Calculation of mean spectral density for statistically uniform treelike random models

Area of Science:

• * Statistical mechanics
• * Random matrix theory
• * Graph theory

Background:

• * Recursive relations for local Green functions are key for analyzing treelike random matrices.
• * These relations simplify calculations in the limit of infinite matrix dimensions.

Purpose of the Study:

• * To investigate and compare different expressions for the spectral density of random regular graphs.
• * To evaluate approximations of recursive relation solutions for treelike structures.

Main Methods:

• * Solving recursive relations for local Green functions in treelike random matrices.
• * Developing and applying approximations for real solutions, particularly for large coordination numbers.
• * Comparing derived formulas with numerical calculation results.

Main Results:

• * Formulas for spectral density were derived based on approximations of recursive relation solutions.
• * The obtained formulas show good agreement with numerical calculations.
• * The approximations are effective even for random regular graphs with small coordination numbers.

Conclusions:

• * The study provides accurate methods for determining the spectral density of random regular graphs.
• * The findings highlight the utility of approximations for recursive relations in analyzing complex matrix structures.
• * The research bridges theoretical approximations with numerical validation in random matrix theory.