Dimensional crossover and effective exponents

We investigate the critical behavior of the lambda phi(4) theory defined on S-1 x R-d having two finite length scales beta, the circumference of S-1, and k(-1), the blocking scale introduced by the renormalization group transformation. By numerically solving the coupled differential RG equations for the finite-temperature blocked potential U-beta,U-k(Phi) and the wave-function renormalization constant Z(beta,k)(Phi), we demonstrate how the finite-size scaling variable <(beta)over bar> = beta k determines whether the phase transition is (d + 1)- or d-dimensional in the limits <(beta)over bar> much greater than 1 and <(beta)over bar> much less than 1, respectively. For the intermediate values of p, finite-size effects play an important role. We also discuss the failure of the polynomial expansion of the effective potential near criticality. (C) 1997 Elsevier Science B.V.