Enumeration of simple random walks and tridiagonal matrices

We present some old and new results in the enumeration of random walks in one dimension, mostly developed in works of enumerative combinatorics. The relation between the trace of the $n$-th power of a tridiagonal matrix and the enumeration of weighted paths of $n$ steps allows an easier combinatorial enumeration of the paths. It also seems promising for the theory of tridiagonal random matrices .Comment: several ref.and comments added, misprints correcte

Schwinger-boson approach to quantum spin systems: Gaussian fluctuactions in the "natural" gauge

We compute the Gaussian-fluctuation corrections to the saddle-point Schwinger-boson results using collective coordinate methods. Concrete application to investigate the frustrated J1-J2 antiferromagnet on the square lattice shows that, unlike the saddle-point predictions, there is a quantum nonmagnetic phase for 0.53 < J2/J1 < 0.64. This result is obtained by considering the corrections to the spin stiffness on large lattices and extrapolating to the thermodynamic limit, which avoids the infinite-lattice infrared divergencies associated to Bose condensation. The very good agreement of our results with exact numerical values on finite clusters lends support to the calculational scheme employed.Comment: 4 pages, Latex, 3 figures included as eps files,minor correction

Superconductivity and Quantum Spin Disorder in Cuprates

A fundamental connection between superconductivity and quantum spin fluctuations in underdoped cuprates, is revealed. A variational calculation shows that {\em Cooper pair hopping} strongly reduces the local magnetization $m_0$. This effect pertains to recent neutron scattering and muon spin rotation measurements in which $m_0$ varies weakly with hole doping in the poorly conducting regime, but drops precipitously above the onset of superconductivity

Finite time and asymptotic behaviour of the maximal excursion of a random walk

We evaluate the limit distribution of the maximal excursion of a random walk in any dimension for homogeneous environments and for self-similar supports under the assumption of spherical symmetry. This distribution is obtained in closed form and is an approximation of the exact distribution comparable to that obtained by real space renormalization methods. Then we focus on the early time behaviour of this quantity. The instantaneous diffusion exponent $\nu_n$ exhibits a systematic overshooting of the long time exponent. Exact results are obtained in one dimension up to third order in $n^{-1/2}$. In two dimensions, on a regular lattice and on the Sierpi\'nski gasket we find numerically that the analytic scaling $\nu_n \simeq \nu+A n^{-\nu}$ holds.Comment: 9 pages, 4 figures, accepted J. Phys.

The 1/N1/N Expansion and Spin Correlations in Constrained Wavefunctions

We develop a large-N expansion for Gutzwiller projected spin states. We consider valence bonds singlets, constructed by Schwinger bosons or fermions, which are variational ground states for quantum antiferromagnets. This expansion is simpler than the familiar expansions of the quantum Heisenberg model, and thus more instructive. The diagrammatic rules of this expansion allow us to prove certain identities to all orders in 1/N. We derive the on-site spin fluctuations sum rule for arbitrary N. We calculate the correlations of the one dimensional Valence Bonds Solid states and the Gutzwiller Projected Fermi Gas upto order 1/N. For the bosons case, we are surprised to find that the mean field, the order 1/N and the exact correlations are simply proportional. For the fermions case, the 1/N correction enhances the zone edge singularity. The comparison of our leading order terms to known results for N=2, enhances our understanding of large-N approximations in general.Comment: 36 pages, LaTe

Regional mechanical and biochemical properties of the porcine cortical meninges

peer-reviewedThe meninges are pivotal in protecting the brain against traumatic brain injury (TBI), an ongoing issue in most mainstream sports. Improved understanding of TBI biomechanics and pathophysiology is desirable to improve preventative measures, such as protective helmets, and advance our TBI diagnostic/prognostic capabilities. This study mechanically characterised the porcine meninges by performing uniaxial tensile testing on the dura mater (DM) tissue adjacent to the frontal, parietal, temporal, and occipital lobes of the cerebellum and superior sagittal sinus region of the DM. Mechanical characterisation revealed a significantly higher elastic modulus for the superior sagittal sinus region when compared to other regions in the DM. The superior sagittal sinus and parietal regions of the DM also displayed local mechanical anisotropy. Further, fatigue was noted in the DM following ten preconditioning cycles, which could have important implications in the context of repetitive TBI. To further understand differences in regional mechanical properties, regional variations in protein content (collagen I, collagen III, fibronectin and elastin) were examined by immunoblot analysis. The superior sagittal sinus was found to have significantly higher collagen I, elastin, and fibronectin content. The frontal region was also identified to have significantly higher collagen I and fibronectin content while the temporal region had increased elastin and fibronectin content. Regional differences in the mechanical and biochemical properties along with regional tissue thickness differences within the DM reveal that the tissue is a non-homogeneous structure. In particular, the potentially influential role of the superior sagittal sinus in TBI biomechanics warrants further investigation