The lace expansion has been a powerful tool for investigating mean-field behavior for various stochastic-geometrical models, such as self-avoiding walk and percolation, above their respective upper-critical dimension. In this paper, we prove the lace expansion for the Ising model that is valid for any spin-spin coupling. For the ferromagnetic case, we also prove that the expansion coefficients ...

Texture is one of the most important visual attributes for image analysis. It has been widely used in image analysis and pattern recognition. A partially self-avoiding deterministic walk has recently been proposed as an approach for texture analysis with promising results. This approach uses walkers (called tourists) to exploit the gray scale image contexts in several levels. Here, we present a...

We define a class of weakly self-avoiding walks on the integers by conditioning a simple random walk of length n to have a p-fold self-intersection local time smaller than nβ, where 1 < β < (p+1)/2. We show that the conditioned paths grow of order nα, where α = (p− β)/(p− 1), and also prove a coarse large deviation principle for the order of growth.