Scaling behavior of explosive percolation on the square lattice

Clusters generated by the product-rule growth model of Achlioptas, D'Souza, and Spencer on a two-dimensional square lattice are shown to obey qualitatively different scaling behavior than standard (random growth) percolation. The threshold with unrestricted bond placement (allowing loops) is found precisely using several different criteria based upon both moments and wrapping probabilities, yielding p_c = 0.526565 +/- 0.000005, consistent with the recent result of Radicchi and Fortunato. The correlation-length exponent nu is found to be close to 1. The qualitative difference from regular percolation is shown dramatically in the behavior of the percolation probability P_(infinity) (size of largest cluster), the susceptibility, and of the second moment of finite clusters, where discontinuities appears at the threshold. The critical cluster-size distribution does not follow a consistent power-law for the range of system sizes we study L 2 for larger L.Comment: v2: Updated results in original version with new data; expanded discussion. v3: Resubmitted version. New figures, reference

Hermitian Dirac Hamiltonian in time dependent gravitational field

It is shown by a straightforward argument that the Hamiltonian generating the time evolution of the Dirac wave function in relativistic quantum mechanics is not hermitian with respect to the covariantly defined inner product whenever the background metric is time dependent. An alternative, hermitian, Hamiltonian is found and is shown to be directly related to the canonical field Hamiltonian used in quantum field theory.Comment: 9 pages, final version, to appear in Class. Quant. Gra

Treatment of the infrared contribution: NLO QED evolution as a pedagogic example

We show that the conventional prescription used for DGLAP parton evolution at NLO has an inconsistent treatment of the contribution from the infrared (IR) region. We illustrate the problem by studying the simple example of QED evolution, treating the electron and photon as partons. The deficiency is not present in a physical approach which removes the IR divergency and allows calculation in the normal 4-dimensional space.Comment: 15 pages, 2 figures, erratum at the end of the articl

Corrections to Finite Size Scaling in Percolation

A 1/L-expansion for percolation problems is proposed, where L is the lattice finite length. The square lattice with 27 different sizes L = 18, 22 ... 1594 is considered. Certain spanning probabilities were determined by Monte Carlo simulations, as continuous functions of the site occupation probability p. We estimate the critical threshold pc by applying the quoted expansion to these data. Also, the universal spanning probability at pc for an annulus with aspect ratio r=1/2 is estimated as C = 0.876657(45)

Improving the Drell-Yan probe of small x partons at the LHC via a k_t cut

We show that the observation of the Drell-Yan production of low-mass lepton-pairs (M 3) at the LHC can make a direct measurement of parton distribution functions (PDFs) in the low x region, x < 10^{-4}. We describe a procedure that greatly reduces the sensitivity of the predictions to the choice of the factorization scale and, in particular, show how, by imposing a cutoff on the transverse momentum of the lepton-pair, the data are able to probe PDFs in the important low scale, low x domain. We include the effects of the Sudakov suppression factor.Comment: 14 pages, 5 figures, version to be published in EPJC, with expanded explanatio