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

Developments in perfect simulation of Gibbs measures through a new result for the extinction of Galton-Watson-like processes

This paper deals with the problem of perfect sampling from a Gibbs measure with infinite range interactions. We present some sufficient conditions for the extinction of processes which are like supermartingales when large values are taken. This result has deep consequences on perfect simulation, showing that local modifications on the interactions of a model do not affect simulability. We also pose the question to optimize over a class of sequences of sets that influence the sufficient condition for the perfect simulation of the Gibbs measure. We completely solve this question both for the long range Ising models and for the spin models with finite range interactions.Comment: 28 page

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

A combinatorial approach to jumping particles II: general boundary conditions

International audienceWe consider a model of particles jumping on a row, the TASEP. From the point of view of combinatorics a remarkable feauture of this Markov chain is that Catalan numbers are involved in several entries of its stationary distribution. In a companion paper, we gave a combinatorial interpretaion and a simple proof of these observations in the simplest case where the particles enter, jump and exit at the same rate. In this paper we show how to deal with general rates

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

On 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

Discrete approximations to vector spin models

We strengthen a result of two of us on the existence of effective interactions for discretised continuous-spin models. We also point out that such an interaction cannot exist at very low temperatures. Moreover, we compare two ways of discretising continuous-spin models, and show that, except for very low temperatures, they behave similarly in two dimensions. We also discuss some possibilities in higher dimensions.Comment: 12 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

A Covariant OBE Model for η\eta Production in NN Collisions

A relativistic covariant one boson exchange model, previously applied to describe elastic nucleon-nucleon scattering, is extended to study $\eta$ production in NN collisions. The transition amplitude for the elementary BN->$\eta$N process with B being the meson exchanged (B=$\pi$, $|sigma$,$\eta$, $\rho$, $\omega$ and $\delta$) are taken to be the sum of four terms corresponding to s and u-channels with a nucleon or a nucleon isobar N*(1535MeV) in the intermediate states. Taking the relative phases of the various exchange amplitudes to be +1, the model reproduces the cross sections for the $NN\to X\eta$ reactions in a consistent manner. In the limit where all $\eta$'s are produced via N^* excitations, interference terms between the overall contributions from the exchange of pseudoscalart and scalar mesons with that of vector mesons cancel out. Consequently, much of the ambiguities in the model predictions due to unknown relative phases of different vector pseudoscalar exchanges are strongly reduced.Comment: 40 pages, 15 figure

Dynamical percolation on general trees

H\"aggstr\"om, Peres, and Steif (1997) have introduced a dynamical version of percolation on a graph $G$. When $G$ is a tree they derived a necessary and sufficient condition for percolation to exist at some time $t$. In the case that $G$ is a spherically symmetric tree, H\"aggstr\"om, Peres, and Steif (1997) derived a necessary and sufficient condition for percolation to exist at some time $t$ in a given target set $D$. The main result of the present paper is a necessary and sufficient condition for the existence of percolation, at some time $t\in D$, in the case that the underlying tree is not necessary spherically symmetric. This answers a question of Yuval Peres (personal communication). We present also a formula for the Hausdorff dimension of the set of exceptional times of percolation.Comment: 24 pages; to appear in Probability Theory and Related Field