Size-Dependent Tile Self-Assembly: Constant-Height Rectangles and Stability

We introduce a new model of algorithmic tile self-assembly called size-dependent assembly. In previous models, supertiles are stable when the total strength of the bonds between any two halves exceeds some constant temperature. In this model, this constant temperature requirement is replaced by an nondecreasing temperature function $\tau : \mathbb{N} \rightarrow \mathbb{N}$ that depends on the size of the smaller of the two halves. This generalization allows supertiles to become unstable and break apart, and captures the increased forces that large structures may place on the bonds holding them together. We demonstrate the power of this model in two ways. First, we give fixed tile sets that assemble constant-height rectangles and squares of arbitrary input size given an appropriate temperature function. Second, we prove that deciding whether a supertile is stable is coNP-complete. Both results contrast with known results for fixed temperature.Comment: In proceedings of ISAAC 201

Optimal Staged Self-Assembly of General Shapes

We analyze the number of tile types $t$, bins $b$, and stages necessary to assemble $n \times n$ squares and scaled shapes in the staged tile assembly model. For $n \times n$ squares, we prove $\mathcal{O}(\frac{\log{n} - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})$ stages suffice and $\Omega(\frac{\log{n} - tb - t\log t}{b^2})$ are necessary for almost all $n$. For shapes $S$ with Kolmogorov complexity $K(S)$, we prove $\mathcal{O}(\frac{K(S) - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})$ stages suffice and $\Omega(\frac{K(S) - tb - t\log t}{b^2})$ are necessary to assemble a scaled version of $S$, for almost all $S$. We obtain similarly tight bounds when the more powerful flexible glues are permitted.Comment: Abstract version appeared in ESA 201

The structure and assembly history of cluster-size haloes in Self-Interacting Dark Matter

We perform dark-matter-only simulations of 28 relaxed massive cluster-sized haloes for Cold Dark Matter (CDM) and Self-Interacting Dark Matter (SIDM) models, to study structural differences between the models at large radii, where the impact of baryonic physics is expected to be very limited. We find that the distributions for the radial profiles of the density, ellipsoidal axis ratios, and velocity anisotropies ($\beta$) of the haloes differ considerably between the models (at the $\sim1\sigma$ level), even at $\gtrsim10\%$ of the virial radius, if the self-scattering cross section is $\sigma/m_\chi=1$ cm$^2$ gr$^{-1}$. Direct comparison with observationally inferred density profiles disfavours SIDM for $\sigma/m_\chi=1$ cm$^2$ gr$^{-1}$, but in an intermediate radial range ($\sim3\%$ of the virial radius), where the impact of baryonic physics is uncertain. At this level of the cross section, we find a narrower $\beta$ distribution in SIDM, clearly skewed towards isotropic orbits, with no SIDM (90\% of CDM) haloes having $\beta>0.12$ at $7\%$ of the virial radius. We estimate that with an observational sample of $\sim30$ ($\sim10^{15}$ M$_\odot$) relaxed clusters, $\beta$ can potentially be used to put competitive constraints on SIDM, once observational uncertainties improve by a factor of a few. We study the suppression of the memory of halo assembly history in SIDM clusters. For $\sigma/m_\chi=1$ cm$^2$ gr$^{-1}$, we find that this happens only in the central halo regions ($\sim1/4$ of the scale radius of the halo), and only for haloes that assembled their mass within this region earlier than a formation redshift $z_f\sim2$. Otherwise, the memory of assembly remains and is reflected in ways similar to CDM, albeit with weaker trends.Comment: 15 pages, 15 figures. Submitted to MNRAS. Revisions: added new figure with an observational comparison of density profiles, improvements and corrections to the section on velocity anisotropie