The Gamma-disordered Aztec diamond

TL;DR

This work introduces a Gamma-disordered dimer model on the Aztec diamond and proves there is no phase transition in the free energy, while the annealed-quenched energy gap remains positive with a universal $n^2$-scaling. It builds exact distributional bridges between random dimer configurations and integrable directed polymers (log-Gamma, Beta, and strict-weak), enabling transfer of KPZ-class fluctuation results to turning points on the Aztec diamond. The authors develop a unique shuffle-invariant full-plane Gamma family via Lukacs' theorem and extend the analysis to hybrid polymers along vertical and horizontal slices, yielding a unified framework that connects dimer models with multiple polymer universality classes. The results provide precise turning-point statistics, dynamical behavior under the shuffling algorithm, and a robust methodology to port exact polymer results to discrete dimer systems, advancing our understanding of glassy phases and integrable structures in random tilings.

Abstract

We introduce a multi-parameter family of random edge weights on the Aztec diamond graph, given by certain Gamma variables, and prove several results about the corresponding random dimer measures. Firstly, we show there is no phase transition at the level of the free energy. This provides rigorous backing for the physics predictions of Zeng-Leath-Hwa and later works that dimer models with random weights are in the glassy `super-rough' phase at all temperatures with no phase transition. Secondly, we show that the random dimer covers themselves enjoy exact distributional equalities of certain marginals with path locations in new `hybrid' integrable polymers. These reduce to the stationary log-Gamma, strict-weak, and Beta polymer in random environment in certain cases, allowing transfer of known results from integrable polymers to dimers with random weights. As an example application, we prove that the turning points at the boundaries of the Aztec diamond exhibit fluctuations of order $n^{2/3}$, in contrast to the $n^{1/2}$ fluctuations for deterministic weights. Underlying all these is a key integrability property of the weights: they are the unique family for which independence is preserved under the shuffling algorithm.

Figures (42)

• Figure 1: A perfect matching of the Aztec diamond graph $G_n^{\mathop{\mathrm{Az}}\nolimits}$ with $n=3$, where each included edge is colored in red, blue, yellow or green according to its orientation---edges in the same direction are differentiated by whether the leftmost vertex is white or black. The probability of the matching is proportional to the product of the weights of the edges, in this case $a_{1,1}a_{2,1}a_{2,3} a_{3,2}b_{1,2}b_{1,3}$.
• Figure 2: Perfect matchings of $G_{500}^{\mathop{\mathrm{Az}}\nolimits}$ taken from the dimer measure with random weights as in \ref{['def:gamma_weights_intro']} with $\alpha=.2,\beta=.25$ (left), and deterministic weights $a_{i,j} = .2,b_{i,j}=.25$ (right). Top figures depict a single matching with edges colored as in \ref{['fig:our_weights_intro']}. Bottom figures depict a pair of matchings taken from the dimer measure with the same weights, overlaid, with the edges in both matchings removed to leave a collection of loops. It is important that on the bottom left, both matchings are taken from the same sample of random weights rather than two independent samples.
• Figure 3: A path $\pi \in \Pi_{n,n}$ with $n=20$. Here $x_{mid}(\pi) = 10$.
• Figure 4: Part of a matching $M$ of $G_7^{\mathop{\mathrm{Az}}\nolimits}$ (left), and a path configuration $\pi_1,\ldots,\pi_5$ on the directed graph $G^{\beta \Gamma}_{5,3}$ (right). These satisfy $X_3^{\mathop{\mathrm{Az}}\nolimits}(M) = \{2,4,5,6,8\} = X^{poly}(\pi_1,\ldots,\pi_5)$ (see Definitions \ref{['def:aztec_slices_intro']} and \ref{['def:X_polymer_intro']}). \ref{['thm:multi-path_intro_vert']} in this case $n=7,\ell=3$ implies that $M$ and $\pi_1,\ldots,\pi_5$ may be coupled so that this equality $X_3^{\mathop{\mathrm{Az}}\nolimits}(M) = X^{poly}(\pi_1,\ldots,\pi_5)$ always holds. The parts of $\pi_1,\ldots,\pi_5$ to the left and right of the central line correspond to the history of $M$ under the shuffling algorithm, as we discuss later in \ref{['sec:dynamical_vert']}.
• Figure 5: Illustration of the shuffling algorithm starting from the dimer configuration in Figure \ref{['fig:our_weights_intro']}. The deletion step is illustrated from top-left to top-right, where one pair of a Northwest (red) and Southeast (blue) edge is removed. From top-right to bottom-left, we perform the slide step. This yields a partial matching of an Aztec diamond graph of size $4$, with some empty faces indicated by red boxes. For each box, there are two possible pairs that complete the matching: a pair of Northwest/Southeast (red/blue) edges or a pair of Northeast/Southwest (green/yellow) edges, and each one is chosen independently according to the probabilities \ref{['eq:creation_probabilities']}.

Theorems & Definitions (168)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2: Special case of \ref{['prop:CLT_free_energy']}
• Definition 1.2
• Example 1.1
• Theorem 1.3
• Definition 1.3
• Theorem 1.4
• Remark 1.1
• Definition 1.4