45,113 research outputs found
Geometrical organization of solutions to random linear Boolean equations
The random XORSAT problem deals with large random linear systems of Boolean
variables. The difficulty of such problems is controlled by the ratio of number
of equations to number of variables. It is known that in some range of values
of this parameter, the space of solutions breaks into many disconnected
clusters. Here we study precisely the corresponding geometrical organization.
In particular, the distribution of distances between these clusters is computed
by the cavity method. This allows to study the `x-satisfiability' threshold,
the critical density of equations where there exist two solutions at a given
distance.Comment: 20 page
Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs
We consider the problem of coloring Erdos-Renyi and regular random graphs of
finite connectivity using q colors. It has been studied so far using the cavity
approach within the so-called one-step replica symmetry breaking (1RSB) ansatz.
We derive a general criterion for the validity of this ansatz and, applying it
to the ground state, we provide evidence that the 1RSB solution gives exact
threshold values c_q for the q-COL/UNCOL phase transition. We also study the
asymptotic thresholds for q >> 1 finding c_q = 2qlog(q)-log(q)-1+o(1) in
perfect agreement with rigorous mathematical bounds, as well as the nature of
excited states, and give a global phase diagram of the problem.Comment: 23 pages, 10 figures. Replaced with accepted versio
Environmental Effects on Real-Space and Redshift-Space Galaxy Clustering
Galaxy formation inside dark matter halos, as well as the halo formation
itself, can be affected by large-scale environments. Evaluating the imprints of
environmental effects on galaxy clustering is crucial for precise cosmological
constraints with data from galaxy redshift surveys. We investigate such an
environmental impact on both real-space and redshift-space galaxy clustering
statistics using a semi-analytic model derived from the Millennium Simulation.
We compare clustering statistics from original SAM galaxy samples and shuffled
ones with environmental influence on galaxy properties eliminated. Among the
luminosity-threshold samples examined, the one with the lowest threshold
luminosity (~0.2L_*) is affected by environmental effects the most, which has a
~10% decrease in the real-space two-point correlation function (2PCF) after
shuffling. By decomposing the 2PCF into five different components based on the
source of pairs, we show that the change in the 2PCF can be explained by the
age and richness dependence of halo clustering. The 2PCFs in redshift space are
found to change in a similar manner after shuffling. If the environmental
effects are neglected, halo occupation distribution modeling of the real-space
and redshift-space clustering may have a less than 6.5% systematic uncertainty
in constraining beta from the most affected SAM sample and have substantially
smaller uncertainties from the other, more luminous samples. We argue that the
effect could be even smaller in reality. In the Appendix, we present a method
to decompose the 2PCF, which can be applied to measure the two-point
auto-correlation functions of galaxy sub-samples in a volume-limited galaxy
sample and their two-point cross-correlation functions in a single run
utilizing only one random catalog.Comment: 13 pages, 6 figures, Accepted by AP
Phase Transitions and Computational Difficulty in Random Constraint Satisfaction Problems
We review the understanding of the random constraint satisfaction problems,
focusing on the q-coloring of large random graphs, that has been achieved using
the cavity method of the physicists. We also discuss the properties of the
phase diagram in temperature, the connections with the glass transition
phenomenology in physics, and the related algorithmic issues.Comment: 10 pages, Proceedings of the International Workshop on
Statistical-Mechanical Informatics 2007, Kyoto (Japan) September 16-19, 200
- …