53,261 research outputs found
Saturation in the Hypercube and Bootstrap Percolation
Let denote the hypercube of dimension . Given , a spanning
subgraph of is said to be -saturated if it does not
contain as a subgraph but adding any edge of
creates a copy of in . Answering a question of Johnson and Pinto, we
show that for every fixed the minimum number of edges in a
-saturated graph is .
We also study weak saturation, which is a form of bootstrap percolation. A
spanning subgraph of is said to be weakly -saturated if the
edges of can be added to one at a time so that each
added edge creates a new copy of . Answering another question of Johnson
and Pinto, we determine the minimum number of edges in a weakly
-saturated graph for all . More generally, we
determine the minimum number of edges in a subgraph of the -dimensional grid
which is weakly saturated with respect to `axis aligned' copies of a
smaller grid . We also study weak saturation of cycles in the grid.Comment: 21 pages, 2 figures. To appear in Combinatorics, Probability and
Computin
Instantons in the Saturation Environment
We show that instanton calculations in QCD become theoretically well defined
in the gluon saturation environment which suppresses large size instantons. The
effective cutoff scale is determined by the inverse of the saturation scale. We
concentrate on two most important cases: the small-x tail of a gluon
distribution of a high energy hadron or a large nucleus and the central
rapidity region in a high energy hadronic or heavy ion collision. In the
saturation regime the gluon density in a single large ultrarelativistic nucleus
is high and gluonic fields are given by the classical solutions of the
equations of motion. We show that these strong classical fields do not affect
the density of instantons in the nuclear wave function compared to the
instanton density in the vacuum. A classical solution with non-trivial
topological charge is found for the gluon field of a single nucleus at the
lowest order in the instanton perturbation theory. In the case of
ultrarelativistic heavy ion collisions a strong classical gluonic field is
produced in the central rapidity region. We demonstrate that this field
introduces a suppression factor of exp{-c \rho^4 Q_s^4 / [8 \alpha^2 N_c (Q_s
\tau)^2]} in the instanton size distribution, where Q_s is the saturation scale
of both (identical) nuclei, \tau is the proper time and c = 1 is the gluon
liberation coefficient. This factor suggests that gluonic saturation effects at
the early stages of nuclear collisions regulate the instanton size distribution
in the infrared region and make the instanton density finite by suppressing
large size instantons.Comment: 20 pages, 8 figures, REVTeX, some discussion added including a
possible scenario for unitarization of the soft pomero
Threshold Saturation in Spatially Coupled Constraint Satisfaction Problems
We consider chains of random constraint satisfaction models that are
spatially coupled across a finite window along the chain direction. We
investigate their phase diagram at zero temperature using the survey
propagation formalism and the interpolation method. We prove that the SAT-UNSAT
phase transition threshold of an infinite chain is identical to the one of the
individual standard model, and is therefore not affected by spatial coupling.
We compute the survey propagation complexity using population dynamics as well
as large degree approximations, and determine the survey propagation threshold.
We find that a clustering phase survives coupling. However, as one increases
the range of the coupling window, the survey propagation threshold increases
and saturates towards the phase transition threshold. We also briefly discuss
other aspects of the problem. Namely, the condensation threshold is not
affected by coupling, but the dynamic threshold displays saturation towards the
condensation one. All these features may provide a new avenue for obtaining
better provable algorithmic lower bounds on phase transition thresholds of the
individual standard model
Sandwiching saturation number of fullerene graphs
The saturation number of a graph is the cardinality of any smallest
maximal matching of , and it is denoted by . Fullerene graphs are
cubic planar graphs with exactly twelve 5-faces; all the other faces are
hexagons. They are used to capture the structure of carbon molecules. Here we
show that the saturation number of fullerenes on vertices is essentially
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