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Fixed-point equations in the functional renormalization group approach areintegrated from large to vanishing field, where an asymptotic potential in thelimit of large field is implemented as initial conditions. This approach allowsus to obtain a global fixed-point potential with high numerical accuracy, thatincorporates the correct asymptotic behavior in the limit of large field. Ourcalculated global potential is in good agreement with the Taylor expansion inthe region of small field, and it also coincides with the Laurent expansion inthe regime of large field. Laurent expansion of the potential in the limit oflarge field for general case, that the spatial dimension $d$ is a continuousvariable in the range $2\leq d \leq 4$, is obtained. Eigenfunctions andeigenvalues of perturbations near the Wilson-Fisher fixed point are computedwith the method of eigenperturbations. Critical exponents for different valuesof $d$ and $N$ of the $O(N)$ universality class are calculated.

Criticality of the $O(N)$ universality via global solutions to nonperturbative fixed-point equations