Frequent points for random walks in two dimensions

For a symmetric random walk in $Z^2$ which does not necessarily have bounded jumps we study those points which are visited an unusually large number of times. We prove the analogue of the Erd\H{o}s-Taylor conjecture and obtain the asymptotics for the number of visits to the most visited site. We also obtain the asymptotics for the number of points which are visited very frequently by time $n$. Among the tools we use are Harnack inequalities and Green's function estimates for random walks with unbounded jumps; some of these are of independent interest

Large deviations for renormalized self-intersection local times of stable processes

We study large deviations for the renormalized self-intersection local time of d-dimensional stable processes of index \beta \in (2d/3,d]. We find a difference between the upper and lower tail. In addition, we find that the behavior of the lower tail depends critically on whether \beta <d or \beta =d.Comment: Published at http://dx.doi.org/10.1214/009117904000001099 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Mineralogy and composition of the upper mantle

Seismic velocities are calculated for two petrological models of the upper mantle, an olivine-rich assemblage, pyrolite, and a garnet-clinopyroxene rich, olivine bearing, assemblage, piclogite. These are compared with recent seismic profiles for various tectonic provinces. The shield data is most consistent with a cold olivine and orthopyroxene-rich LID (the seismic lithosphere) extending to 150 km followed by a high temperature gradient and/or a change in mineralogy that serves to decrease the velocity. From 200 to 400 km the velocities follow a 1400°C adiabat. The rise-tectonic mantle is much slower, presumably hotter and is likely to be above the solidus to depths of at least 300 km. The high V_p/V_s ratio of the lower oceanic lithosphere in the western Pacific is most consistent with eclogite

Composition of the upper mantle: Geophysical tests of two petrological models

The elastic properties of candidate mantle phases are used to test the viability of olivine-rich (pyrolitic) and CaO + Al_2O_3-rich (eclogitic) assemblages for the mantle. High temperature adiabats for each phase of interest are constructed and compared to mantle seismic properties. Both pyrolitic and eclogitic assemblages satisfy the seismic properties between ∼ 200 and 400 km. Between 400 and 670 km depth an eclogitic assemblage yields a superior match to velocities and velocity gradients. The 400 km seismic discontinuity may represent a chemical boundary between pyrolite and picritic eclogite (“piclogite”) or phase transformations in the olivine + orthopyroxene components of a piclogitic assemblage containing about 16% olivine. High velocity gradients in the transition zone may be explained by the transformation of Ca-rich cpx to majorite garnet. Seismic properties at the top of the lower mantle are consistent with pyrolite, piclogite or perovskite, implying that the 670 km discontinuity may be a chemical boundary

Sound velocities and elasticity of aluminous MgSiO_3 perovskite: Implications for aluminum heterogeneity in Earth's lower mantle

Aluminum has been reported to have a remarkably strong effect on the thermoelastic properties of MgSiO_3 perovskite. However, the sound velocities of aluminous MgSiO_3 perovskite have not been previously measured, even though this phase likely dominates most of the chemistry in Earth's lower mantle. Here we report the first sound velocity measurements on aluminous MgSiO_3 perovskite using Brillouin spectroscopy and obtain the following values for the room-pressure room-temperature adiabatic bulk and shear moduli: K_S = 252 ± 5 GPa and μ = 165 ± 2 GPa, respectively. The presence of 5.1 ± 0.2 wt.% Al_(2)O_3 in MgSiO_3 perovskite decreases the shear modulus by 5.6%. However, within experimental uncertainties, there is no discernable effect of aluminum on the bulk modulus. We find that variations in the aluminum content of MgSiO_3 perovskite may provide an explanation for some observed lateral heterogeneity in Earth's lower mantle

Holographic in situ stress measurements

A new instrument for measuring the in situ level of stress in boreholes has been developed. The instrument operates on the principle of locally relieving the stresses acting on a rock mass by drilling a small hole into the borehole surface and recording the resultant displacement field by holographic interferometry. Because the recording technique is optical, the entire displacement field due to stress relief is obtained. A description of the stressmeter, theory of the interferometric technique, data reduction methodology, and results of laboratory stress relief calibration tests are presented. In addition, we present results from a field deployment of the instrument in an underground shale mine in Garfield County, Colorado using a test borehole within a support pillar. Sufficient data were obtained to constrain five of six stress components at a shallow level of the test borehole, thereby demonstrating the viability of the holographic technique. The holographic stress-relief data yield an approximate EW maximum horizontal stress direction. By comparison with previous hydrofracture measurements of Bredehoeft et al., our results indicate substantial stress-relief near the pillar face, thus masking any relicts of the far-field tectonic stress