Nonuniversality of compact support probability distributions in random matrix theory

Area of Science:

• Mathematics
• Theoretical Physics
• Statistical Mechanics

Background:

• The study of random matrix theory (RMT) is crucial for understanding complex systems.
• Canonical Gaussian ensembles are well-understood benchmarks in RMT.
• Generalized trace ensembles present a more complex framework for analysis.

Purpose of the Study:

• To compute the two-point resolvent for generalized fixed and bounded trace ensembles in the large-n limit.
• To identify and characterize the nonuniversal differences between these ensembles and the canonical Gaussian ensemble.
• To investigate the behavior of k-point resolvents across different generalized trace ensembles.

Main Methods:

• Calculation of the two-point resolvent in the large-n limit.
• Explicit derivation of the nonuniversal part for monomial potentials V(M)=M(2p).
• Mathematical proof for the agreement of all k-point resolvents in the large-n limit.

Main Results:

• The two-point resolvent for generalized fixed and bounded trace ensembles differs from the canonical Gaussian ensemble.
• A specific nonuniversal part causing this disagreement is explicitly determined for monomial potentials.
• All k-point resolvents for the generalized fixed and bounded trace ensemble exhibit agreement in the large-n limit.

Conclusions:

• Generalized fixed and bounded trace ensembles exhibit distinct behavior compared to canonical Gaussian ensembles.
• The nonuniversal part is explicitly characterized, providing deeper insight into ensemble differences.
• The agreement of all k-point resolvents signifies a unifying property within generalized trace ensembles in the large-n limit.