Spectral Statistics of Lattice Graph Percolation Models

In graph signal processing, the graph adjacency matrix or the graph Laplacian commonly define the shift operator. The spectral decomposition of the shift operator plays an important role in that the eigenvalues represent frequencies and the eigenvectors provide a spectral basis. This is useful, for example, in the design of filters. However, the graph or network may be uncertain due to stochastic influences in construction and maintenance, and, under such conditions, the eigenvalues of the shift matrix become random variables. This paper examines the spectral distribution of the eigenvalues of random networks formed by including each link of a D-dimensional lattice supergraph independently with identical probability, a percolation model. Using the stochastic canonical equation methods developed by Girko for symmetric matrices with independent upper triangular entries, a deterministic distribution is found that asymptotically approximates the empirical spectral distribution of the scaled adjacency matrix for a model with arbitrary parameters. The main results characterize the form of the solution to an important system of equations that leads to this deterministic distribution function and significantly reduce the number of equations that must be solved to find the solution for a given set of model parameters. Simulations comparing the expected empirical spectral distributions and the computed deterministic distributions are provided for sample parameters.