Abstract
We use the lace expansion to prove that in five or more dimensions the standard selfavoiding walk on the hypercubic (integer) lattice behaves in many respects like the simple random walk. In particular, it is shown that the leading asymptotic behaviour of the number of nstep selfavoiding walks is purely exponential, that the mean square displacement is asymptotically linear in the number of steps, and that the scaling limit is Gaussian, in the sense of convergence in distribution to Brownian motion. Some related facts are also proved. These results are optimal, according to the widely believed conjecture that the selfavoiding walk behaves unlike the simple random walk in dimensions two, three and four.
Original language  English 

Pages (fromto)  417423 
Number of pages  7 
Journal  Bulletin of the American Mathematical Society 
Volume  25 
Issue number  2 
DOIs 

Publication status  Published  Oct 1991 
Externally published  Yes 
All Science Journal Classification (ASJC) codes
 General Mathematics
 Applied Mathematics