Spins, percolation and height functions

Percolation of the Minority Spins in High-Dimensional Ising Models

We present some results on the shape and size of the percolative region of the nearest neighbor ferromagnetic Ising model on yd. The percolat ion properties of the connected clusters of (say) the + spins in equilibrium states of these models have been studied because of interest in the properties of correlated percolat ion (see Refs. 1 and 2) and because of some questions related to the ferroma...

Height Functions

Definition 1. An (archimedean) absolute value on a field k is a real valued function ‖ · ‖ : k → [0,∞) with the following three properties: (1) ‖x‖ = 0 if and only if x = 0. (2) ‖xy‖ = ‖x‖ · ‖y‖. (3) ‖x+ y‖ ≤ ‖x‖+ ‖y‖. A nonarchimedean absolute value satisfies the extra condition that (3’) ‖x+ y‖ ≤ max{‖x‖, ‖y‖}. If k is a field, we denote Mk the set of absolute values on k. As an abuse of nota...

Quadratic Forms and Height Functions

Noise sensitivity of Boolean functions and percolation

The notion of noise sensitivity was introduced by Benjamini, Kalai, and Schramm in 1998 [3], in the context of percolation theory. Since then noise sensitivity has found applications in many fields, including concentration of measure, social choice theory, and theoretical computer science. Say we have n people voting for one of two candidates, D or R. One can write their choices as ω = (ω(1), ....

Distribution functions in percolation problems.

Percolation clusters are random fractals whose geometrical and transport properties can be characterized with the help of probability distribution functions. Using renormalized field theory, we determine the asymptotic form of various such distribution functions in the limits where certain scaling variables become small or large. Our study includes the pair-connection probability, the distribut...