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We present the Wilsonian effective action as a solution of the exact RGequation for the critical $O(N)$ vector model in the large $N$ limit. Belowfour dimensions, the exact effective action can be expressed in a closed formas a transcendental function of two leading scaling operators with infinitelymany derivatives. From the exact solution that describes the RG flow from a UVtheory to the fixed point theory in the IR, we obtain the mapping between UVoperators and IR scaling operators. It is shown that IR scaling operators aregiven by sums of infinitely many UV operators with infinitely many derivatives.

Exact effective action for the O(N) vector model in the large N limit