A Periodic Hexagon Tiling Model and Non-Hermitian Orthogonal Polynomials

Area of Science:

• Probability theory
• Statistical mechanics
• Combinatorics

Background:

• Lozenge tilings of hexagons are a key model in statistical mechanics.
• Understanding the asymptotic behavior of these tilings is crucial for statistical physics.
• A one-parameter family interpolates between uniform and frozen tilings.

Purpose of the Study:

• To compute the disordered regions and limiting lozenge densities in asymptotic regimes.
• To analyze the transition between low and high temperature behaviors.
• To investigate the role of non-Hermitian orthogonal polynomials.

Main Methods:

• Utilizing a double integral representation for the correlation kernel.
• Employing Riemann-Hilbert analysis for asymptotic behavior of orthogonal polynomials.
• Applying steepest descent arguments for integral evaluation.

Main Results:

• Characterization of disordered regions as two disjoint ellipses (low temperature) or a single region (high temperature).
• Identification of a tacnode at the transition point.
• Computation of limiting densities of lozenges in disordered regions.

Conclusions:

• The study provides a comprehensive analysis of lozenge tiling behavior across different temperature regimes.
• The methods used offer a powerful framework for studying similar statistical mechanics models.
• The geometric transitions observed offer insights into critical phenomena.