Monwhea Jeng

On the nonlocality of the fractional Schrödinger equation

A number of papers over the past eight years have claimed to solve the fractional Schrodinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schrodinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported ground state, which is based on a piecewise approach, is definitely not a solution of the fractional Schrodinger equation for the general fractional parameter α. On a more positive note, we present a solution to the fractional Schrodinger equation for the one-dimensional harmonic oscillator with α=1.

Height variables in the Abelian sandpile model: Scaling fields and correlations

We compute the lattice 1-site probabilities, on the upper half-plane, of the four height variables in the two-dimensional Abelian sandpile model. We find their exact scaling form when the insertion point is far from the boundary, and when the boundary is either open or closed. Comparing with the predictions of a logarithmic conformal theory with central charge c=-2, we find a full compatibility with the following field assignments: the heights 2, 3 and 4 behave like (an unusual realization of) the logarithmic partner of a primary field with scaling dimension 2, the primary field itself being associated with the height 1 variable. Finite size corrections are also computed and successfully compared with numerical simulations. Relying on these field assignments, we formulate a conjecture for the scaling form of the lattice 2-point correlations of the height variables on the plane, which remain as yet unknown. The way conformal invariance is realized in this system points to a local field theory with c=-2 which is different from the triplet theory. Comment: 68 pages, 17 figures; v2: published version (minor corrections, one comment added)

Error in statistical tests of error in statistical tests

BackgroundA recent paper found that terminal digits of statistical values in Nature deviated significantly from an equiprobable distribution, indicating errors or inconsistencies in rounding. This finding, as well as the discovery that a large percentage of p values were inconsistent with reported test statistics, led to a great deal of concern in the popular press and scientific community. The findings ultimately led to new guidelines for all Nature Research Journals.MethodsWe checked the statistical analysis behind the original papers tests of equiprobability.ResultsThe original paper tested equiprobability with the Kolmogorov-Smirnov test outside its regime of validity. Correct tests find no statistically significant deviations from equiprobability for the statistical values in Nature.ConclusionStatistical tests should be used correctly.

Force-balance percolation

We study models of correlated percolation where there are constraints on the occupation of sites that mimic force balance, i.e., for a site to be stable requires occupied neighboring sites in all four compass directions in two dimensions. We prove rigorously that p(c) < 1 for the two-dimensional models studied. Numerical data indicate that the force-balance percolation transition is discontinuous with a growing crossover length, with perhaps the same form as the jamming percolation models, suggesting that all models belong to the same universality class with the same underlying mechanism driving the transition in both cases. We find a lower bound for the correlation length in the connected phase and that the correlation function does not appear to be a power law at the transition. Finally, we study the dynamics of the culling procedure invoked to obtain the force-balance configurations and find a dynamical exponent similar to that found in sandpile models.