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- Raphaël Cerf
- 2012

The goal of this work is to propose a finite population counterpart to Eigen's model, which incorporates stochastic effects. We consider a Moran model describing the evolution of a population of size m of chromosomes of length ℓ over an alphabet of cardinality κ. The mutation probability per locus is q. We deal only with the sharp peak landscape: the… (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)

We consider the Moran model on the sharp peak landscape, in the asymptotic regime studied in [3], where a quasispecies is formed. We find explicitly the distribution of this quasispecies.

- Ben Gérard, Arous, Gerard Benarous@ens, Fr, Raphaël Cerf
- 1996

We study the metastability of the stochastic three dimensional Ising model on a finite torus under a small positive magnetic field at very low temperatures. Abstract. We study the metastability of the stochastic three dimensional Ising model on a finite torus under a small positive magnetic field at very low temperatures.

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)

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.

- Raphaël Cerf, Joseba Dalmau, Dma, Ecole, Normale Supérieure
- 2016

We give a probabilistic representation of the stationary solutions of Eigen's model, when the set of possible genotypes is finite and the mutation matrix is primitive. In the long chain regime, we perform a formal passage to the limit to obtain a probabilistic representation of the quasispecies distribution. We prove rigorously the validity of this… (More)

- Raphaël Cerf, Joseba Dalmaú
- 2015

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