Halo Assembly Bias in the Quasi-linear Regime

We address the question of whether or not assembly bias arises in the absence of highly non-linear effects such as tidal stripping of halos near larger mass concentrations. Therefore, we use a simplified dynamical scheme where these effects are not modeled. We choose the punctuated Zel'dovich (PZ) approximation, which prevents orbit mixing by coalescing particles coming within a critical distance of each other. A numerical implementation of this approximation is fast, allowing us to run a large number of simulations to study assembly bias. We measure an assembly bias from 60 PZ simulations, each with 512^3 cold particles in a 128h^-1 Mpc cubic box. The assembly bias estimated from the correlation functions at separations < 5h^-1 Mpc for objects (halos) at z=0 is comparable to that obtained in full N-body simulations. For masses 4x10^11 h^-1 Mo the "oldest" 10% haloes are 3-5 times more correlated than the "youngest" 10%. The bias weakens with increasing mass, also in agreement with full N-body simulations. We find that that halo ages are correlated with the dimensionality of the surrounding linear structures as measured by the parameter (\lambda_1+\lambda_2+\lambda_3)/ (\lambda_1^2+\lambda_2^2+\lambda_3^2)^{1/2} where \lambda_i are proportional to the eigenvalues of the velocity deformation tensor. Our results suggest that assembly bias may already be encoded in the early stages of the evolution.Comment: 7 pages, 5 figures; Minor revision; Accepted for publication in MNRA

From one-dimensional charge conserving superconductors to the gapless Haldane phase

We develop a framework to analyze one-dimensional topological superconductors with charge conservation. In particular, we consider models with $N$ flavors of fermions and $(\mathbb{Z}_2)^N$ symmetry, associated with the conservation of the fermionic parity of each flavor. For a single flavor, we recover the result that a distinct topological phase with exponentially localized zero modes does not exist due to absence of a gap to single particles in the bulk. For $N>1$, however, we show that the ends of the system can host low-energy, exponentially-localized modes. The analysis can readily be generalized to systems in other symmetry classes. To illustrate these ideas, we focus on lattice models with $SO\left(N\right)$ symmetric interactions, and study the phase transition between the trivial and the topological gapless phases using bosonization and a weak-coupling renormalization group analysis. As a concrete example, we study in detail the case of $N=3$. We show that in this case, the topologically non-trivial superconducting phase corresponds to a gapless analogue of the Haldane phase in spin-1 chains. In this phase, although the bulk is gapless to single particle excitations, the ends host spin-$1/2$ degrees of freedom which are exponentially localized and protected by the spin gap in the bulk. We obtain the full phase diagram of the model numerically, using density matrix renormalization group calculations. Within this model, we identify the self-dual line studied by Andrei and Destri [Nucl. Phys. B, 231(3), 445-480 (1984)], as a first-order transition line between the gapless Haldane phase and a trivial gapless phase. This allows us to identify the propagating spin-$1/2$ kinks in the Andrei-Destri model as the topological end-modes present at the domain walls between the two phases

Per Family or Familywise Type I Error Control: Eether, Eyether, Neether, Nyther, Let\u27s Call the Whole Thing Off!

Frane (2015) pointed out the difference between per-family and familywise Type I error control and how different multiple comparison procedures control one method but not necessarily the other. He then went on to demonstrate in the context of a two group multivariate design containing different numbers of dependent variables and correlations between variables how the per-family rate inflates beyond the level of significance. In this article I reintroduce other newer better methods of Type I error control. These newer methods provide more power to detect effects than the per-family and familywise techniques of control yet maintain the overall rate of Type I error at a chosen level of significance. In particular, I discuss the False Discovery Rate due to Benjamini and Hochberg (1995) and k-Familywise Type I error control enumerated by Lehmann and Romano (2005), Romano and Shaikh (2006), and Sarkar (2008). I conclude the article by referring readers to articles by Keselman, et al. (2011, 2012) which presented R computer code for determining critical significance levels for these newer methods of Type I error control