures constructed by Duminil-Copin–Peled–Samotij–Spinka, i.e. whether these are the only extremal ones; (b) in the dilute regime ($n x^6\le c$ for some constant $c$), determine whether the Gibbs measure is unique among all boundary conditions, in particular ruling out limiting Gibbs measures having an infinite interface passing near the origin with positive probability.

These three components jointly determine the predicted phase diagram; the common objective is a complete characterization of the transition structure and the set of Gibbs states of the model for large $n$, together with its limiting relation to Baxter's exactly solved hard-hexagon model. In particular, the limit in component (2) is posed conditional on a critical value $x_c(n)$ as in component (1) existing.

References

1. Hugo Duminil-Copin, Ron Peled, Wojciech Samotij, Yinon Spinka, Exponential decay of loop lengths in the loop $O(n)$ model with large $n$, Communications in Mathematical Physics (2016), DOI 10.1007/s00220-016-2815-4, arXiv:1412.8326, https://arxiv.org/abs/1412.8326
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