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Raphael Rossignol

Publications and source records attributed to Raphael Rossignol.

7 recordsLinked to original sources
arXiv2012

Greedy Polyominoes and first-passage times on random Voronoi tilings

Let N be distributed as a Poisson random set on R^d with intensity comparable to the Lebesgue measure. Consider the Voronoi tiling of R^d, (C_v)_{v\in N}, where C_v is composed by points x in R^d that are closer to v than to any other v' in N. A polyomino P of size n is a connected union (in the usual R^d topological sense) of n tiles, and we denote by Pi_n the collection of all polyominos P of size n containing the origin. Assume that the weight of a Voronoi tile C_v is given by F(C_v), where F is a nonnegative functional on Voronoi tiles. In this paper we investigate the tail behavior of the maximal weight among polyominoes in Pi_n for some functionals F, mainly when F(C_v) is the number of faces of C_v. Next we apply our results to study self-avoiding paths, first-passage percolation models and the stabbing number on the dual graph, named the Delaunay triangulation. As the main application we show that first passage percolation has at most linear variance.

math.PR↗
arXiv2009

The threshold for random (1,2)-QSAT

The QSAT problem is the quantified version of the SAT problem. We show the existence of a threshold effect for the phase transition associated with the satisfiability of random quantified extended 2-CNF formulas. We consider boolean CNF formulas of the form $\forall X \exists Y φ(X,Y)$, where $X$ has $m$ variables, $Y$ has $n$ variables and each clause in $φ$ has one literal from $X$ and two from $Y$. For such formulas, we show that the threshold phenomenon is controlled by the ratio between the number of clauses and the number $n$ of existential variables. Then we give the exact location of the associated critical ratio $c^{*}$. Indeed, we prove that $c^{*}$ is a decreasing function of $ α$, where $α$ is the limiting value of $m / \log (n)$ when $n$ tends to infinity.

cs.DM↗
arXiv2007

Submean variance bound for effective resistance of random electric networks

We study a model of random electric networks with Bernoulli resistances. In the case of the lattice Z^2, we show that the point-to-point effective resistance between 0 and a vertex v has a variance of order at most (log |v|)^(2/3) whereas its expected value is of order log |v|, when v goes to infinity. When the dimension of Z^d is different than 2, expectation and variance are of the same order. Similar results are obtained in the context of p-resistance. The proofs rely on a modified Poincare inequality due to Falik and Samorodnitsky.

math.PR↗
arXiv2007

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

We revisit the work of Bourgain, Kahn, Kalai, Katznelson and Linial (1992) -- referred to as ``BKKKL'' in the title -- about influences on Boolean functions in order to give a precise statement of threshold phenomenon on the product space $\{1,..., r\}^\NN$, generalizing one of the main results of a paper by Talagrand (1994).

math.PR↗
arXiv2007

Effective resistance on random electrical networks

This paper has been withdrawn. See v1 still available to understand the problem: Proposition 2.2 is false. The error in the proof is in claim (3). Then, the whole paper collapses. We do not have any correction for now. We apologize to everyone.

math.PR↗
arXiv2006

A modified Poincare inequality and its application to First Passage Percolation

We extend a Gaussian functional inequality to a countable product of Gaussian measures. This inequality improves on the classical Poincare inequality for Gaussian measures. As an application, we prove that First Passage Percolation has sublinear variance when the edge times distribution belongs to a wide class of continuous distributions, including the exponential one. This extends a result by Benjamini, Kalai and Schramm, valid for positive Bernoulli edge times.

math.PR↗