Disorder-induced regions and the height-function scaling limit in the square-lattice dimer model with layered quenched disorder

The dimer model on the square lattice with row-constant i.i.d. quenched random edge weights (layered disorder, in the spirit of the McCoy-Wu random-bond Ising model) was solved at the level of the quenched free energy by Moulard and Toninelli (2025), who found an essential singularity at the origin where the homogeneous model is analytic, disorder-induced regions $C_{\pm}$ and $G_{\gamma}$ in the phase diagram where the free energy is flat in one magnetic field, a liquid/gas transition order varying continuously with the staggering parameter $\gamma$, and stretched-exponential correlations at the singularity. This problem asks for the pointwise characterization that those free-energy results leave open for the same model: the local thermodynamic and correlation behavior inside the disorder-induced regions $C_{\pm}$ and $G_{\gamma}$ (in particular whether the system is frozen on every compact subset of them, and the geometry and regularity of their boundaries), the scaling limit of the height function in the liquid region (whether, as in the homogeneous model, it converges to a Gaussian free field, and if so with which effective coupling, or whether layered disorder changes the universality class), and the sharp pointwise asymptotics $\Delta_{\gamma}(H_2)-\log\gamma\sim c_{\alpha(\gamma)}|H_2|^{2\min(\alpha(\gamma),1)}$ conjectured by the authors, of which only an integrated version is proven.

Background

The dimer model is a classical exactly solvable lattice model: a configuration is a perfect matching of a graph (a set of edges covering every vertex exactly once; each edge is a 'dimer'), weighted by the product of its edge weights. On planar bipartite graphs such as the square lattice $\mathbb{Z}^2$ it is solvable by Kasteleyn's theory, which expresses the partition function and correlations through a signed adjacency matrix. A dimer covering of a planar graph is encoded by an integer-valued height function on the faces, changing by prescribed increments across each edge according to whether the edge carries a dimer, so the model is equivalently a random surface. Two 'magnetic fields' $(H_1,H_2)$ are exponential reweightings that pin the two components of the average height gradient, i.e., the density imbalance between edge types. For the homogeneous (non-random) model, Kenyon (2001) proved that in the 'liquid' (massless) phase the rescaled height function converges to the Gaussian free field (GFF), the massless scalar field of two-dimensional field theory characterized by logarithmically growing covariances, with dimer-dimer correlations decaying like the inverse squared distance. Kenyon, Okounkov and Sheffield (2006) completed the phase diagram: unbounded 'frozen' (solid) regions where the system locks into one periodic covering and correlations vanish identically, possible bounded 'gaseous' (massive) regions with exponential correlation decay, and the liquid region with strictly convex real-analytic free energy; at generic liquid/frozen and liquid/gaseous boundaries the free energy has a singularity of order $3/2$ in the distance to the boundary (the Pokrovsky-Talapov law). How quenched randomness — disorder chosen once and for all, with 'quenched' quantities computed for a typical fixed realization and 'annealed' ones averaged over the disorder — modifies this picture is a classic and largely open problem; the archetypal example is the McCoy-Wu random-bond Ising model, where couplings that are random but constant along rows produce an essential singularity (a singularity weaker than any power law) in place of the homogeneous critical point. Moulard and Toninelli (2025) introduced the analogous layered disorder for the square-grid dimer model: horizontal edges in row $y$ carry weight $w_1(y)$ and vertical edges $w_2(y)$ (modulated by $H_1,H_2$), with $(w_1(y),w_2(y))$ i.i.d. across rows under their Assumption 1 (positive compactly supported laws, $w_2$ non-degenerate and independent of $w_1$, $\mathbb{E}\log w_2=0$); a variant with vertical weights $w_2(y)\gamma^{2(y\bmod 2)-1}$, $\gamma>1$, gives the pure model a gaseous phase. Writing the quenched free energy through the top Lyapunov exponent (the almost-sure exponential growth rate of products of the i.i.d. $2\times 2$ random matrices obtained by Fourier transform along the horizontal direction, one per row), they proved: an essential singularity at $(H_1,H_2)=(0,0)$, a point where the homogeneous free energy is analytic — $F(H_1)-F(0)=\exp(-(1+o(1))\mathrm{Var}(\log w_2)/H_1)$; disorder-induced regions $C_{\pm}$ (for $\gamma=1$; $G_{\gamma}$ for $\gamma>1$) where the free energy is constant in $H_1$ although not affine, of width $\Delta_{\gamma}(H_2)$ with $\Delta_{\gamma}(0)=\log\gamma$ and $\Delta(H_2)\sim\mathrm{Var}(\log w_2)/|\log H_2|$, tangent to the singularity point and with no analogue in the homogeneous phase diagram; for $\gamma>1$ a liquid/gas transition at $H_1=\log\gamma$ whose order varies continuously with $\gamma$ through exponents $\beta(\gamma)=\max(1+1/(2\alpha(\gamma)),3/2)\in[3/2,\infty)$, where $\alpha(\gamma)$ is the unique strictly positive solution of $\gamma^{4\alpha}=\mathbb{E}[w_2^{2\alpha}]\mathbb{E}[w_2^{-2\alpha}]$, or $+\infty$ if no positive solution exists; robustness of the Pokrovsky-Talapov exponent $3/2$ at liquid/frozen boundaries; and, for correlations, stretched-exponential decay like $\exp(-r\sqrt{y})$ across rows at the singularity, with inverse-squared-distance decay restored in the liquid region along the $H_2=0$ axis. All of this concerns the free energy and axis-restricted correlations: inside $C_{\pm}$ and $G_{\gamma}$ nothing is known beyond flatness of the free energy in $H_1$, the boundary shape near $(\log\gamma,0)$ is controlled only by an integrated (averaged) version of the conjectured law, and the height-function scaling limit in the liquid region is not addressed at all.

