Third-order phase transition in random tilings

• Main Authors: Colomo, F., Pronko, A. G.
• Format: Text
• Language: unknown
• Published: arXiv 2013
• Subjects: Mathematical Physics math-ph; Statistical Mechanics cond-mat.stat-mech; High Energy Physics - Theory hep-th; Combinatorics math.CO; FOS Physical sciences; FOS Mathematics; Arctic
• Online Access: https://dx.doi.org/10.48550/arxiv.1306.6207; https://arxiv.org/abs/1306.6207

We consider the domino tilings of an Aztec diamond with a cut-off corner of macroscopic square shape and given size, and address the bulk properties of tilings as the size is varied. We observe that the free energy exhibits a third-order phase transition when the cut-off square, increasing in size, reaches the arctic ellipse---the phase separation curve of the original (unmodified) Aztec diamond. We obtain this result by studying the thermodynamic limit of certain nonlocal correlation function of the underlying six-vertex model with domain wall boundary conditions, the so-called emptiness formation probability (EFP). We consider EFP in two different representations: as a tau-function for Toda chains and as a random matrix model integral. The latter has a discrete measure and a linear potential with hard walls; the observed phase transition shares properties with both Gross-Witten-Wadia and Douglas-Kazakov phase transitions. : 21 pages, 6 figures; v3: journal version with misprints in text and Fig. 3 corrected; footnote added at page 3