5 research outputs found
Sudden emergence of q-regular subgraphs in random graphs
We investigate the computationally hard problem whether a random graph of
finite average vertex degree has an extensively large -regular subgraph,
i.e., a subgraph with all vertices having degree equal to . We reformulate
this problem as a constraint-satisfaction problem, and solve it using the
cavity method of statistical physics at zero temperature. For , we find
that the first large -regular subgraphs appear discontinuously at an average
vertex degree c_\reg{3} \simeq 3.3546 and contain immediately about 24% of
all vertices in the graph. This transition is extremely close to (but different
from) the well-known 3-core percolation point c_\cor{3} \simeq 3.3509. For
, the -regular subgraph percolation threshold is found to coincide with
that of the -core.Comment: 7 pages, 5 figure
A discrete model of water with two distinct glassy phases
We investigate a minimal model for non-crystalline water, defined on a Husimi
lattice. The peculiar random-regular nature of the lattice is meant to account
for the formation of a random 4-coordinated hydrogen-bond network. The model
turns out to be consistent with most thermodynamic anomalies observed in liquid
and supercooled-liquid water. Furthermore, the model exhibits two glassy phases
with different densities, which can coexist at a first-order transition. The
onset of a complex free-energy landscape, characterized by an exponentially
large number of metastable minima, is pointed out by the cavity method, at the
level of 1-step replica symmetry breaking.Comment: expanded version: 6 pages, 7 figure
The number of matchings in random graphs
We study matchings on sparse random graphs by means of the cavity method. We
first show how the method reproduces several known results about maximum and
perfect matchings in regular and Erdos-Renyi random graphs. Our main new result
is the computation of the entropy, i.e. the leading order of the logarithm of
the number of solutions, of matchings with a given size. We derive both an
algorithm to compute this entropy for an arbitrary graph with a girth that
diverges in the large size limit, and an analytic result for the entropy in
regular and Erdos-Renyi random graph ensembles.Comment: 17 pages, 6 figures, to be published in Journal of Statistical
Mechanic