Erdős-Rényi G(n,p): adjacency spectrum and the Wigner semicircle
Parameters
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Spectral radius λ₁
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Predicted √(np(1-p))·√(2n)
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Edge count
Wigner's theorem: for large symmetric random matrices with i.i.d. entries (mean 0, variance σ²), the empirical spectral distribution converges to the semicircle law on [−2σ√n, 2σ√n].
For adjacency matrices of G(n,p), the leading eigenvalue λ₁ ≈ np + O(√(np)) — the spectral radius tracks connectivity.