Problem Statement

Work in the layered-disordered dimer model of Moulard and Toninelli on the square grid (their Definition 1.1 and Assumption 1: row-i.i.d. weights $(w_1(y),w_2(y))$, positive and compactly supported, $w_2$ non-degenerate and independent of $w_1$, $\mathbb{E}\log w_2=0$; magnetic fields $(H_1,H_2)$; the $\gamma>1$ variant with vertical weights $w_2(y)\gamma^{2(y\bmod 2)-1}$ where relevant; thermodynamic limit on the periodized torus). Unless stated otherwise, statements are quenched, i.e., they hold almost surely in the disorder. The objective is to complete the pointwise characterization of this model's phases — extending the proven free-energy results to local behavior, boundary geometry, and scaling limits — via three linked components: (1) Local behavior in the disorder-induced regions. Determine the thermodynamic and correlation behavior inside $C_{\pm}$ and $G_{\gamma}$: the decay of dimer-dimer correlations, the local edge-occupation statistics, whether the system is frozen on every compact subset of $C_{\pm}$ and $G_{\gamma}$ (in the sense that correlations vanish identically and local statistics concentrate on one or finitely many periodic coverings) or realizes a phase with no counterpart in the homogeneous model — settling the authors' suggestion that 'possibly, the model is frozen also deep inside the regions $C_{\pm}$' — and the geometry and regularity of the boundaries of $C_{\pm}$ and $G_{\gamma}$. (2) Height-function scaling limit in the liquid region. Determine the scaling limit of the height function for $(H_1,H_2)$ in the liquid region (where dimer-dimer correlations decay like the inverse squared distance, proven along the $H_2=0$ axis and conjectured throughout): does the rescaled height function converge to a Gaussian free field as in the homogeneous model — and if so, with which covariance, i.e., does the disorder renormalize the effective coupling constants — or does layered disorder change the universality class of the massless phase? (3) Sharp pointwise boundary asymptotics. Prove or refute the conjecture that, as $H_2\to 0$, $\Delta_{\gamma}(H_2)-\log\gamma\sim c_{\alpha(\gamma)}|H_2|^{2\min(\alpha(\gamma),1)}$ with $c_{\alpha}>0$ if $\alpha(\gamma)<1/2$ and $c_{\alpha}<0$ if $\alpha(\gamma)>1/2$ (with $\alpha(\gamma)$ the unique strictly positive solution of $\gamma^{4\alpha}=\mathbb{E}[w_2^{2\alpha}]\mathbb{E}[w_2^{-2\alpha}]$, or $+\infty$ if no positive solution exists), which sharpens the integrated statement proven by the authors and pins the relation $\epsilon(H_2)=\Delta_{\gamma}(H_2)-\log\gamma$ between $H_2$ and the width of $G_{\gamma}$ near $(\log\gamma,0)$. The common scientific objective is a pointwise phase diagram of the same model: components (1) and (3) together determine the microscopic nature of the disorder-induced regions and the exact shape of their boundaries, while component (2) determines the universality class of the liquid phase; each fills exactly the gap left by the source paper's free-energy-level results. All three components are jointly required, and no single component alone resolves the task. For context, the source paper additionally raised the distinct question whether the same phenomenology persists under bulk (non-layered, i.i.d.) disorder, and encouraged numerical verification of non-analytic limit shapes near slope zero in domains with boundary; those directions concern a different model class and an auxiliary computational task respectively, and are not part of this problem's completion boundary.

References

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