Zero-temperature crystallization of the doubly periodic Aztec diamond dimer model beyond smooth tropical curves and iterated limits

Berggren and Borodin proved that the dimer model on the Aztec diamond with doubly periodic edge weights $\nu(\cdot)^\beta$ crystallizes as the inverse temperature $\beta\to\infty$: for generic weights — equivalently, when the associated tropical curve $\mathcal{A}_t$ is smooth — the limit shape converges to a piecewise-linear tropical limit shape, the arctic curve degenerates to a tropical arctic curve described by a tropical action function solving a Kirchhoff problem on $\mathcal{A}_t$, and local statistics converge to explicit measures $\mathbb{P}_\mu$ supported on periodic subgraphs $G_\mu$ of the square lattice built from the maximizers of the energy. The authors themselves formulate five interlinked questions that must be settled to complete this theory in full generality: (a) the possible behaviors of the limit shape when $\mathcal{A}_t$ is not smooth; (b) whether strict concavity of the tropical surface tension, or piecewise linearity of the limit shape, forces all connected components of $G_\mu$ to be bounded (prove or disprove); (c) whether $\beta^{-1}\sigma_\beta$ converges uniformly to $-\mathcal{E}^*$ on the whole Newton polygon without restrictions on the edge weights (prove or disprove); (d) the joint double limit in which $\beta$ and the diamond's size tend to infinity simultaneously; and (e) convergence of the finite-$\beta$ arctic curves to the tropical arctic curve as planar sets. No resolution of any of these five items is known in the literature.

Background

A domino tiling of an Aztec diamond — the diamond-shaped region of the square lattice consisting of the squares $|x|+|y|\le N$ — is equivalent to a dimer cover (a perfect matching: a set of edges covering every vertex exactly once) of the associated Aztec diamond graph $G_{\mathrm{Az}}$. Equipping edges with positive weights makes the set of tilings into a statistical-mechanical model: a random dimer model in which each dimer cover $\mathcal{D}$ has probability proportional to the product of its edge weights. Berggren and Borodin consider doubly periodic edge weights of the form $\nu(\cdot)^\beta$, where $\nu$ is a fixed periodic function on the edges with a $k\times\ell$ fundamental domain and $\beta$ plays the role of inverse temperature: as $\beta\to\infty$ the measure concentrates on the covers maximizing the energy $\sum_{e\in\mathcal{D}}\log\nu(e)$.

Two classical results frame the subject. Cohn–Kenyon–Propp proved a variational principle for domino tilings: the macroscopic height function (the surface encoding a tiling, whose local gradient records which domino orientations are used) converges for large regions to the maximizer of a concave functional of the dimer entropy. Kenyon–Okounkov–Sheffield classified periodic dimer models through their spectral data: the characteristic polynomial $P(z,w)$ of the Kasteleyn matrix — the signed adjacency matrix, introduced in the 1960s solution of the planar dimer model, whose minors count dimer covers — defines a Harnack curve, and its amoeba (the image of the curve under $(z,w)\mapsto(\log|z|,\log|w|)$) governs the phase diagram: frozen facets (where the tiling is asymptotically deterministic), a disordered rough region, and possibly smooth phases, separated by the arctic curve. The entropy is expressed through the surface tension $\sigma_\beta$, the Legendre transform of the Ronkin function $R_\beta$ of the polynomial $P_\beta$ (the $\log$-modulus average of $P_\beta$ over the torus $|z|=|w|=1$).

Berggren–Borodin (2023) carried out this program for doubly periodic Aztec diamonds at fixed $\beta$, expressing the arctic curve, the limit shape, and the local statistics in terms of the spectral curve and an action function. The 2024 paper "Crystallization of the Aztec diamond" analyzes the zero-temperature limit $\beta\to\infty$. The (rescaled) amoeba degenerates to a tropical curve $\mathcal{A}_t$: a planar graph with straight edges of rational slope, dual to the subdivision $N_S(P_t)$ of the Newton polygon $N(P)$ (the convex hull of the exponent vectors of $P$), the subdivision induced by the tropical surface tension $\mathcal{E}^*(\mu)=\max_{\mathcal{D}}\sum_{e\in\mathcal{D}}\log\nu(e)$, where the maximum is over dimer covers of the fundamental domain with slope $\mu$ (the average height gradient of the periodically lifted cover). The tropical curve is called smooth when $N_S(P_t)$ is a triangulation into triangles of area $1/2$ (equivalently, all $2k\ell$ vertices of $\mathcal{A}_t$ are trivalent); the authors prove this holds outside finitely many hyperplanes in the $k\ell$-dimensional space of weights $\log\nu(\cdot)$, i.e. generically.

