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Area of Science:

  • Statistical Mechanics
  • Complex Analysis
  • Phase Transitions

Background:

  • The Yang-Lee theory investigates phase transitions through the distribution of zeros of the partition function in the complex activity plane.
  • Random allocation models provide a framework for studying statistical mechanics phenomena with tunable parameters.

Purpose of the Study:

  • To derive an exact formula for the limiting Yang-Lee zero distribution in the random allocation model with general weights.
  • To analyze the resulting phase transition and its critical behavior.
  • To explore the connection between zero distribution and physical quantities in the mesoscopic regime.

Main Methods:

  • Utilizing an electrostatic analogy to derive the exact formula.
  • Analyzing the properties of the derived zero distribution.
  • Investigating the conformal mapping of complex phases to Yang-Lee zeros.

Main Results:

  • An exact formula for the limiting Yang-Lee zero distribution was derived.
  • A real-space condensation phase transition, induced by pressure change, was identified.
  • The scaling of zero density and the angle of zero locus at the critical point were determined.
  • Yang-Lee zeros were shown to be asymptotically images of a conformal mapping of uniformly distributed complex phases.

Conclusions:

  • The model serves as a valuable testbed for verifying relations between zero distribution and critical behavior.
  • It facilitates the exploration of physical quantities in large but finite systems (mesoscopic regime).
  • The critical exponents and phase transition order can be tuned by a single parameter for various weight families.