Abstract
We continue Joel Friedman's demonstration of almost certain strong spectral convergence : it is done by the trace method. We perform an asymptotic expansion of the expectation of the trace of a large power of the adjacency matrix, and attempt a precise geometric description of the coefficients of this expansion. One of the key points of Friedman's method is to show that these coefficients are " Ramanujan functions ".
References
- J. Friedman. On the second eigenvalue and random walks in random d-regular graphs. Combinatorica, volume 11, pages 331-362, (1991).
- J. Friedman. A proof of Alon's second eigenvalue conjecture and related problems. Mem. Amer. Math. Soc. 195(910), 2008.
- J. Friedman, D. Kohler. On the relativized Alon eigenvalue conjecture II: Asymptotic expansion theorems for walks. arXiv:1911.05705, 2019.
