The spectral density of various ensembles of sparse symmetric random matrices is analyzed using the cavity method. We consider two cases: matrices whose associated graphs are locally treelike, and sparse covariance matrices. We derive a closed set of equations from which the density of eigenvalues can be efficiently calculated. Within this approach, the Wigner semicircle law for Gaussian matrices and the Marcenko-Pastur law for covariance matrices are recovered easily. Our results are compared with numerical diagonalization, finding excellent agreement.
Abstract. We compute the spectral density for ensembles of of sparse symmetric random matrices using replica, managing to circumvent difficulties that have been encountered in earlier approaches along the lines first suggested in a seminal paper by Rodgers and Bray. Due attention is payed to the issue of localization. Our approach is not restricted to matrices defined on graphs with Poissonian degree distribution. Matrices defined on regular random graphs or on scale-free graphs, are easily handled. We also look at matrices with row constraints such as discrete graph Laplacians. Our approach naturally allows to unfold the total density of states into contributions coming from vertices of different local coordination.
The evolution of liquid water and its transport through the porous gas diffusion media in an operating fuel cell were investigated applying an experimental setup for high spatial resolution of 3 m. Fundamental aspects of cluster formation in hydrophobic/hydrophilic porous materials as well as processes of multiphase flow are addressed. The obtained water distributions provide a detailed insight in the membrane electrode assembly and the porous electrode with regard on the existence and transport of liquid water. In addition, the results approve transport theories used within the framework of percolation theory and demonstrate the need for adapted modeling approaches.
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