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Third-order phase transition in random tilings
1INFN, Sezione di Firenze Via G. Sansone 1, 50019 Sesto Fiorentino (FI), Italy.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 16, 2013
Summary
Researchers found a third-order phase transition in domino tilings of a modified Aztec diamond. This transition occurs when the cut-off corner reaches the arctic ellipse, impacting bulk properties.
Area of Science:
- Combinatorics
- Statistical Mechanics
- Mathematical Physics
Background:
- Domino tilings are a key area in combinatorics and statistical mechanics.
- The Aztec diamond is a well-studied model with rich properties, including its arctic curve.
- Understanding bulk properties and phase transitions is crucial for statistical models.
Purpose of the Study:
- To investigate the bulk properties of domino tilings of an Aztec diamond with a cut-off corner.
- To identify and characterize phase transitions in this modified tiling model.
- To analyze the relationship between the cut-off size and the observed phase transitions.
Main Methods:
- Studying the thermodynamic limit of the emptiness formation probability (EFP) in the underlying six-vertex model.
- Representing EFP as a τ function for Toda chains.
- Analyzing EFP as a random matrix model integral with a discrete measure and linear potential.
Main Results:
- A third-order phase transition in free energy was observed.
- The transition occurs when the cut-off square size reaches the arctic ellipse.
- The emptiness formation probability (EFP) exhibits properties similar to Gross-Witten-Wadia and Douglas-Kazakov phase transitions.
Conclusions:
- The cut-off corner's interaction with the arctic ellipse drives a significant phase transition in domino tilings.
- The study connects domino tilings to advanced concepts in random matrix theory and integrable systems.
- This work provides new insights into phase transitions in exactly solvable models.
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