Abstract
We analyse the eigenvectors of the adjacency matrix of a critical Erdős–Rényi graph {mathbb {G}}(N,d/N), where d is of order log N. We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices. In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function. The transition between the phases is sharp and is manifested in a discontinuity in the localization exponent gamma (varvec{mathrm {w}}) of an eigenvector varvec{mathrm {w}}, defined through Vert varvec{mathrm {w}} Vert _infty / Vert varvec{mathrm {w}} Vert _2 = N^{-gamma (varvec{mathrm {w}})}. Our results remain valid throughout the optimal regime sqrt{log N} ll d leqslant O(log N).
Highlights
We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices
In this paper we address the question of spatial localization for the random Erdos– Rényi graph G(N, d/N )
Up to now we have focused on the Erdos–Rényi graph on the critical scale d log N d (log N)
Summary
We remark that the exponent ρb(λ) describes the density of states at energy λ: under the above assumptions on b and λ, for any interval I containing λ and satisfying ξ |I | 1, the number of eigenvalues in I is equal to N ρb(λ)+o(1)|I | with probability 1 − o(1), as can be seen from Lemma A.12 (i) and Theorem 1.7. D-tuning forks provide a simple way of constructing localized states Note that this is a very basic form of concentration of mass, supported at the periphery of the graph on special graph structures, and is unrelated to the much more subtle concentration in the semilocalized phase described in Sect. In the language of Definition 4.6 below, it is linked to the property that most neighbours of any vertex are typical (see Proposition 4.8 (ii) below)
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