On the asymptotics of dimers on tori

We study asymptotics of the dimer model on large toric graphs. Let $\mathbb L$ be a weighted $\mathbb{Z}^2$-periodic planar graph, and let $\mathbb{Z}^2 E$ be a large-index sublattice of $\mathbb{Z}^2$. For $\mathbb L$ bipartite we show that the dimer partition function on the quotient $\mathbb{L}/(\mathbb{Z}^2 E)$ has the asymptotic expansion $\exp[A f_0 + \text{fsc} + o(1)]$, where $A$ is the area of $\mathbb{L}/(\mathbb{Z}^2 E)$, $f_0$ is the free energy density in the bulk, and $\text{fsc}$ is a finite-size correction term depending only on the conformal shape of the domain together with some parity-type information. Assuming a conjectural condition on the zero locus of the dimer characteristic polynomial, we show that an analogous expansion holds for $\mathbb{L}$ non-bipartite. The functional form of the finite-size correction differs between the two classes, but is universal within each class. Our calculations yield new information concerning the distribution of the number of loops winding around the torus in the associated double-dimer models.Comment: 48 pages, 18 figure

The star cluster mass--galactocentric radius relation: Implications for cluster formation

Whether or not the initial star cluster mass function is established through a universal, galactocentric-distance-independent stochastic process, on the scales of individual galaxies, remains an unsolved problem. This debate has recently gained new impetus through the publication of a study that concluded that the maximum cluster mass in a given population is not solely determined by size-of-sample effects. Here, we revisit the evidence in favor and against stochastic cluster formation by examining the young ($\lesssim$ a few $\times 10^8$ yr-old) star cluster mass--galactocentric radius relation in M33, M51, M83, and the Large Magellanic Cloud. To eliminate size-of-sample effects, we first adopt radial bin sizes containing constant numbers of clusters, which we use to quantify the radial distribution of the first- to fifth-ranked most massive clusters using ordinary least-squares fitting. We supplement this analysis with an application of quantile regression, a binless approach to rank-based regression taking an absolute-value-distance penalty. Both methods yield, within the $1\sigma$ to $3\sigma$ uncertainties, near-zero slopes in the diagnostic plane, largely irrespective of the maximum age or minimum mass imposed on our sample selection, or of the radial bin size adopted. We conclude that, at least in our four well-studied sample galaxies, star cluster formation does not necessarily require an environment-dependent cluster formation scenario, which thus supports the notion of stochastic star cluster formation as the dominant star cluster-formation process within a given galaxy.Comment: ApJ, in press, 39 pages in AAS preprint format, 10 multi-panel figures (some reduced in size to match arXiv compilation routines

Interpreting Labor Supply Regressions in a Model of Full and Part-Time Work

We construct a family model of labor supply that features adjustment along both the intensive and extensive margin. Intensive margin adjustment is restricted to two values: full time work and part-time work. Using simulated data from the steady state of the calibrated model, we examine whether standard labor supply regressions can uncover the true value of the intertemporal elasticity of labor supply parameter. We find positive estimated elasticities that are larger for women and that are highly significant, but they bear virtually no relationship to the underlying preference parameters.