Expected spectral measures of unimodular graph operators satisfy µ(I) = O(1/ln(1/|I|)) on intervals, extending Craig–Simon's log-Hölder regularity beyond Z^d to indicable groups, Anderson models, percolation, and quasi-transitive graphs.
Sparse graphs and their Benjamini-Schramm limits: a spectral tour.arXiv preprint arXiv:2510.10299, 2025
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In amenable unimodular random rooted networks, positive point mass in expected spectral measure implies positive probability of finite-support eigenfunction.
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Logarithmic regularity of spectral measures on infinite graphs
Expected spectral measures of unimodular graph operators satisfy µ(I) = O(1/ln(1/|I|)) on intervals, extending Craig–Simon's log-Hölder regularity beyond Z^d to indicable groups, Anderson models, percolation, and quasi-transitive graphs.
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Localization of eigenfunctions in amenable unimodular random networks
In amenable unimodular random rooted networks, positive point mass in expected spectral measure implies positive probability of finite-support eigenfunction.