Information geometry of the ising model on planar random graphs

Area of Science:

• Statistical Mechanics
• Information Geometry
• Phase Transitions

Background:

• Information geometry offers an alternative view of statistical mechanics by introducing a metric onto parameter space.
• Scalar curvature (R) of the information metric often diverges at phase transition points.
• Previous studies on two-parameter models suggested R scales as |beta-beta(c)|^(alpha-2).

Purpose of the Study:

• To investigate the scaling behavior of scalar curvature (R) for the Ising model on planar random graphs.
• To analyze the phase structure using an information geometric approach.
• To reconcile discrepancies in scaling relations observed in different models.

Main Methods:

• Utilized the field solution of the Ising model on an ensemble of planar random graphs.
• Calculated the scalar curvature (R) of the information metric.
• Evaluated the scaling behavior of R near the phase transition point (beta(c)).

Main Results:

• The scalar curvature R was found to scale as approximately |beta-beta(c)|^(-2) on planar random graphs.
• This observed scaling differs from the previously postulated relation R approximately |beta-beta(c)|^(alpha-2).
• The discrepancy was attributed to the influence of a negative alpha parameter.

Conclusions:

• The study demonstrates a specific scaling behavior of scalar curvature for the Ising model on planar random graphs.
• The findings highlight the importance of the alpha parameter in determining the scaling relations of scalar curvature.
• Information geometry provides valuable insights into the phase structure of statistical mechanical models.