Raphaël Rossignol

Exponential concentration for first passage percolation through modified Poincaré inequalities

We provide a new exponential concentration inequality for first passage percolation valid for a wide class of edge times distributions. This improves and extends a result by Benjamini, Kalai and Schramm (Ann. Probab. 31 (2003)) which gave a variance bound for Bernoulli edge times. Our approach is based on some functional inequalities extending the work of Rossignol (Ann. Probab. 35 (2006)), Falik and Samorodnitsky (Combin. Probab. Comput. 16 (2007)). Resume. On obtient une nouvelle inegalite de concentration exponentielle pour la percolation de premier passage, valable pour une large classe de distributions des temps d’aretes. Ceci ameliore et etend un resultat de Benjamini, Kalai et Schramm (Ann. Probab. 31 (2003)) qui donnait une borne sur la variance pour des temps d’aretes suivant une loi de Bernoulli. Notre approche se fonde sur des inegalites fonctionnelles etendant les travaux de Rossignol (Ann. Probab. 35 (2006)), Falik et Samorodnitsky (Combin. Probab. Comput. 16 (2007)). AMS 2000 subject classifications: Primary 60E15; secondary 60K35

Threshold for monotone symmetric properties through a logarithmic Sobolev inequality

Threshold phenomena are investigated using a general approach, following Talagrand [Ann. Probab. 22 (1994) 1576-1587] and Friedgut and Kalai [Proc. Amer. Math. Soc. 12 (1999) 1017-10541. The general upper bound for the threshold width of symmetric monotone properties is improved. This follows from a new lower bound on the maximal influence of a variable on a Boolean function. The method of proof is based on a well-known logarithmic Sobolev inequality on {0, 1} n . This new bound is shown to be asymptotically optimal.

Noise-stability and central limit theorems for effective resistance of random electric networks

We investigate the (generalized) Walsh decomposition of point-to-point effective resistances on countable random electric networks with i.i.d. resistances. We show that it is concentrated on low levels, and thus point-to-point effective resistances are uniformly stable to noise. For graphs that satisfy some homogeneity property, we show in addition that it is concentrated on sets of small diameter. As a consequence, we compute the right order of the variance and prove a central limit theorem for the effective resistance through the discrete torus of side length n in Zd, when n goes to infinity.

Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation

We consider the standard first passage percolation model in

Sharp threshold for percolation on expanders

\mathbb{Z}^d

Exact location of the phase transition for random (1, 2)-QSAT

for

Threshold phenomena on product spaces: BKKKL revisited (once more)

d\geq 2

Submean Variance Bound for Effective Resistance of Random Electric Networks

. We are interested in two quantities, the maximal flow

Greedy polyominoes and first-passage times on random Voronoi tilings

\tau

Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation

between the lower half and the upper half of the box, and the maximal flow