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Dive into the research topics where Raphaël Cerf is active.

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Featured researches published by Raphaël Cerf.


Stochastic Processes and their Applications | 2002

The threshold regime of finite volume bootstrap percolation

Raphaël Cerf; Francesco Manzo

We prove that the threshold regime for bootstrap percolation in a d-dimensional box of diameter L with parameters p and l, where 3[less-than-or-equals, slant]l[less-than-or-equals, slant]d, is L~ exp°(l-1)(Cp-1/(d-l+1)), where exp°(l-1) is the exponential iterated l-1 times and C is bounded from above and from below by two positive constants depending on d, l only.


Annals of Probability | 2013

Nucleation and growth for the Ising model in

Raphaël Cerf; Francesco Manzo

This work extends to dimension d≥3 the main result of Dehghanpour and Schonmann. We consider the stochastic Ising model on Zd evolving with the Metropolis dynamics under a fixed small positive magnetic field h starting from the minus phase. When the inverse temperature β goes to ∞, the relaxation time of the system, defined as the time when the plus phase has invaded the origin, behaves like exp(βκd). The value κd is equal to κd=1d+1(Γ1+⋯+Γd), where Γi is the energy of the i-dimensional critical droplet of the Ising model at zero temperature and magnetic field h.


Proceedings of the American Mathematical Society | 1999

d

Raphaël Cerf

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 Cramer 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 the central limit theorem by Cressie [3], Lyashenko [10] and Weil [16]. For generalizations to non compact sets, see also Hess [8]. These limit theorems were generalized to the case of random compact sets in a Banach space by Gine, Hahn and Zinn [7] and Puri and Ralescu [11]. Our aim is to prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is, to prove the analog of the Cramer theorem. We consider a separable Banach space F with norm || ||. We denote by K(F ) the collection of all non empty compact subsets of F . For an element A of K(F ), we denote by coA the closed convex hull of A. Mazur’s theorem [5, p 416] implies that, for A in K(F ), coA belongs to coK(F ), the collection of the non empty compact convex subsets of F . The space K(F ) is equipped with the Minkowski addition and the scalar multiplication: for A1, A2 in K(F ) and λ a real number, A1 +A2 = { a1 + a2 : a1 ∈ A1, a2 ∈ A2 } , λA1 = {λa1 : a1 ∈ A1 } . 1991 Mathematics Subject Classification. 60D05, 60F10.


Annals of Probability | 2016

dimensions at very low temperatures

Raphaël Cerf; Matthias Gorny

We try to design a simple model exhibiting self-organized criticality, which is amenable to a rigorous mathematical analysis. To this end, we modify the generalized Ising Curie-Weiss model by implementing an automatic control of the inverse temperature. With the help of exact computations, we show that, in the case of a centered Gaussian measure with positive variance


Memoirs of the American Mathematical Society | 2015

Large deviations for sums of i.i.d. random compact sets

Raphaël Cerf

\sigma^{2}


Annals of Applied Probability | 2015

A Curie-Weiss Model of Self-Organized Criticality

Raphaël Cerf

, the sum


Transactions of the American Mathematical Society | 2011

Critical population and error threshold on the sharp peak landscape for a Moran model

Raphaël Cerf; Marie Théret

S_n


european conference on artificial evolution | 1995

Critical population and error threshold on the sharp peak landscape for the Wright–Fisher model

Raphaël Cerf

of the random variables has fluctuations of order


Annals of Probability | 2015

Law of large numbers for the maximal flow through a domain of

Raphaël Cerf

n^{3/4}


Annals of Probability | 2010

\mathbb{R}^{d}

Raphaël Cerf; Reda Messikh

and that

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Joseba Dalmau

École Normale Supérieure

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Marie Théret

École Normale Supérieure

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Matthias Gorny

École Normale Supérieure

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Reda Messikh

École Polytechnique Fédérale de Lausanne

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Ágoston Pisztora

Carnegie Mellon University

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