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- Raphaël Cerf, Emilio N M Cirillo
- 1999

We consider the problem of bootstrap percolation on a three dimensional lattice and we study its finite size scaling behavior. Bootstrap percolation is an example of Cellular Automata defined on the d-dimensional lattice {1, 2, ..., L} d in which each site can be empty or occupied by a single particle; in the starting configuration each site is occupied… (More)

The set of the three dimensional polyominoes of minimal area and of volume n contains a polyomino which is the union of a quasicube j × (j + δ) × (j + θ), δ, θ ∈ {0, 1}, a quasisquare l × (l +), ∈ {0, 1}, and a bar k. This shape is naturally associated to the unique decomposition of n = j(j + δ)(j + θ) + l(l +) + k as the sum of a maximal quasicube, a… (More)

Based on speculations coming from statistical mechanics and the conjectured existence of critical states, I propose a simple heuristic in order to control the mutation probability and the population size of a genetic algorithm. Genetic algorithms are widely used nowadays, as well as their cousins evolutionary algorithms. The most cited initial references on… (More)

We study Eigen's quasispecies model in the asymptotic regime where the length of the genotypes goes to [Formula: see text] and the mutation probability goes to 0. We give several explicit formulas for the stationary solutions of the limiting system of differential equations.

We study the simple genetic algorithm with a ranking selection mechanism (linear ranking or tournament). We denote by ℓ the length of the chromosomes, by m the population size, by pC the crossover probability and by pM the mutation probability. We introduce a parameter σ, called the selection drift, which measures the selection intensity of the fittest… (More)

- Raphaël Cerf, Sana Louhichi
- 2004

We consider the 2D stochastic Ising model evolving according to the Glauber dynamics at zero temperature. We compute the initial drift for droplets which are suitable approximations of smooth domains. A specific spatial average of the derivative at time 0 of the volume variation of a droplet close to a boundary point is equal to its curvature multiplied by… (More)

- Raphaël Cerf, Matthias Gorny
- 2015

Let ρ and µ be two probability measures on R which are not the Dirac mass at 0. We denote by H(µ|ρ) the relative entropy of µ with respect to ρ. We prove that, if ρ is symmetric and µ has a finite first moment, then H(µ|ρ) ≥ R z dµ(z) 2 2 R z 2 dµ(z) , with equality if and only if µ = ρ. We give an applicaion to the Curie-Weiss model of self-organized… (More)

We introduce a new parameter to discuss the behavior of a genetic algorithm. This parameter is the mean number of exact copies of the best fit chromosomes from one generation to the next. We argue that the genetic algorithm should operate efficiently when this parameter is slightly larger than 1. We consider the case of the simple genetic algorithm with the… (More)

- Raphaël Cerf
- 2014

We consider the standard site percolation model on the three dimensional cubic lattice. Starting solely with the hypothesis that θ(p) > 0, we prove that, for any α > 0, there exists κ > 0 such that, with probability larger than 1 − 1/n α , every pair of sites inside the box Λ(n) are joined by a path having at most κ(ln n) 2 closed sites.