The asymptotic trace norm of random circulants and the graph energy

TL;DR

This paper analyzes the asymptotic behavior of the normalized trace norm (graph energy) of large random symmetric band circulant matrices and graphs, providing explicit convergence bounds and comparing with related ensembles.

Contribution

It introduces explicit asymptotic formulas and convergence bounds for the trace norm of random band circulant matrices and graphs, extending to Toeplitz matrices.

Findings

Expected normalized trace norm converges to a specific limit as matrix size grows.

Explicit bounds on convergence rate and deviation probabilities are provided.

Symmetric band Toeplitz matrices share the same limit norm under certain conditions.

Abstract

We compute the expected normalized trace norm (matrix/graph energy) of random symmetric band circulant matrices and graphs in the limit of large sizes, and obtain explicit bounds on the rate of convergence to the limit, and on the probabilities of large deviations. We also show that random symmetric band Toeplitz matrices have the same limit norm assuming that their band widths remain small relative to their sizes. We compare the limit norms across a range of related random matrix and graph ensembles.