Overview
This research introduces a new methodological approach for the study of large deviations principles (LDPs) in the context of random matrices. The methodology leverages a partially lifted variant of random duality theory (RDT) to construct a generic LDP framework. The framework was specifically applied to classical statistical ensembles, namely Wishart and Wigner GOE matrices, to characterize their upper and lower spectral edges.
Research Context
The study of large deviations principles (LDPs) for random matrices traditionally relies on methods within random matrix theory (RMT). One such traditional approach is the Coulomb gas methodology, which has been utilized to derive LDP characterizations for spectral edges in established random matrix ensembles. Prior work, cited as [85,95], has established these results using Coulomb gas techniques.
Approach
The investigators developed a novel methodology centered on a partially lifted variant of random duality theory (RDT). This approach was designed to facilitate the study of large deviations principles (LDPs) for random matrices. The framework explicitly avoids traditional random matrix theory (RMT) methods. For demonstrating its capabilities, the methodology was applied to two specific classical statistical ensembles: Wishart matrices and Wigner GOE matrices. The application focused on characterizing the LDPs of their upper and lower spectral edges.
Findings
The application of the RDT-based framework to Wishart and Wigner GOE classical statistical ensembles resulted in LDP characterizations for their upper and lower spectral edges. These characterizations were found to fully match results previously obtained through traditional Coulomb gas methodologies, as referenced in [85,95]. The developed methodology is noted for its simplicity and accuracy in achieving these results.