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- Raphaël Cerf
- 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)

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.

- 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 p , the law of large numbers was initially proved by Artstein and Vitale [1]… (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 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)

- 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)