Third-order phase transition in random tilings

Abstract

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

References in corpus (3)

• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• The Heisenberg XX spin chain and low-energy QCD

Cited by in corpus (28)

• Top eigenvalue of a random matrix: large deviations and third order phase transition
• Exact short-time height distribution in 1D KPZ equation and edge fermions at high temperature
• Exact extremal statistics in the classical $1d$ Coulomb gas
• Lower tail of the KPZ equation
• A shortcut through the Coulomb gas method for spectral linear statistics on random matrices
• Extreme statistics and index distribution in the classical $1d$ Coulomb gas
• Universality of the third-order phase transition in the constrained Coulomb gas
• Universality of the weak pushed-to-pulled transition in systems with repulsive interactions
• Edge fluctuations and third-order phase transition in harmonically confined long-range systems
• Thermodynamics of the six-vertex model in an L-shaped domain
• Large-N ground state of the Lieb-Liniger model and Yang-Mills theory on a two-sphere
• Arctic curve of the free-fermion six-vertex model in an L-shaped domain
• Third-order phase transition: random matrices and screened Coulomb gas with hard walls
• Exact equivalences and phase discrepancies between random matrix ensembles
• Critical exponents for higher order phase transitions: Landau theory and RG flow
• The hard-edge tacnode process for Brownian motion
• Limit shape phase transitions: a merger of arctic circles
• Generalized emptiness formation probability in the six-vertex model
• Emptiness formation probability of the six-vertex model and the sixth Painlevé equation
• Toric Quiver Asymptotics and Mahler Measure: $\mathcal{N}=2$ BPS States
• Phase transition in complex-time Loschmidt echo of short and long range spin chain
• Semi-classical quantisation of magnetic solitons in the anisotropic Heisenberg quantum chain
• Limit distributions for KPZ growth models with spatially homogeneous random initial conditions
• Importance Sampling for counting statistics in one-dimensional systems
• Evaluation of integrals for the emptiness formation probability in the square-ice model
• Large deviations of spread measures for Gaussian matrices
• Phase transition in the diffusion and bootstrap percolation models on regular random and Erdős-Rényi networks
• Scaling limit of domino tilings on a pentagonal domain