The Gamma-disordered Aztec diamond

TL;DR

This paper studies a new class of random dimer models on the Aztec diamond with Gamma-distributed weights, revealing no phase transition, distributional identities with integrable polymers, and boundary fluctuation behaviors.

Contribution

It introduces Gamma-distributed weights on the Aztec diamond, proves the absence of phase transition, and establishes connections with integrable polymers and boundary fluctuation results.

Findings

No phase transition at the free energy level.

Distributional equalities with integrable polymers.

Boundary fluctuations of order n^{2/3}.

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.