The rank of diluted random graphs

We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs converging locally to a tree, we give new formulas for the asymptotic of the multiplicity of the eigenvalue 0. In particular, the result depends only on the limiting tree structure, showing that the normalized rank is 'continuous at infinity'. Our work also gives a new formula for the mass at zero of the spectral measure of a Galton-Watson tree. Our techniques of proofs borrow ideas from analysis of algorithms, random matrix theory, statistical physics and analysis of Schrödinger operators on trees.

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