Finite-size corrections to the spectrum of regular random graphs: An analytical solution

Area of Science:

• Graph Theory
• Statistical Physics
• Random Matrix Theory

Background:

• Understanding the spectral properties of large random graphs is crucial in various fields.
• Previous studies have focused on the leading order behavior of graph spectra.
• Finite-size effects and corrections beyond the leading order remain an active area of research.

Purpose of the Study:

• To analytically derive the O(1/N) correction to the spectrum of regular random graphs.
• To provide an explicit formula for finite-size fluctuations of the resolvent.
• To validate the derived analytical expressions through comparison with numerical simulations.

Main Methods:

• Development of a detailed analytical framework for random graph spectra.
• Calculation of resolvent fluctuations using a weighted series of loop contributions.
• Derivation of the O(1/N) correction to both isolated eigenvalues and the continuous spectrum.
• Direct diagonalization of random graph instances for numerical validation.

Main Results:

• An analytical expression for the O(1/N) correction to the spectrum of regular random graphs was obtained.
• The finite-size fluctuations of the resolvent were characterized by loop contributions of all lengths.
• The derived analytical formula for spectral corrections demonstrated excellent agreement with direct diagonalization results.

Conclusions:

• The analytical study successfully determined the O(1/N) spectral correction for regular random graphs.
• The findings confirm the accuracy of the derived analytical expressions for spectral properties.
• This work provides a foundation for understanding more refined spectral behaviors in random graph ensembles.