On the largest component of a random graph with a subpower-law degree sequence in a subcritical phase

A uniformly random graph on $n$ vertices with a fixed degree sequence, obeying a $\gamma$ subpower law, is studied. It is shown that, for $\gamma>3$, in a subcritical phase with high probability the largest component size does not exceed $n^{1/\gamma+\varepsilon_n}$, $\varepsilon_n=O(\ln\ln n/\ln n)$, $1/\gamma$ being the best power for this random graph. This is similar to the best possible $n^{1/(\gamma-1)}$ bound for a different model of the random graph, one with independent vertex degrees, conjectured by Durrett, and proved recently by Janson.Comment: Published in at http://dx.doi.org/10.1214/07-AAP493 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Satisfiability Threshold for k-XORSAT

We consider "unconstrained" random $k$-XORSAT, which is a uniformly random system of $m$ linear non-homogeneous equations in $\mathbb{F}_2$ over $n$ variables, each equation containing $k \geq 3$ variables, and also consider a "constrained" model where every variable appears in at least two equations. Dubois and Mandler proved that $m/n=1$ is a sharp threshold for satisfiability of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform hypergraph to extend this result to find the threshold for unconstrained 3-XORSAT. We show that $m/n=1$ remains a sharp threshold for satisfiability of constrained $k$-XORSAT for every $k\ge 3$, and we use standard results on the 2-core of a random $k$-uniform hypergraph to extend this result to find the threshold for unconstrained $k$-XORSAT. For constrained $k$-XORSAT we narrow the phase transition window, showing that $m-n \to -\infty$ implies almost-sure satisfiability, while $m-n \to +\infty$ implies almost-sure unsatisfiability.Comment: Version 2 adds sharper phase transition result, new citation in literature survey, and improvements in presentation; removes Appendix treating k=

The Density Matrix Renormalization Group and the Nuclear Shell Model

We summarize recent efforts to develop an angular-momentum-conserving variant of the Density Matrix Renormalization Group method into a practical truncation strategy for large-scale shell model calculations of atomic nuclei. Following a brief description of the key elements of the method, we report the results of test calculations for $^{48}$Cr and $^{56}$Ni. In both cases we consider nucleons limited to the 2p-1f shell and interacting via the KB3 interaction. Both calculations produce a high level of agreement with the exact shell-model results. Furthermore, and most importantly, the fraction of the complete space required to achieve this high level of agreement goes down rapidly as the size of the full space grows

Density Matrix Renormalization Group study of 48^{48}Cr and 56^{56}Ni

We discuss the development of an angular-momentum-conserving variant of the Density Matrix Renormalization Group (DMRG) method for use in large-scale shell-model calculations of atomic nuclei and report a first application of the method to the ground state of $^{56}$Ni and improved results for $^{48}$Cr. In both cases, we see a high level of agreement with the exact results. A comparison of the two shows a dramatic reduction in the fraction of the space required to achieve accuracy as the size of the problem grows.Comment: 4 pages. Published in PRC Rapi

Proton-neutron pairing correlations in the nuclear shell model

A shell-model study of proton-neutron pairing in f - p shell nuclei using a parametrized hamiltonian that includes deformation and spin-orbit effects as well as isoscalar and isovector pairing is reported. By working in a shell-model framework we are able to assess the role of the various modes of proton-neutron pairing in the presence of nuclear deformation without violating symmetries. Results are presented for $^{44}$Ti, $^{46}$Ti and $^{48}$Cr.Comment: Presented at "XXXIII Symposium on Nuclear Physics" 05 Jan 2010 - 08 Jan 2010; Hacienda Cocoyoc, Morelos, Mexic

k-core organization of complex networks

We analytically describe the architecture of randomly damaged uncorrelated networks as a set of successively enclosed substructures -- k-cores. The k-core is the largest subgraph where vertices have at least k interconnections. We find the structure of k-cores, their sizes, and their birth points -- the bootstrap percolation thresholds. We show that in networks with a finite mean number z_2 of the second-nearest neighbors, the emergence of a k-core is a hybrid phase transition. In contrast, if z_2 diverges, the networks contain an infinite sequence of k-cores which are ultra-robust against random damage.Comment: 5 pages, 3 figure

Exactly-solvable models of proton and neutron interacting bosons

We describe a class of exactly-solvable models of interacting bosons based on the algebra SO(3,2). Each copy of the algebra represents a system of neutron and proton bosons in a given bosonic level interacting via a pairing interaction. The model that includes s and d bosons is a specific realization of the IBM2, restricted to the transition regime between vibrational and gamma-soft nuclei. By including additional copies of the algebra, we can generate proton-neutron boson models involving other boson degrees of freedom, while still maintaining exact solvability. In each of these models, we can study not only the states of maximal symmetry, but also those of mixed symmetry, albeit still in the vibrational to gamma-soft transition regime. Furthermore, in each of these models we can study some features of F-spin symmetry breaking. We report systematic calculations as a function of the pairing strength for models based on s, d, and g bosons and on s, d, and f bosons. The formalism of exactly-solvable models based on the SO(3,2) algebra is not limited to systems of proton and neutron bosons, however, but can also be applied to other scenarios that involve two species of interacting bosons.Comment: 8 pages, 3 figures. Submitted to Phys.Rev.

Systematic study of proton-neutron pairing correlations in the nuclear shell model

A shell-model study of proton-neutron pairing in $2p1f$ shell nuclei using a parametrized hamiltonian that includes deformation and spin-orbit effects as well as isoscalar and isovector pairing is reported. By working in a shell-model framework we are able to assess the role of the various modes of proton-neutron pairing in the presence of nuclear deformation without violating symmetries. Results are presented for $^{44}$Ti, $^{45}$Ti, $^{46}$Ti, $^{46}$V and $^{48}$Cr to assess how proton-neutron pair correlations emerge under different scenarios. We also study how the presence of a one-body spin-obit interaction affects the contribution of the various pairing modes.Comment: 12 pages, 16 figure

On the dominance of J(P)=0(+) ground states in even-even nuclei from random two-body interactions

Recent calculations using random two-body interactions showed a preponderance of J(P)=0(+) ground states, despite the fact that there is no strong pairing character in the force. We carry out an analysis of a system of identical particles occupying orbits with j=1/2, 3/2 and 5/2 and discuss some general features of the spectra derived from random two-body interactions. We show that for random two-body interactions that are not time-reversal invariant the dominance of 0(+) states in this case is more pronounced, indicating that time-reversal invariance cannot be the origin of the 0(+) dominance.Comment: 8 pages, 3 tables and 3 figures. Phys. Rev. C, in pres