Asymptotic Expansions for lambda_d of the Dimer and Monomer-Dimer Problems

In the past few years we have derived asymptotic expansions for lambda_d of the dimer problem and lambda_d(p) of the monomer-dimer problem. The many expansions so far computed are collected herein. We shine a light on results in two dimensions inspired by the work of M. E. Fisher. Much of the work reported here was joint with Shmuel Friedland.Comment: 4 page

An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem

Let (lambda_d)(p) be the p monomer-dimer entropy on the d-dimensional integer lattice Z^d, where p in [0,1] is the dimer density. We give upper and lower bounds for (lambda_d)(p) in terms of expressions involving (lambda_(d-1))(q). The upper bound is based on a conjecture claiming that the p monomer-dimer entropy of an infinite subset of Z^d is bounded above by (lambda_d)(p). We compute the first three terms in the formal asymptotic expansion of (lambda_d)(p) in powers of 1/d. We prove that the lower asymptotic matching conjecture is satisfied for (lambda_d)(p).Comment: 15 pages, much more about d=1,2,

Detection of the old stellar component of the major Galactic bar

We present near-IR colour--magnitude diagrams and star counts for a number of regions along the Galactic plane. It is shown that along the l=27 b=0 line of sight there is a feature at 5.7 +-0.7kpc with a density of stars at least a factor two and probably more than a factor five times that of the disc at the same position. This feature forms a distinct clump on an H vs. J-H diagram and is seen at all longitudes from the bulge to about l=28, but at no longitude greater than this. The distance to the feature at l=20 is about 0.5kpc further than at l=27 and by l=10 it has merged with, or has become, the bulge. Given that at l=27 and l=21 there is also a clustering of very young stars, the only component that can reasonably explain what is seen is a bar with half length of around 4kpc and a position angle of about 43+-7.Comment: 5 pages, 5 figures accepted as a letter in MNRA

Old stellar Galactic disc in near-plane regions according to 2MASS: scales, cut-off, flare and warp

We have pursued two different methods to analyze the old stellar population near the Galactic plane, using data from the 2MASS survey. The first method is based on the isolation of the red clump giant population in the color-magnitude diagrams and the inversion of its star counts to obtain directly the density distribution along the line of sight. The second method fits the parameters of a disc model to the star counts in 820 regions. Results from both independent methods are consistent with each other. The qualitative conclusions are that the disc is well fitted by an exponential distribution in both the galactocentric distance and height. There is not an abrupt cut-off in the stellar disc (at least within R<15 kpc). There is a strong flare (i.e. an increase of scale-height towards the outer Galaxy) which begins well inside the solar circle, and hence there is a decrease of the scale-height towards the inner Galaxy. Another notable feature is the existence of a warp in the old stellar population whose amplitude is coincident with the amplitude of the gas warp. It is shown for low latitude stars (mean height: |z|~300 pc) in the outer disc (galactocentric radius R>6 kpc) that: the scale-height in the solar circle is h_z(R_sun)=3.6e-2 R_sun, the scale-length of the surface density is h_R=0.42 R_sun and the scale-length of the space density in the plane (i.e. including the effect of the flare) is H=0.25 R_sun. The variation of the scale-height due to the flare follows roughly a law h_z(R) =~ h_z(R_sun) exp [(R-R_\odot)/([12-0.6R(kpc)] kpc)] (for R<~15 kpc; R_sun=7.9 kpc). The warp moves the mean position of the disc to a height z_w=1.2e-3 R(kpc)^5.25 sin(phi+(5 deg.)) pc (for R<~13 kpc; R_sun=7.9 kpc).Comment: LaTEX, 20 pages, 23 figures, accepted to be published in A&

