Heavy tails in last passage percolation

We consider last-passage percolation models in two dimensions, in which the underlying weight distribution has a heavy tail of index $\alpha<2$. We prove scaling laws and asymptotic distributions, both for the passage times and for the shape of optimal paths; these are expressed in terms of a family (indexed by $\alpha$) of ``continuous last-passage percolation'' models in the unit square. In the extreme case $\alpha=0$ (corresponding to a distribution with slowly varying tail) the asymptotic distribution of the optimal path can be represented by a random self-similar measure on [0,1], whose multifractal spectrum we compute. By extending the continuous last-passage percolation model to $\mathbb{R}^2$ we obtain a heavy-tailed analogue of the Airy process, representing the limit of appropriately scaled vectors of passage times to different points in the plane. We give corresponding results for a directed percolation problem based on $\alpha$-stable Levy processes, and indicate extensions of the results to higher dimensions

Quantum fields, dark matter and non-standard Wigner classes

The Elko field of Ahluwalia and Grumiller is a quantum field for massive spin-1/2 particles. It has been suggested as a candidate for dark matter. We discuss our attempts to interpret the Elko field as a quantum field in the sense of Weinberg. Our work suggests that one should investigate quantum fields based on representations of the full Poincar\'e group which belong to one of the non-standard Wigner classes.Comment: 6 pages. Submitted to proceedings of Dark2009, Christchurch, New Zealand, January 200

On Tits' Centre Conjecture for Fixed Point Subcomplexes

We give a short and uniform proof of a special case of Tits' Centre Conjecture using a theorem of J-P. Serre and a result from our earlier work. We consider fixed point subcomplexes $X^H$ of the building $X = X(G)$ of a connected reductive algebraic group $G$, where $H$ is a subgroup of $G$.Comment: 4 pages; minor changes, to appear in C. R. Acad. Sci. Paris Ser. I Mat

Effective Three-Body Interactions in Jaynes-Cummings-Hubbard Systems

A generalisation of the Jaynes-Cummings-Hubbard model for coupled-cavity arrays is introduced, where the embedded two-level system in each cavity is replaced by a $\Xi$-type three-level system. We demonstrate that the resulting effective polariton-polariton interactions at each site are both two-body and three-body. By tuning the ratio of the two transition dipole matrix elements, we show that the strength and sign of the two-body interaction can be controlled whilst maintaining a three-body repulsion. We then proceed to demonstrate how different two-body and three-body interactions alter the mean field superfluid-Mott insulator phase diagram, with the possible emergence of a pair superfluid phase in the two-body attractive regime.Comment: 10 pages, 4 figures, accepted for publication in Scientific Reports; v3 - revised manuscrip

Complete Reducibility and Commuting Subgroups

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p. We study J.-P. Serre's notion of G-complete reducibility for subgroups of G. In particular, for a subgroup H and a normal subgroup N of H, we look at the relationship between G-complete reducibility of N and of H, and show that these properties are equivalent if H/N is linearly reductive, generalizing a result of Serre. We also study the case when H = MN with M a G-completely reducible subgroup of G which normalizes N. We show that if G is connected, N and M are connected commuting G-completely reducible subgroups of G, and p is good for G, then H = MN is also G-completely reducible.Comment: 21 pages; to appear in J. Reine Angew. Math. final for

Effect of soil waterlogging on below-ground biomass allometric relations in Norway spruce

An increasing importance is assigned to the estimation and verification of carbon stocks in forests. Forestry practice has several long-established and reliable methods for the assessment of aboveground biomass; however we still miss accurate predictors of belowground biomass. A major windthrow event exposing the coarse root systems of Norway spruce trees allowed us to assess the effects of contrasting soil stone and water content on belowground allocation. Increasing stone content decreases root/shoot ratio, while soil waterlogging leads to an increase in this ratio. We constructed allometric relationships for belowground biomass prediction and were able to show that only soil waterlogging significantly impacts model parameters. We showed that diameter at breast height is a reliable predictor of belowground biomass and, once site-specific parameters have been developed, it is possible to accurately estimate belowground biomass in Norway spruce

Wind-tunnel acoustic results of two rotor models with several tip designs

A three-phase research program has been undertaken to study the acoustic signals due to the aerodynamic interaction of rotorcraft main rotors and tail rotors. During the first phase, two different rotor models with several interchangeable tips were tested in the Langley 4- by 7-Meter Tunnel on the U.S. Army rotor model system. An extensive acoustic data base was acquired, with special emphasis on blade-vortex interaction (BVI) noise. The details of the experimental procedure, acoustic data acquisition, and reduction are documented. The overall sound pressure level (OASPL) of the high-twist rotor systems is relatively insensitive to flight speed but generally increases with rotor tip-path-plane angle. The OASPL of the high-twist rotors is dominated by acoustic energy in the low-frequency harmonics. The OASPL of the low-twist rotor systems shows more dependence on flight speed than the high-twist rotors, in addition to being quite sensitive to tip-path-plane angle. An integrated band-limited sound pressure level, limited by 500 to 3000 Hz, is a useful metric to quantify the occurrence of BVI noise. The OASPL of the low-twist rotors is strongly influenced by the band-limited sound levels, indicating that the blade-vortex impulsive noise is a dominant noise source for this rotor design. The midfrequency acoustic levels for both rotors show a very strong dependence on rotor tip-path-plane angle. The tip-path-plane angle at which the maximum midfrequency sound level occurs consistently decreases with increasing flight speed. The maximum midfrequency sound level measured at a given location is constant regardless of the flight speed