Under this smoothness assumption the paper proves crystallization: the limit shape converges to the tropical limit shape $\bar h_t$, a continuous piecewise-linear function; the tropical arctic curve (the union of the polygonal boundaries of the facets $R_\mu$) is described through the tropical action function $F_t$, which solves a Kirchhoff-type problem on $\mathcal{A}_t$ (the classical network problem of finding a harmonic function or edge current with prescribed residues at the unbounded edges, in the tropical limit of the imaginary-normalized differentials studied by Lang); for each facet slope $\mu$, local statistics converge (Aztec diamond size $\to\infty$ at fixed sufficiently large $\beta$) to the ergodic translation-invariant Gibbs measure of slope $\mu$ (the extremal infinite-volume dimer measure on $\mathbb{Z}^2$ with fixed average height gradient); these Gibbs measures converge as $\beta\to\infty$ to explicit measures $\mathbb{P}_\mu$ supported on the periodic subgraph $G_\mu\subset\mathbb{Z}^2$ spanned by the energy-maximizing dimer covers of the fundamental domain; and $\beta^{-1}\sigma_\beta(\mu)\to-\mathcal{E}^*(\mu)$ at slopes where $\mathcal{E}^*$ is strictly concave.

The authors are explicit that this picture is complete only under the smoothness/genericity hypothesis and under iterated limits, and they single out five gaps: (i) after Theorem 1.1, "It would be very interesting to understand what can happen to the limit shape in the $\beta\to\infty$ limit if $\mathcal{A}_t$ is not a smooth tropical curve, and we leave this question for future research" (e.g. when all weights are $1$ the model does not depend on $\beta$ at all, so a piecewise-linear limit is not automatic); (ii) after Theorem 1.4, "One may hope that the same measures $\mathbb{P}_\mu$ should arise as local limits as both $\beta$ and the size of the Aztec diamond tend to infinity. Such double limit transitions are beyond the scope of the present paper"; (iii) the finite-$\beta$ theory shows the $\beta\to\infty$ limit of the Gibbs measures is the product of uniform dimer models on the connected components of $G_\mu$ whenever those components are finite, and "It seems likely that this should always happen when the zero-temperature limit shape is piecewise linear, or at least when the surface tension is strictly concave, but we have not been able to verify that so far" — with neither a proof nor a counterexample found (Remark 4.6), the paper proving only that strict concavity together with exactly two maximizers implies bounded components (Corollary 4.10); (iv) Remark A.2, "we expect that $\beta^{-1}\sigma_\beta\to-\mathcal{E}^*$ uniformly as $\beta\to\infty$, and this limit holds without any restrictions on the edge weights", with the pointwise statement proved only under strict concavity (Proposition A.1); (v) after Theorem 3.14 (which proves pointwise convergence of the macroscopic regions $R_{\beta,\mu}\to R_\mu$), "While convergence of the arctic curves themselves might be an interesting avenue for future investigation, this is not the focus of this paper".

Problem Statement

Complete the zero-temperature crystallization theory of Berggren–Borodin for the Aztec diamond dimer model with doubly periodic edge weights $\nu(\cdot)^\beta$, by removing its genericity hypotheses and iterated-limit structure. The common objective is a crystallization picture valid for arbitrary doubly periodic weights, in the natural joint scaling limit, with each convergence statement at its natural strength. Concretely, establish the following five interlinked results.

