Within the exact renormalisation group, the scaling solutions for O(N)
symmetric scalar field theories are studied to leading order in the derivative
expansion. The Gaussian fixed point is examined for d>2 dimensions and
arbitrary infrared regularisation. The Wilson-Fisher fixed point in d=3 is
studied using an optimised flow. We compute critical exponents and subleading
corrections-to-scaling to high accuracy from the eigenvalues of the stability
matrix at criticality for all N. We establish that the optimisation is
responsible for the rapid convergence of the flow and polynomial truncations
thereof. The scheme dependence of the leading critical exponent is analysed.
For all N > 0, it is found that the leading exponent is bounded. The upper
boundary is achieved for a Callan-Symanzik flow and corresponds, for all N, to
the large-N limit. The lower boundary is achieved by the optimised flow and is
closest to the physical value. We show the reliability of polynomial
approximations, even to low orders, if they are accompanied by an appropriate
choice for the regulator. Possible applications to other theories are outlined.Comment: 34 pages, 15 figures, revtex, to appear in NP
