A Long Range Dependence Stable Process and an Infinite Variance Branching System

We prove a functional limit theorem for the rescaled occupation time fluctuations of a (d, , )- branching particle system (particles moving in Rd according to a symmetric -stable L´evy process, branching law in the domain of attraction of a (1 + )-stable law, 0 d/(d + ), which coincides with the case of finite variance branching ( = 1), and another one for d/(d + ), where the long range dependence depends on the value of . The long range dependence is characterized by a dependence exponent which describes the asymptotic behavior of the codierence of increments of on intervals far apart, and which is d/ for the first case and (1 + - d/(d + ))d/ for the second one. The convergence proofs use techniques of S0(Rd)-valued processes.Branching particle system, occupation time fluctuation, functional limit theorem, stable process, long range dependence.

Power-free values, large deviations, and integer points on irrational curves

Let $f\in \mathbb{Z}\lbrack x\rbrack$ be a polynomial of degree $d\geq 3$ without roots of multiplicity $d$ or $(d-1)$. Erd\H{o}s conjectured that, if $f$ satisfies the necessary local conditions, then $f(p)$ is free of $(d-1)$th powers for infinitely many primes $p$. This is proved here for all $f$ with sufficiently high entropy. The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov's theorem from the theory of large deviations.Comment: 39 pages; rather major revision, with strengthened and generalized statement

Occupation Time Fluctuations of an Infinite Variance Branching System in Large Dimensions

We prove limit theorems for rescaled occupation time fluctuations of a (d, , )-branching particle system (particles moving in Rd according to a spherically symmetric -stable L´evy process, (1 + )- branching, 0 (1 + )/. The fluctuation processes are continuous but their limits are stable processes with independent increments, which have jumps. The convergence is in the sense of finite-dimensional distributions, and also of space-time random fields (tightness does not hold in the usual Skorohod topology). The results are in sharp contrast with those for intermediate dimensions, /Branching particle system, critical and large dimensions, limit theorem, occupation time fluctuation, stable process.

Upper large deviations for the maximal flow in first passage percolation

We consider the standard first passage percolation in $\mathbb{Z}^{d}$ for $d\geq 2$ and we denote by $\phi_{n^{d-1},h(n)}$ the maximal flow through the cylinder $]0,n]^{d-1} \times ]0,h(n)]$ from its bottom to its top. Kesten proved a law of large numbers for the maximal flow in dimension three: under some assumptions, $\phi_{n^{d-1},h(n)} / n^{d-1}$ converges towards a constant $\nu$. We look now at the probability that $\phi_{n^{d-1},h(n)} / n^{d-1}$ is greater than $\nu + \epsilon$ for some $\epsilon >0$, and we show under some assumptions that this probability decays exponentially fast with the volume of the cylinder. Moreover, we prove a large deviations principle for the sequence $(\phi_{n^{d-1},h(n)} / n^{d-1}, n\in \mathbb{N})$.Comment: 27 pages, 4 figures; small changes of notation