Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fully localized and partially delocalized states in the tails of
  Erd\"os-R\'enyi graphs in the critical regime   In this work we study the spectral properties of the adjacency matrix of
critical Erd\"os-R\'enyi (ER) graphs, i.e. when the average degree is of order
\log N. In a series of recent inspiring papers Alt, Ducatez, and Knowles have
rigorously shown that these systems exhibit a "semilocalized" phase in the
tails of the spectrum where the eigenvectors are exponentially localized on a
sub-extensive set of nodes with anomalously large degree. We propose two
approximate analytical strategies to analyze this regime based respectively on
the simple "rules of thumb" for localization and ergodicity and on an
approximate treatment of the self-consistent cavity equation for the resolvent.
Both approaches suggest the existence of two different regimes: a fully
Anderson localized phase at the spectral edges, in which the eigenvectors are
localized around a unique center, and an intermediate partially delocalized but
non-ergodic phase, where the eigenvectors spread over many resonant
localization centers. In this phase the exponential decay of the effective
tunneling amplitudes between the localization centers is counterbalanced by the
large number of nodes towards which tunneling can occur, and the system
exhibits mini-bands in the local spectrum over which the Wigner-Dyson
statistics establishes. We complement these results by a detailed numerical
study of the finite-size scaling behavior of several observables that supports
the theoretical predictions and allows us to determine the critical properties
of the two transitions. Critical ER graphs provide a pictorial representation
of the Hilbert space of a generic many-body Hamiltonian with short range
interactions. In particular we argue that their phase diagram can be mapped
onto the out-of-equilibrium phase diagram of the quantum random energy model.

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The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
30
The categories of this paper are: q-alg
32
The categories of this paper are: atom-ph
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int