Raphaël Cerf

The threshold regime of finite volume bootstrap percolation

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.

Nucleation and growth for the Ising model in

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.

d

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.

dimensions at very low temperatures

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

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

\sigma^{2}

A Curie-Weiss Model of Self-Organized Criticality

, the sum

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

S_n

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

of the random variables has fluctuations of order

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

n^{3/4}

\mathbb{R}^{d}

and that