The Page-R{\'e}nyi parking process

In the Page parking (or packing) model on a discrete interval (also known as the discrete R{\'e}nyi packing problem or the unfriendly seating problem), cars of length two successively park uniformly at random on pairs of adjacent places, until only isolated places remain. We give a probabilistic proof of the (known) fact that the proportion of the interval covered by cars goes to 1-exp(-2) , when the length of the interval goes to infinity. We obtain some new consequences, and also study a version of this process defined on the infinite line

On the algebraic numbers computable by some generalized Ehrenfest urns

This article deals with some stochastic population protocols, motivated by theoretical aspects of distributed computing. We modelize the problem by a large urn of black and white balls from which at every time unit a fixed number of balls are drawn and their colors are changed according to the number of black balls among them. When the time and the number of balls both tend to infinity the proportion of black balls converges to an algebraic number. We prove that, surprisingly enough, not every algebraic number can be "computed" this way

Construction of a short path in high dimensional First Passage Percolation

For First Passage Percolation in Z^d with large d, we construct a path connecting the origin to {x_1 =1}, whose passage time has optimal order \log d/d. Besides, an improved lower bound for the "diagonal" speed of the cluster combined with a result by Dhar (1988) shows that the limiting shape in FPP with exponential passage times (and thus that of Eden model) is not the euclidian ball in dimension larger than 35

Distances in the highly supercritical percolation cluster

On the supercritical percolation cluster with parameter p, the distances between two distant points of the axis are asymptotically increased by a factor 1+(1-p)/2+o(1-p) with respect to the usual distance. The proof is based on an apparently new connection with the TASEP (totally asymmetric simple exclusion process).Comment: 15 page

HCO, c-C3H and CF+ : three new molecules in diffuse, translucent and "spiral-arm'' clouds

%methods {We used the EMIR receiver and FTS spectrometer at the IRAM 30m to construct absorption spectra toward bright extra-galactic background sources at 195 kHz spectral resolution ($\approx$ 0.6 \kms). We used the IRAM Plateau de Bure interferometer to synthesize absorption spectra of \hthcop\ and HCO toward the galactic HII region W49.} %results {HCO, \cc3h\ and CF\p\ were detected toward the blazars \bll\ and 3C111 having \EBV\ = 0.32 and 1.65 mag. HCO was observed in absorption from ``spiral-arm'' clouds in the galactic plane occulting W49. The complement of detectable molecular species in the 85 - 110 GHz absorption spectrum of diffuse/translucent gas is now fully determined at rms noise level $\delta_\tau \approx 0.002$ at \EBV\ = 0.32 mag (\AV\ = 1 mag) and $\delta_\tau$/\EBV\ $\approx\ 0.003$ mag$^{-1}$ overall.} %conclusions {As with OH, \hcop\ and \cch, the relative abundance of \cc3h\ varies little between diffuse and dense molecular gas, with N(\cc3h)/N({\it o-c}-\c3h2) $\approx$ 0.1. We find N(CF\p)/N(H$^{13}$CO\p) $\approx 5$, N(CF\p)/N(\cch) $\approx$ 0.005-0.01 and because N(CF\p) increases with \EBV\ and with the column densities of other molecules we infer that fluorine remains in the gas phase as HF well beyond \AV\ = 1 mag. We find N(HCO)/N(H$^{13}$CO\p) = 16 toward \bll, 3C111 and the 40 km/s spiral arm cloud toward W49, implying X(HCO) $\approx 10^{-9}$, about 10 times higher than in dark clouds. The behaviour of HCO is consistent with previous suggestions that it forms from C\p\ and \HH, even when \AV\ is well above 1 mag. The survey can be used to place useful upper limits on some species, for instance N(\hhco)/N(\HH CS) $>$ 32 toward 3C111, compared to 7 toward TMC-1, confirming the possibility of a gas phase formation route to \hhco.}Comment: A\%A in pres

A Branching-selection process related to censored Galton-Walton processes

We obtain the asymptotics for the speed of a particular case of a particle system with branching and selection introduced by B\'erard and Gou\'er\'e (2010). The proof is based on a connection with a supercritical Galton-Watson process censored at a certain level.Comment: 16 pages, 2 figure

From Hammersley's lines to Hammersley's trees

We construct a stationary random tree, embedded in the upper half plane, with prescribed offspring distribution and whose vertices are the atoms of a unit Poisson point process. This process which we call Hammersley's tree process extends the usual Hammersley's line process. Just as Hammersley's process is related to the problem of the longest increasing subsequence, this model also has a combinatorial interpretation: it counts the number of heaps (i.e. increasing trees) required to store a random permutation. This problem was initially considered by Byers et. al (2011) and Istrate and Bonchis (2015) in the case of regular trees. We show, in particular, that the number of heaps grows logarithmically with the size of the permutation

The Brownian limit of separable permutations

We study random uniform permutations in an important class of pattern-avoiding permutations: the separable permutations. We describe the asymptotics of the number of occurrences of any fixed given pattern in such a random permutation in terms of the Brownian excursion. In the recent terminology of permutons, our work can be interpreted as the convergence of uniform random separable permutations towards a "Brownian separable permuton".Comment: 45 pages, 14 figures, incorporating referee's suggestion