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.

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

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