(a) Non-smooth tropical regimes: classify the possible behaviors of the zero-temperature ($\beta\to\infty$) limit shape of the doubly periodic Aztec diamond dimer model when the tropical curve $\mathcal{A}_t$ is not smooth, i.e. when the Newton subdivision $N_S(P_t)$ contains faces that are not area-$1/2$ triangles or $\mathcal{A}_t$ has vertices of valency other than three. In particular: does the normalized height function still converge to a limit (piecewise linear or otherwise), and what is the structure of the limiting phases and of the arctic curve? A complete answer to (a) is a general classification of the possible non-smooth behaviors, of which the two questions it poses are the concrete instances to settle; exhibiting a single non-smooth example does not by itself constitute that classification.

(b) Boundedness of $G_\mu$: for the periodic subgraph $G_\mu\subset\mathbb{Z}^2$ built from the maximizers of the energy $\sum_e\log\nu(e)$ over slope-$\mu$ dimer covers of the fundamental domain, prove or disprove each of the following two claims, which are to be settled separately. (b1) If the tropical surface tension $\mathcal{E}^*$ is strictly concave at the slope $\mu$, then all connected components of $G_\mu$ are bounded. (b2) If the zero-temperature limit shape is piecewise linear, then all connected components of $G_\mu$ are bounded. A complete answer to (b) is a complete answer to each of (b1) and (b2) separately, and each of (b1) and (b2) is settled by either a proof or an explicit counterexample of that claim. (The paper proves this only under the additional assumption of exactly two maximizers, Corollary 4.10, and records that neither a proof nor a counterexample is known in general.)

(c) Uniform convergence of the surface tension: prove or disprove that $\beta^{-1}\sigma_\beta\to-\mathcal{E}^*$ uniformly on the Newton polygon $N(P)$ as $\beta\to\infty$, with no restrictions on the doubly periodic edge weights, where $\sigma_\beta$ is the surface tension (the Legendre transform of the Ronkin function of $P_\beta$) and $\mathcal{E}^*$ is extended piecewise linearly to $N(P)$; the paper proves only pointwise convergence at slopes of strict concavity.

(d) Joint double limit: determine the local limit of the dimer model on the growing Aztec diamond as both the size and $\beta$ tend to infinity simultaneously — in particular, whether the local statistics at points of the eventual facets converge to the measures $\mathbb{P}_\mu$ (the joint version of Theorems 1.3 and 1.4); identify whether restrictions on the relative rates are needed, and any new scaling phenomena at coupled rates. A complete answer to (d) is the general identification of the joint limit, of which convergence to the measures $\mathbb{P}_\mu$ and the relative-rate statements it names are components.

(e) Arctic curves as planar sets: formulate and prove convergence of the finite-$\beta$ arctic curves to the tropical arctic curve as subsets of the plane, in a precise standard sense (for instance Hausdorff convergence on compacta, or convergence of the closures of the rough regions), going beyond the pointwise convergence of the phase regions $R_{\beta,\mu}\to R_\mu$ proved in Theorem 3.14. Here the finite-$\beta$ arctic curve is the boundary of the rough region: the locus of macroscopic points at which the differential $\mathrm{d}F_\beta$ of the action function $F_\beta$ on the spectral curve (the function whose zeros encode the local phase) has zeros of higher order; the rough region is where $\mathrm{d}F_\beta$ has non-real zeros, and $R_{\beta,\mu}$ is the region of the frozen or smooth phase corresponding to the slope $\mu$. A complete answer to (e) is a precise formulation of the convergence — choosing and justifying a specific sense of set convergence — together with its proof.

The five parts are linked as one objective: (a) and (c) both remove the genericity restriction — (c) is the variational-data counterpart that underlies the limit shapes in (a); (b) determines the structure of the limiting measures $\mathbb{P}_\mu$ appearing in (d); (d) merges the two iterated limits into the physically natural crystallization statement; and (e) is the geometric counterpart of (a)–(d) at the phase boundaries. Each of (a)–(e) may be answered independently, and a complete answer to any single part counts as a full answer to that part; in particular, for the claims in (b) and (c), an explicit counterexample to a stated implication counts as a full answer to that claim.

References

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