In the large N limit, we show that the Local Potential Approximation to the
flow equation for the Legendre effective action, is in effect no longer an
approximation, but exact - in a sense, and under conditions, that we determine
precisely. We explain why the same is not true for the Polchinski or Wilson
flow equations and, by deriving an exact relation between the Polchinski and
Legendre effective potentials (that holds for all N), we find the correct large
N limit of these flow equations. We also show that all forms (and all parts) of
the renormalization group are exactly soluble in the large N limit, choosing as
an example, D dimensional O(N) invariant N-component scalar field theory.Comment: 13 pages, uses harvmac; Added: one page with further clarification of
the main results, discussion of earlier work, and new references. To be
published in Phys. Lett.
