n--> infinity limit of O(n) ferromagnetic models on graphs

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Graph Theory

Background:

• H. E. Stanley's 30-year-old finding: O(n) spin models on lattices converge to spherical models as n approaches infinity.
• This convergence implies identical free energies at all temperatures.
• The result is not directly applicable to general discrete structures like graphs due to lack of translation invariance.

Purpose of the Study:

• To investigate if the singular parts of free energies, crucial for critical behavior, coincide for O(n) and spherical models on graphs.
• To determine if critical exponents for O(n) models on graphs (as n approaches infinity) align with spherical model exponents.
• To explore the dependence of these critical exponents on graph properties.

Main Methods:

• Analysis of ferromagnetic O(n) spin models on general graphs.
• Focus on the thermodynamic limit to study critical phenomena.
• Examination of the singular components of free energies.

Main Results:

• Demonstrated that the singular parts of free energies for O(n) and spherical models coincide on graphs.
• Established that critical exponents for O(n) models (n-->infinity) on graphs converge to spherical model exponents.
• Showed that these critical exponents are determined by the graph's spectral dimension.

Conclusions:

• The universality of critical behavior between O(n) and spherical models extends to graphs, despite the breakdown of direct free energy coincidence.
• Graph spectral dimension is a key factor governing critical exponents in the large-n limit.
• This finding offers insights into phase transitions on complex network structures.