Universal microscopic correlation functions for products of independent Ginibre matrices

We consider the product of n complex non-Hermitian, independent random matrices, each of size NxN with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Ginibre matrix, but with a more complicated weight given by a Meijer G-function depending on n. Using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite N and fixed n. They are given by the determinant of the corresponding kernel which we construct explicitly. In the large-N limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum. After unfolding they are identical to that of the Ginibre ensemble with n=1 and thus universal. In contrast the microscopic correlations we find at the origin differ for each n>1 and generalise the known Bessel-law in the complex plane for n=2 to a new hypergeometric kernel 0_F_n-1.Comment: 20 pages, v2 published version: typos corrected and references adde

Badger macrophages fail to produce nitric oxide, a key anti-mycobacterial effector molecule

The European badger is recognised as a wildlife reservoir for bovine tuberculosis (bTB); the control of which is complex, costly and controversial. Despite the importance of badgers in bTB and the well-documented role for macrophages as anti-mycobacterial effector cells, badger macrophage (bdMij) responses remain uncharacterised. Here, we demonstrate that bdMij fail to produce nitric oxide (NO) or upregulate inducible nitric oxide synthase (iNOS) mRNA following Toll-like receptor (TLR) agonist treatment. BdMij also failed to make NO after stimulation with recombinant badger interferon gamma (bdIFNȖ) or a combination of bdIFNȖ and lipopolysaccharide. Exposure of bdMij to TLR agonists and/or bdIFNȖ resulted in upregulated cytokine (IL1ȕ, IL6, IL12 and TNFĮ) mRNA levels indicating that these critical pathways were otherwise intact. Although stimulation with most TLR agonists resulted in strong cytokine mRNA responses, weaker responses were evident after exposure to TLR9 agonists, potentially due to very low expression of TLR9 in bdMij. Both NO and TLR9 are important elements of innate immunity to mycobacteria, and these features of bdMij biology would impair their capacity to resist bTB infection. These findings have significant implications for the development of bTB management strategies, and support the use of vaccination to reduce bTB infection in badgers