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In independent bond percolation on Z d with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p does the remaining graph contain an infinite connected component? Grimmett-Holroyd-Kozma used the triangle condition to show that for d ≥ 19, the set of such p contains values strictly larger than the percolation threshold p c . With the work of Fitzner-van der Hofstad, this has been reduced to d ≥ 11. We improve this result by showing that for d ≥ 10 and some p > p c , there are infinite paths consisting of "shielded" vertices -vertices all whose adjacent edges are closed -which must be in the complement of the infinite cluster. Using values of p c obtained from computer simulations, this bound can be reduced to d ≥ 7. Our methods are elementary and do not require the triangle condition.

This paper considers a hybrid risky asset price model given by a constant elasticity of variance multiplied by a stochastic volatility factor. A multiscale analysis leads to an asymptotic pricing formula for both European vanilla option and a Barrier option near the zero elasticity of variance. The accuracy of the approximation is provided in a rigorous manner. A numerical experiment for implied volatilities shows that the hybrid model improves some of the well-known models in view of fitting the data for different maturities.

Invasion percolation is a stochastic growth model that follows a greedy algorithm. After assigning i.i.d. uniform random variables (weights) to all edges of Z d , the growth starts at the origin. At each step, we adjoin to the current cluster the edge of minimal weight from its boundary. In '85, Chayes-Chayes-Newman studied the "acceptance profile" of the invasion: for a given p ∈ [0, 1], it is the ratio of the expected number of invaded edges until time n with weight in [p, p+dp] to the expected number of observed edges (those in the cluster or its boundary) with weight in the same interval. They showed that in all dimensions, the acceptance profile a n (p) converges to one for p < p c and to zero for p > p c . In this paper, we consider a n (p) at the critical point p = p c in two dimensions and show that it is bounded away from zero and one as n → ∞.