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原文摘要

Using the recent strong convergence results of Bordenave and Collins (Annals of Mathematics) and Chen et al. (Annals of Mathematics), we study the average case behavior of rotating random walks of Cohen and Maor (STOC 2023) and the derandomized square construction of Rozenman and Vadhan (RANDOM 2005). Concretely, we prove a conjecture of Cohen et al. (ITCS 2025) regarding the average case behavior of derandomized squaring for random graphs and show that the existential bounds of Cohen and Maor for their rotating random walk construction hold with high-probability for random graphs. Using the results of O'Donnell and Wu (FOCS 2020), we can in polynomial time construct a family of graphs and rotations which almost saturate the bounds of Cohen and Maor, and meet the bounds conjectured by Cohen et al. for all vertex-transitive auxiliary graphs of fixed degree in the derandomized square. Finally, we study independent random permutation conjugates of deterministic regular graph adjacency matrices. We prove quantitative weak convergence to freely independent tree adjacency operators when the degrees are fixed and the girths tend to infinity. Separately, a covering argument gives deterministic spectral lower bounds for polynomials with non-negative coefficients, recovering the lower bound of Cohen et al. for rotating walks.