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- Raphaël Cerf
- Artificial Evolution
- 1995

- Laurent Alonso, Raphaël Cerf
- Electr. J. Comb.
- 1996

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 maximal… (More)

- Raphaël Cerf, Pierre Petit
- The American Mathematical Monthly
- 2011

We give a short proof of Cramér’s large deviations theorem based on convex duality. This proof does not resort to the law of large numbers or any other limit theorem. The most fundamental result in probability theory is the law of large numbers for a sequence (Xn)n≥1 of independent and identically distributed real-valued random variables. Define the… (More)

- RAPHAËL CERF
- 1999

We prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is, the analog of Cramér theorem for random compact sets. Several works have been devoted to deriving limit theorems for random sets. For i.i.d. random compact sets in R, the law of large numbers was initially proved by Artstein and Vitale [1] and… (More)

- Raphaël Cerf, Joseba Dalmau
- Bulletin of mathematical biology
- 2016

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.

- Raphaël Cerf
- ArXiv
- 2010

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)

- Raphaël Cerf
- ArXiv
- 2014

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)

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)

There has been a great deal of interest and activity in percolation theory since the two Saint-Flour courses, [76, 111], of 1984 and 1996 reprinted in this new edition. We present here a summary of progress since the first publications of our lecture notes. The second edition of Percolation, [77], was published in 1999 as a fairly complete, contemporary… (More)

We consider the classical Wright–Fisher model of population genetics. We prove the existence of an error threshold for the mutation probability, below which a quasispecies is formed. We show a new phenomenon, specific to a finite population model, namely the existence of a population threshold: to ensure the stability of the quasispecies, the population… (More)