Raphaël Cerf
University of Paris-Sud
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Featured researches published by Raphaël Cerf.
Stochastic Processes and their Applications | 2002
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
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
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
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
Raphaël Cerf
\sigma^{2}
Annals of Applied Probability | 2015
Raphaël Cerf
, the sum
Transactions of the American Mathematical Society | 2011
Raphaël Cerf; Marie Théret
S_n
european conference on artificial evolution | 1995
Raphaël Cerf
of the random variables has fluctuations of order
Annals of Probability | 2015
Raphaël Cerf
n^{3/4}
Annals of Probability | 2010
Raphaël Cerf; Reda Messikh
and that