Average characteristic polynomials in the two-matrix model

The two-matrix model is defined on pairs of Hermitian matrices $(M_1,M_2)$ of size $n\times n$ by the probability measure $$\frac{1}{Z_n} \exp\left(\textrm{Tr} (-V(M_1)-W(M_2)+\tau M_1M_2)\right)\ dM_1\ dM_2,$$ where $V$ and $W$ are given potential functions and \tau\in\er. We study averages of products and ratios of characteristic polynomials in the two-matrix model, where both matrices $M_1$ and $M_2$ may appear in a combined way in both numerator and denominator. We obtain determinantal expressions for such averages. The determinants are constructed from several building blocks: the biorthogonal polynomials $p_n(x)$ and $q_n(y)$ associated to the two-matrix model; certain transformed functions $\P_n(w)$ and \Q_n(v); and finally Cauchy-type transforms of the four Eynard-Mehta kernels $K_{1,1}$, $K_{1,2}$, $K_{2,1}$ and $K_{2,2}$. In this way we generalize known results for the $1$-matrix model. Our results also imply a new proof of the Eynard-Mehta theorem for correlation functions in the two-matrix model, and they lead to a generating function for averages of products of traces.Comment: 28 pages, references adde

Tracing the origin of dissolved silicon transferred from various soil-plant systems towards rivers: a review

Silicon (Si) released as H4SiO4 by weathering of Si-containing solid phases is partly recycled through vegetation before its land-to-rivers transfer. By accumulating in terrestrial plants to a similar extent as some major macronutrients (0.1–10% Si dry weight), Si becomes largely mobile in the soil-plant system. Litter-fall leads to a substantial reactive biogenic silica pool in soil, which contributes to the release of dissolved Si (DSi) in soil solution. Understanding the biogeochemical cycle of silicon in surface environments and the DSi export from soils into rivers is crucial given that the marine primary bio-productivity depends on the availability of H4SiO4 for phytoplankton that requires Si. Continental fluxes of DSi seem to be deeply influenced by climate (temperature and runoff) as well as soil-vegetation systems. Therefore, continental areas can be characterized by various abilities to transfer DSi from soil-plant systems towards rivers. Here we pay special attention to those processes taking place in soil-plant systems and controlling the Si transfer towards rivers. We aim at identifying relevant geochemical tracers of Si pathways within the soil-plant system to obtain a better understanding of the origin of DSi exported towards rivers. In this review, we compare different soil-plant systems (weathering-unlimited and weathering-limited environments) and the variations of the geochemical tracers (Ge/Si ratios and d30Si) in DSi outputs. We recommend the use of biogeochemical tracers in combination with Si mass-balances and detailed physico-chemical characterization of soil-plant systems to allow better insight in the sources and fate of Si in these biogeochemical systems

Sub-micrometer distribution of Fe oxides and organic matter in Podzol horizons

The spatial distribution of soil constituents at the micrometer scale is of great importance to understand processes controlling the formation of micro-aggregates and the stabilization of organic carbon. Here, the spatial distribution of organic and mineral constituents in Podzol horizons is studied by concerted measurements of (i) the content of various forms of Fe, Al, Si and C determined by selective extraction in the fine earth fraction of soil (f < 2 mm); (ii) the elemental composition of the clay fraction (f < 2 um) with lateral resolution using scanning electron microscopy with energy-dispersive X-ray spectroscopy (SEM-EDS), and with surface selectivity using X-ray photoelectron spectroscopy (XPS); (iii) the specific surface area (SSA) of fine earth and clay fractions by krypton physisorption. The SSA of the fine earth in illuvial horizons is predominantly due to finely divided Fe oxides, including goethite, characterized by an equivalent particle size of about 10 mu m. Kaolinite platelets of about 2 gm size account for a large volume proportion in the clay fraction but have a minor contribution to SSA. Fe oxides and organic matter (OM) are intimately associated. Heterogeneity at the um scale is created by local variations in the relative amounts of kaolinite and Fe-OM associations. These two kinds of physical entities are in random mixture. Moreover, variation of C/Fe atomic ratios reveals sub-mu m scale heterogeneity. The latter is due to variation in the relative proportion of organic compounds and Fe oxides, indicating that aggregation of nanoparticles, and not only mere adsorption or pore filling, plays a role in these associations. In this regard, our results highlight that OM associated with Fe protects Fe oxides against physical displacement and that part of this associated OM is oxidizable by NaOCl treatment. These findings demonstrate that the concept of OM stabilization through association with Fe must be revisited when considering the sub-mu m scale level because fine Fe oxide particles can be easily dispersed during oxidation of associated carbon. Combination of physical fractionation and microanalysis (e.g. SEM-EDS, vibrational spectroscopy) offer promising perspectives to clarify the relationship between chemical composition and sub-inn scale architecture, and to better understand soil processes

Non-colliding Brownian Motions and the extended tacnode process

We consider non-colliding Brownian motions with two starting points and two endpoints. The points are chosen so that the two groups of Brownian motions just touch each other, a situation that is referred to as a tacnode. The extended kernel for the determinantal point process at the tacnode point is computed using new methods and given in a different form from that obtained for a single time in previous work by Delvaux, Kuijlaars and Zhang. The form of the extended kernel is also different from that obtained for the extended tacnode kernel in another model by Adler, Ferrari and van Moerbeke. We also obtain the correlation kernel for a finite number of non-colliding Brownian motions starting at two points and ending at arbitrary points.Comment: 38 pages. In the revised version a few arguments have been expanded and many typos correcte

Gamma-Ray Bursts Trace UV Metrics of Star Formation over 3 < z < 5

We present the first uniform treatment of long duration gamma-ray burst (GRB) host galaxy detections and upper limits over the redshift range 3<z<5, a key epoch for observational and theoretical efforts to understand the processes, environments, and consequences of early cosmic star formation. We contribute deep imaging observations of 13 GRB positions yielding the discovery of eight new host galaxies. We use this dataset in tandem with previously published observations of 31 further GRB positions to estimate or constrain the host galaxy rest-frame ultraviolet (UV; 1600 A) absolute magnitudes M_UV. We then use the combined set of 44 M_UV estimates and limits to construct the M_UV luminosity function (LF) for GRB host galaxies over 3<z<5 and compare it to expectations from Lyman break galaxy (LBG) photometric surveys with the Hubble Space Telescope. Adopting standard prescriptions for the luminosity dependence of galaxy dust obscuration (and hence, total star formation rate), we find that our LF is compatible with LBG observations over a factor of 600x in host luminosity, from M_UV = -22.5 mag to >-15.6 mag, and with extrapolations of the assumed Schechter-type LF well beyond this range. We review proposed astrophysical and observational biases for our sample, and find they are for the most part minimal. We therefore conclude, as the simplest interpretation of our results, that GRBs successfully trace UV metrics of cosmic star formation over the range 3<z<5. Our findings suggest GRBs are providing an accurate picture of star formation processes from z ~3 out to the highest redshifts.Comment: publ. ApJ 809 (2015) 76; 14 figures; replacement to reflect changes to v1 (rounding effects, diff. LF from Bouwens