Determinantal formulae and loop equations

We prove that the correlations functions, generated by the determinantal process of the Christoffel-Darboux kernel of an arbitrary order 2 ODE, do satisfy loop equations.Comment: Latex, 35 pages few misprints correcte

The sine-law gap probability, Painlev\'e 5, and asymptotic expansion by the topological recursion

The goal of this article is to rederive the connection between the Painlev\'e $5$ integrable system and the universal eigenvalues correlation functions of double-scaled hermitian matrix models, through the topological recursion method. More specifically we prove, \textbf{to all orders}, that the WKB asymptotic expansions of the $\tau$-function as well as of determinantal formulas arising from the Painlev\'e $5$ Lax pair are identical to the large $N$ double scaling asymptotic expansions of the partition function and correlation functions of any hermitian matrix model around a regular point in the bulk. In other words, we rederive the "sine-law" universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in the bulk with only algebraic methods. Eventually we exhibit the first orders of the series expansion up to $O(N^{-5})$.Comment: 37 pages, 1 figure, published in Random Matrices: Theory and Application

Discrete Thermodynamic Bethe Ansatz

We propose discrete TBA equations for models with discrete spectrum. We illustrate our construction on the Calogero-Moser model and determine the discrete 2-body TBA function which yields the exact N-body Calogero-Moser thermodynamics. We apply this algorithm to the Lieb-Liniger model in a harmonic well, a model which is relevant for the microscopic description of harmonically trapped Bose-Einstein condensates in one dimension. We find that the discrete TBA reproduces correctly the N-body groundstate energy of the Lieb-Liniger model in a harmonic well at first order in perturbation theory, but corrections do appear at second order

Trypanosoma vivax Infections: Pushing Ahead with Mouse Models for the Study of Nagana. II. Immunobiological Dysfunctions

Trypanosoma vivax is the main species involved in trypanosomosis, but very little is known about the immunobiology of the infective process caused by this parasite. Recently we undertook to further characterize the main parasitological, haematological and pathological characteristics of mouse models of T. vivax infection and noted severe anemia and thrombocytopenia coincident with rising parasitemia. To gain more insight into the organism's immunobiology, we studied lymphocyte populations in central (bone marrow) and peripherical (spleen and blood) tissues following mouse infection with T. vivax and showed that the immune system apparatus is affected both quantitatively and qualitatively. More precisely, after an initial increase that primarily involves CD4+ T cells and macrophages, the number of splenic B cells decreases in a step-wise manner. Our results show that while infection triggers the activation and proliferation of Hematopoietic Stem Cells, Granulocyte-Monocyte, Common Myeloid and Megacaryocyte Erythrocyte progenitors decrease in number in the course of the infection. An in-depth analysis of B-cell progenitors also indicated that maturation of pro-B into pre-B precursors seems to be compromised. This interferes with the mature B cell dynamics and renewal in the periphery. Altogether, our results show that T. vivax induces profound immunological alterations in myeloid and lymphoid progenitors which may prevent adequate control of T. vivax trypanosomosis

Trypanosoma vivax Infections: Pushing Ahead with Mouse Models for the Study of Nagana. I. Parasitological, Hematological and Pathological Parameters

African trypanosomiasis is a severe parasitic disease that affects both humans and livestock. Several different species may cause animal trypanosomosis and although Trypanosoma vivax (sub-genus Duttonella) is currently responsible for the vast majority of debilitating cases causing great economic hardship in West Africa and South America, little is known about its biology and interaction with its hosts. Relatively speaking, T. vivax has been more than neglected despite an urgent need to develop efficient control strategies. Some pioneering rodent models were developed to circumvent the difficulties of working with livestock, but disappointedly were for the most part discontinued decades ago. To gain more insight into the biology of T. vivax, its interactions with the host and consequently its pathogenesis, we have developed a number of reproducible murine models using a parasite isolate that is infectious for rodents. Firstly, we analyzed the parasitical characteristics of the infection using inbred and outbred mouse strains to compare the impact of host genetic background on the infection and on survival rates. Hematological studies showed that the infection gave rise to severe anemia, and histopathological investigations in various organs showed multifocal inflammatory infiltrates associated with extramedullary hematopoiesis in the liver, and cerebral edema. The models developed are consistent with field observations and pave the way for subsequent in-depth studies into the pathogenesis of T. vivax - trypanosomosis

Rational differential systems, loop equations, and application to the q-th reductions of KP

49 pagesTo any solution of a linear system of differential equations, we associate a kernel, correlators satisfying a set of loop equations, and in presence of isomonodromic parameters, a Tau function. We then study their semiclassical expansion (WKB type expansion in powers of the weight hbar per derivative) of these quantities. When this expansion is of topological type (TT), the coefficients of expansions are computed by the topological recursion with initial data given by the semiclassical spectral curve of the linear system. This provides an efficient algorithm to compute them at least when the semiclassical spectral curve is of genus 0. TT is a non trivial property, and it is an open problem to find a criterion which guarantees it is satisfied. We prove TT and illustrate our construction for the linear systems associated to the q-th reductions of KP - which contain the (p,q) models as a specialization

Counting mobiles by integrable systems

Mobiles are a particular class of decorated plane trees which serve as codings for planar maps. Here we address the question of enumerating mobiles in their most general flavor, in correspondence with planar Eulerian (i.e., bicolored) maps. We show that the generating functions for such mobiles satisfy a number of recursive equations which lie in the field of integrable systems, leading us to explicit expressions for these generating functions as ratios of particular determinants. In particular we recover known results for mobiles associated with uncolored maps and prove some conjectured formulas for the generating functions of mobiles associated with $p$-constellations