Critical percolation of free product of groups

In this article we study percolation on the Cayley graph of a free product of groups. The critical probability $p_c$ of a free product $G_1*G_2*...*G_n$ of groups is found as a solution of an equation involving only the expected subcritical cluster size of factor groups $G_1,G_2,...,G_n$. For finite groups these equations are polynomial and can be explicitly written down. The expected subcritical cluster size of the free product is also found in terms of the subcritical cluster sizes of the factors. In particular, we prove that $p_c$ for the Cayley graph of the modular group $\hbox{PSL}_2(\mathbb Z)$ (with the standard generators) is $.5199...$, the unique root of the polynomial $2p^5-6p^4+2p^3+4p^2-1$ in the interval $(0,1)$. In the case when groups $G_i$ can be "well approximated" by a sequence of quotient groups, we show that the critical probabilities of the free product of these approximations converge to the critical probability of $G_1*G_2*...*G_n$ and the speed of convergence is exponential. Thus for residually finite groups, for example, one can restrict oneself to the case when each free factor is finite. We show that the critical point, introduced by Schonmann, $p_{\mathrm{exp}}$ of the free product is just the minimum of $p_{\mathrm{exp}}$ for the factors

High resolution general purpose D-layer experiment for EISCAT incoherent scatter radars using selected set of random codes

International audienceThe ionospheric D-layer is a narrow bandwidth radar target often with a very small scattering cross section. The target autocorrelation function can be obtained by transmitting a series of relatively short coded pulses and computing the correlation between data obtained from different pulses. The spatial resolution should be as high as possible and the spatial side lobes of the codes used should be as small as possible. However, due to the short pulse repetition period (in the order of milliseconds) at any instant, the radar receives detectable scattered signals not only from the pulse illuminating the D-region but also from 3?5 ambiguous-range pulses, which makes it difficult to produce a reliable estimate near zero lag of the autocorrelation function. A new experimental solution to this measurement problem, using a selected set of 40-bit random codes with 4 µs elements giving 600 m spatial resolution is presented. The zero lag is approximated by dividing the pulse into two 20-bit codes and computing the correlation between those two pulses. The lowest altitudes of the E-layer are measured by dividing the pulse into 5 pieces of 8 bits, which allows for computation of 4 lags. In addition, coherent integration of data from four pulses is used for obtaining separately the autocorrelation function estimate for the lowest altitudes and in cases when the target contains structures with a long coherence time. Design details and responses of the experiment are given, and analysed test data are shown

Study on spatiotemporal variations of ionospheric trough in auroral/subauroral region

第3回極域科学シンポジウム/第36回極域宙空圏シンポジウム 11月26日（月）、27日（火） 国立極地研究所 2階ラウン

Percolation in invariant Poisson graphs with i.i.d. degrees

Let each point of a homogeneous Poisson process in R^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution mu. Leaving aside degenerate cases, we prove that for any mu there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme that is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components

The dynamical background of polar mesosphere winter echoes from simultaneous EISCAT and ESRAD observations

International audienceOn 30 October 2004 during a strong solar proton event, layers of enhanced backscatter from altitudes between 55 and 75km have been observed by both ESRAD (52MHz) and the EISCAT VHF (224MHz) radars. These echoes have earlier been termed Polar Mesosphere Winter Echoes, PMWE. After considering the morphology of the layers and their relation to observed atmospheric waves, we conclude that the radars have likely seen the same phenomenon even though the radars' scattering volumes are located about 220km apart and that the most long-lasting layer is likely associated with wind-shear in an inertio-gravity wave. An ion-chemistry model is used to determine parameters necessary to relate wind-shear induced turbulent energy dissipation rates to radar backscatter. The model is verified by comparison with electron density profiles measured by the EISCAT VHF radar. Observed radar signal strengths are found to be 2-3 orders of magnitude stronger than the maximum which can be expected from neutral turbulence alone, assuming that previously published results relating radar signal scatter to turbulence parameters, and turbulence parameters to wind shear, are correct. The possibility remains that some additional or alternative mechanism may be involved in producing PMWE, such as layers of charged dust/smoke particles or large cluster ions

Convergence towards an asymptotic shape in first-passage percolation on cone-like subgraphs of the integer lattice

In first-passage percolation on the integer lattice, the Shape Theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape. Here, we address convergence towards an asymptotic shape for cone-like subgraphs of the $\Z^d$ lattice, where $d\ge2$. In particular, we identify the asymptotic shapes associated to these graphs as restrictions of the asymptotic shape of the lattice. Apart from providing necessary and sufficient conditions for $L^p$- and almost sure convergence towards this shape, we investigate also stronger notions such as complete convergence and stability with respect to a dynamically evolving environment.Comment: 23 pages. Together with arXiv:1305.6260, this version replaces the old. The main results have been strengthened and an earlier error in the statement corrected. To appear in J. Theoret. Proba

Palm pairs and the general mass-transport principle

We consider a lcsc group G acting properly on a Borel space S and measurably on an underlying sigma-finite measure space. Our first main result is a transport formula connecting the Palm pairs of jointly stationary random measures on S. A key (and new) technical result is a measurable disintegration of the Haar measure on G along the orbits. The second main result is an intrinsic characterization of the Palm pairs of a G-invariant random measure. We then proceed with deriving a general version of the mass-transport principle for possibly non-transitive and non-unimodular group operations first in a deterministic and then in its full probabilistic form.Comment: 26 page

Temperature and pressure evolution of the crystal structure of Ax(Fe1-ySe)2 (A = Cs, Rb, K) studied by synchrotron powder diffraction

Temperature-dependent synchrotron powder diffraction on Cs0.83(Fe0.86Se)2 revealed first order I4/m to I4/mmm structural transformation around 216{\deg}C associated with the disorder of the Fe vacancies. Irreversibility observed during the transition is likely associated with a mobility of intercalated Alkali atoms. Pressure-dependent synchrotron powder diffraction on Cs0.83(Fe1-ySe)2, Rb0.85(Fe1-ySe)2 and K0.8(Fe1-ySe)2 (y ~ 0.14) indicated that the I4/m superstructure reflections are present up to pressures of 120 kbar. This may indicate that the ordering of the Fe vacancies is present in both superconducting and non-superconductive states.Comment: 11 pages, 5 figures, 1 tabl

Ion-dispersion and rapid electron fluctuations in the cusp: a case study

We present results from co-ordinated measurements with the low altitude REIMEI satellite and the ESR (EISCAT Svalbard Radar), together with other ground-based instruments carried out in February 2006. The results mainly relate to the dayside cusp where clear signatures of so-called ion-dispersion are seen in the satellite data. The cusp ion-dispersion is important for helping to understand the temporal and spatial structure of magnetopause reconnection. Whenever a satellite crosses boundaries of flux tubes or convection cells, cusp structures such as ion-dispersion will always be encountered. In our case we observed 3 distinct steps in the ion energy, but it includes at least 2 more steps as well, which we interpret as temporal features in relation to pulsed reconnection at the magnetopause. In addition, fast variations of the electron flux and energy occurring during these events have been studied in detail. The variations of the electron population, if interpreted as structures crossed by the REIMEI satellite, would map near the magnetopause to similar features as observed previously with the Cluster satellites. These were explained as Alfvén waves originating from an X-line of magnetic reconnection