Critical temperature of finite systems in d dimensions

The critical temperature of an Ising system consisting of n(d-1) dimensional layers in d dimensions is estimated for large n from the properties of random and self-avoiding walks (SAW) in the finite system. Denoting the deviation of T c (n) from T c ( infinity ) by 1/n lambda it is found that there are two contributions, a finite size effect which applies both to a torus and a strip and for which lambda =d-2+ eta , and a free boundary effect which applies to a strip alone and for which lambda =1/ nu d . ( nu d , eta are exponents associated with the spin pair correlation function using the standard notation of Fisher). The results are based on a perturbation expansion for SAW which should be reliable for d>or=4, but which needs further examination when d=3. Some, but not all, of the above features are in agreement with the results of Fisher and Barber for the spherical model.