Self-avoiding walk in 5 or more dimensions
We consider a random walk on a simple cubic lattice of five or more dimensions. The probability for an N step walk, w is proportional to \(\exp (-\beta \sum n^ 2_ x(w))\), where the sum is over all sites x in the lattice, \(n_ x(w)\) is the number of times w visits x and \(\beta\geq 0\) is a parameter. This is the Domb-Joyce model. We prove for small \(\beta\) results equivalent to a central limit theorem for the endpoint of the walk. In particular for \(\beta >0\) sufficiently small E(exp(ikw(st)/\(\sqrt{t}))\to \exp (-const.sk^ 2)\) as \(t\to \infty\) uniformly on compacts in k and s, where w(st) is the endpoint of a walk with (integer part of) st steps. The method of proof combines some ideas from expansions in statistical mechanics with an induction on time-scales reminiscent of the renormalization group philosophy.
scientific article; zbMATH DE number 3919520