A percolation system with extremely long range connections and node dilution

We study the very long-range bond-percolation problem on a linear chain with both sites and bonds dilution. Very long range means that the probability $p_{ij}$ for a connection between two occupied sites $i,j$ at a distance $r_{ij}$ decays as a power law, i.e. $p_{ij} = \rho/[r_{ij}^\alpha N^{1-\alpha}]$ when $0 \le \alpha < 1$, and $p_{ij} = \rho/[r_{ij} \ln(N)]$ when $\alpha = 1$. Site dilution means that the occupancy probability of a site is $0 < p_s \le 1$. The behavior of this model results from the competition between long-range connectivity, which enhances the percolation, and site dilution, which weakens percolation. The case $\alpha=0$ with $p_s =1$ is well-known, being the exactly solvable mean-field model. The percolation order parameter $P_\infty$ is investigated numerically for different values of $\alpha$, $p_s$ and $\rho$. We show that in the ranges $0 \le \alpha \le 1$ and $0 < p_s \le 1$ the percolation order parameter $P_\infty$ depends only on the average connectivity $\gamma$ of sites, which can be explicitly computed in terms of the three parameters $\alpha$, $p_s$ and $\rho$

Fluctuating Commutative Geometry

We use the framework of noncommutative geometry to define a discrete model for fluctuating geometry. Instead of considering ordinary geometry and its metric fluctuations, we consider generalized geometries where topology and dimension can also fluctuate. The model describes the geometry of spaces with a countable number $n$ of points. The spectral principle of Connes and Chamseddine is used to define dynamics.We show that this simple model has two phases. The expectation value , the average number of points in the universe, is finite in one phase and diverges in the other. Moreover, the dimension $\delta$ is a dynamical observable in our model, and plays the role of an order parameter. The computation of is discussed and an upper bound is found, $< 2$. We also address another discrete model defined on a fixed $d=1$ dimension, where topology fluctuates. We comment on a possible spontaneous localization of topology.Comment: 7 pages. Talk at the conference "Spacetime and Fundamental Interactions: Quantum Aspects" (Vietri sul Mare, Italy, 26-31 May 2003), in honour of A. P. Balachandran's 65th birthda

Lifshitz-point critical behaviour to O(ϵ2){\boldsymbol{O(\epsilon^2)}}

We comment on a recent letter by L. C. de Albuquerque and M. M. Leite (J. Phys. A: Math. Gen. 34 (2001) L327-L332), in which results to second order in $\epsilon=4-d+\frac{m}{2}$ were presented for the critical exponents $\nu_{{\mathrm{L}}2}$, $\eta_{{\mathrm{L}}2}$ and $\gamma_{{\mathrm{L}}2}$ of d-dimensional systems at m-axial Lifshitz points. We point out that their results are at variance with ours. The discrepancy is due to their incorrect computation of momentum-space integrals. Their speculation that the field-theoretic renormalization group approach, if performed in position space, might give results different from when it is performed in momentum space is refuted.Comment: Latex file, uses the included iop stylefiles; Uses the texdraw package to generate included figure

Anisotropic Lifshitz Point at O(ϵL2)O(\epsilon_L^2)

We present the critical exponents $\nu_{L2}$, $\eta_{L2}$ and $\gamma_{L}$ for an $m$-axial Lifshitz point at second order in an $\epsilon_{L}$ expansion. We introduced a constraint involving the loop momenta along the $m$-dimensional subspace in order to perform two- and three-loop integrals. The results are valid in the range $0 \leq m < d$. The case $m=0$ corresponds to the usual Ising-like critical behavior.Comment: 10 pages, Revte

Use of water quality index to evaluate the influence of anthropogenic contamination on groundwater chemistry of a shallow aquifer, Loures valley, Lisbon, Portugal

A significant industrial development, associated with a demographic expansion, occurred during the last decades of the XX century, in Loures valley, a region located in the vicinities of Lisbon, the capital city of Portugal. This was accompanied with an important modification of land use and occupation patterns, mainly the decrease of the agricultural land. One of the main consequences was the deterioration of the groundwater chemistry observed in the shallow aquifer associated to Trancão river, a subsidiary of Tagus river. Factorial Correspondence Analysis has been used to build a water quality index, for evaluating the impact of the anthropogenic factors on groundwater of the shallow aquifer. By analysing the kriged maps of the values of the index, it was possible to identify the areas more sensitive to the anthropogenic impact

Neutrino Telescopes as a Direct Probe of Supersymmetry Breaking

We consider supersymmetric models where the scale of supersymmetry breaking lies between 5 $\times 10^6$ GeV and 5 $\times 10^8$ GeV. In this class of theories, which includes models of gauge mediated supersymmetry breaking, the lightest supersymmetric particle is the gravitino. The next to lightest supersymmetric particle is typically a long lived charged slepton with a lifetime between a microsecond and a second, depending on its mass. Collisions of high energy neutrinos with nucleons in the earth can result in the production of a pair of these sleptons. Their very high boost means they typically decay outside the earth. We investigate the production of these particles by the diffuse flux of high energy neutrinos, and the potential for their observation in large ice or water Cerenkov detectors. The relatively small cross-section for the production of supersymmetric particles is partially compensated for by the very long range of heavy particles. The signal in the detector consists of two parallel charged tracks emerging from the earth about 100 meters apart, with very little background. A detailed calculation using the Waxman-Bahcall limit on the neutrino flux and realistic spectra shows that km$^3$ experiments could see as many as 4 events a year. We conclude that neutrino telescopes will complement collider searches in the determination of the supersymmetry breaking scale, and may even give the first evidence for supersymmetry at the weak scale.Comment: 4 pages, 3 figure