7,955 research outputs found
Condensation of degrees emerging through a first-order phase transition in classical random graphs
Due to their conceptual and mathematical simplicity, Erd\"os-R\'enyi or
classical random graphs remain as a fundamental paradigm to model complex
interacting systems in several areas. Although condensation phenomena have been
widely considered in complex network theory, the condensation of degrees has
hitherto eluded a careful study. Here we show that the degree statistics of the
classical random graph model undergoes a first-order phase transition between a
Poisson-like distribution and a condensed phase, the latter characterized by a
large fraction of nodes having degrees in a limited sector of their
configuration space. The mechanism underlying the first-order transition is
discussed in light of standard concepts in statistical physics. We uncover the
phase diagram characterizing the ensemble space of the model and we evaluate
the rate function governing the probability to observe a condensed state, which
shows that condensation of degrees is a rare statistical event akin to similar
condensation phenomena recently observed in several other systems. Monte Carlo
simulations confirm the exactness of our theoretical results.Comment: 8 pages, 6 figure
Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase
We calculate the level compressibility of the energy levels
inside for the Anderson model on infinitely large random regular
graphs with on-site potentials distributed uniformly in . We show
that approaches the limit
for a broad interval of the disorder strength within the extended phase,
including the region of close to the critical point for the Anderson
transition. These results strongly suggest that the energy levels follow the
Wigner-Dyson statistics in the extended phase, consistent with earlier
analytical predictions for the Anderson model on an Erd\"os-R\'enyi random
graph. Our results are obtained from the accurate numerical solution of an
exact set of equations valid for infinitely large regular random graphs.Comment: 7 pages, 3 figure
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