Multiple Meixner-Pollaczek polynomials and the six-vertex model

We study multiple orthogonal polynomials of Meixner-Pollaczek type with respect to a symmetric system of two orthogonality measures. Our main result is that the limiting distribution of the zeros of these polynomials is one component of the solution to a constrained vector equilibrium problem. We also provide a Rodrigues formula and closed expressions for the recurrence coefficients. The proof of the main result follows from a connection with the eigenvalues of block Toeplitz matrices, for which we provide some general results of independent interest. The motivation for this paper is the study of a model in statistical mechanics, the so-called six-vertex model with domain wall boundary conditions, in a particular regime known as the free fermion line. We show how the multiple Meixner-Pollaczek polynomials arise in an inhomogeneous version of this model.Comment: 32 pages, 4 figures. References adde

Heisenberg double as braided commutative Yetter-Drinfel'd module algebra over Drinfel'd double in multiplier Hopf algebra case

Based on a pairing of two regular multiplier Hopf algebras $A$ and $B$, Heisenberg double $\mathscr{H}$ is the smash product $A \# B$ with respect to the left regular action of $B$ on $A$. Let $\mathscr{D}=A\bowtie B$ be the Drinfel'd double, then Heisenberg double $\mathscr{H}$ is a Yetter-Drinfel'd $\mathscr{D}$-module algebra, and it is also braided commutative by the braiding of Yetter-Drinfel'd module, which generalizes the results in [10] to some infinite dimensional cases.Comment: 18 pages. arXiv admin note: text overlap with arXiv:math/0404029 by other author

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

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

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