The influence theorem for product measures on the discrete space {0, 1} N may be extended to probability measures with the property of monotonicity (which is equivalent to 'strong positive-association'). Corresponding results are valid for probability measures on the cube [0, 1] N that are absolutely continuous with respect to Lebesgue measure. These results lead to a sharp-threshold theorem for measures of random-cluster type, and this may be applied to box-crossings in the two-dimensional random-cluster model.

A sharp-threshold theorem is proved for box-crossing probabilities on the square lattice. The models in question are the random-cluster model near the self-dual point $p_{\mathrm {sd}}(q)=\sqrt{q}/(1+\sqrt{q})$, the Ising model with external field, and the colored random-cluster model. The principal technique is an extension of the influence theorem for monotonic probability measures applied to increasing events with no assumption of symmetry.Comment: Published in at http://dx.doi.org/10.1214/10-AAP693 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A range of first-passage percolation type models are believed to demonstrate the related properties of sublinear variance and superdiffusivity. We show that directed last-passage percolation with Gaussian vertex weights has a sublinear variance property. We also consider other vertex weight distributions.Corresponding results are obtained for the ground state of the 'directed polymers in a random environment' model.

Let µ be the self-avoiding walk connective constant on Z d . We show that the asymptotic expansion for βc = 1/µ in powers of 1/(2d) satisfies Borel type bounds. This supports the conjecture that the expansion is Borel summable.

Suppose a binary string x = x1 . . . xn is being broadcast repeatedly over a faulty communication channel. Each time, the channel delivers a fixed number m of the digits (m < n) with the lost digits chosen uniformly at random, and the order of the surviving digits preserved. How large does m have to be to reconstruct the message?