Inversion of stellar statistics equation for the Galactic Bulge

A method based on Lucy (1974, AJ 79, 745) iterative algorithm is developed to invert the equation of stellar statistics for the Galactic bulge and is then applied to the K-band star counts from the Two-Micron Galactic Survey in a number of off-plane regions (10 deg.>|b|>2 deg., |l|<15 deg.). The top end of the K-band luminosity function is derived and the morphology of the stellar density function is fitted to triaxial ellipsoids, assuming a non-variable luminosity function within the bulge. The results, which have already been outlined by Lopez-Corredoira et al.(1997, MNRAS 292, L15), are shown in this paper with a full explanation of the steps of the inversion: the luminosity function shows a sharp decrease brighter than M_K=-8.0 mag when compared with the disc population; the bulge fits triaxial ellipsoids with the major axis in the Galactic plane at an angle with the line of sight to the Galactic centre of 12 deg. in the first quadrant; the axial ratios are 1:0.54:0.33, and the distance of the Sun from the centre of the triaxial ellipsoid is 7860 pc. The major-minor axial ratio of the ellipsoids is found not to be constant. However, the interpretation of this is controversial. An eccentricity of the true density-ellipsoid gradient and a population gradient are two possible explanations. The best fit for the stellar density, for 1300 pc<t<3000 pc, are calculated for both cases, assuming an ellipsoidal distribution with constant axial ratios, and when K_z is allowed to vary. From these, the total number of bulge stars is ~ 3 10^{10} or ~ 4 10^{10}, respectively.Comment: 19 pages, 23 figures, accepted in MNRA

Rearranged Stochastic Heat Equation

The purpose of this work is to provide an explicit construction of a strong Feller semigroup on the space of probability measures over the real line that additionally maps bounded measurable functions into Lipschitz continuous functions, with a Lipschitz constant that blows up in an integrable manner in small time. Our construction relies on a rearranged version of the stochastic heat equation on the circle driven by a coloured noise. Formally, this stochastic equation writes as a reflected equation in infinite dimension, a topic that is known to be challenging. Under the action of the rearrangement, the solution is forced to live in a space of quantile functions that is isometric to the space of probability measures on the real line. We prove the equation to be solvable by means of an Euler scheme in which we alternate flat dynamics in the space of random variables on the circle with a rearrangement operation that projects back the random variables onto the subset of quantile functions. A first challenge is to prove that this scheme is tight. A second one is to provide a consistent theory for the limiting reflected equation and in particular to interpret in a relevant manner the reflection term. The last step in our work is to establish the aforementioned Lipschitz property of the semigroup by adapting earlier ideas from the Bismut-Elworthy-Li formula in stochastic analysis

The Interpretation of Near-Infrared Star Counts at the South Galactic Pole

We present new deep $K'$ counts of stars at the South Galactic Pole (SGP) taken with the NAOJ PICNIC camera to $K'=17.25$. Star-galaxy separation to $K'=17.5$ was accomplished effectively using image profiles because the pixel size we used is 0.509 arcsec. We interpret these counts using the SKY (Cohen 1994) model of the Galactic point source sky and determine the relative normalization of halo-to-disk populations, and the location of the Sun relative to the Galactic plane, within the context of this model. The observed star counts constrain these parameters to be: halo/disk $\sim$ 1/900 and z$_\odot$=16.5$\pm$2.5 pc. These values have been used to correct our SGP galaxy counts for contamination by the point source Galactic foreground.Comment: accepted for publication in AJ, 15 pages with 2 figure

The bulge luminosity functions in the MSX infrared bands

We use an inversion technique to derive the luminosity functions of the Galactic bulge from point source counts extracted from the Midcourse Space Experiment's Point Source Catalog (version 1.2).Comment: 5 pages, 2 figures, to be published in A&

An excess of very bright stars in the inner bulge

From an analysis of the stars remaining in central regions of the Galaxy after subtracting those belonging to the disc and the bulge, we deduce that the inner bulge must have an extra young population with respect the rest of the bulge. It is shown that there is a higher ratio of very bright stars in the central bulge than than there is in the outer bulge. This is interpreted as being an additional young component due to the presence of star formation regions near the Galactic Centre which is absent in the outer bulge.Comment: 6 pages, 12 figures, accepted to be published in MNRA