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We analyze lozenge tilings, revealing distinct disordered regions in low and high temperature regimes. These regions transition from two ellipses to one, with a tacnode appearing at the critical point.

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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.