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Olivier Couronné

Publications and source records attributed to Olivier Couronné.

9 recordsLinked to original sources
arXiv2025

The critical curve of long-range percolation on oriented trees

We consider a long-range percolation model on homogeneous oriented trees with several lengths. We obtain the critical surface as the set of zeros of a specific polynomial with coefficients depending explicitly on the lengths and the degree of the tree. Restricting to the case of two lengths, we obtain new bounds on the critical parameters, monotonicity properties, as well as continuity of the critical curve, plus some partial results concerning its convexity. Our proofs rely on the study of the properties of the characteristic polynomial of the transition matrix of a multi-step Markov chain related to the model.

math.PR↗
arXiv2025

An Improved Lower Bound for the Critical Parameter of the Stavskaya's Process via a Generalized Recurrent Method

Stavskaya's process, a discrete-time version of the contact process on $\Z$, is known to exhibit a phase transition at a critical parameter $\alpc$ whose exact value remains an open problem. Recent work by Ramos et al. established a lower bound by linking the process's survival to the non-percolation of a dual contour. The probability of this contour was estimated using a recurrent method on a state space of weighted random walks with short-term memory. In this paper, we generalize and extend this method by systematically increasing the walk's memory and enriching the set of forbidden path sequences. By increasing the memory up to a length of 20 steps (corresponding to our parameter $n=7$), we formulate the problem with a one-step transition matrix and numerically optimize its spectral radius. We thus establish the new lower bound $\alpc > 0.1370721$.

math.PR↗
arXiv2023

An update on lower bounds for the critical values of oriented percolation models

We obtain new lower bounds on the critical points for various models of oriented percolation. The method is to provide a stochastic domination of the percolation processes by multitype Galton-Watson trees. This can be apply to the classical bond and site oriented percolation on Z^2 , but also on other lattices such as inhomogeneous ones, and on dimension three.

math.PR↗
arXiv2021

Entanglement percolation and spheres in $\mathbb{Z}^d$

We obtain a new lower bound of 0.06576 for the 1-entanglement critical probability (in dimension 3), and prove that the critical point for the existence of a sphere surrounding the origin and intersecting only closed bonds in $\mathbb{Z}^d$ is greater than $\frac1{8(d-1)}$, $d \geq 3$. This substantially improves the previous lower bounds and gives the correct order of magnitude for large $d$.

math.PR↗
arXiv2011

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.

math.PR↗