Emergent ${\text{AdS}}_{d+1}$ Geometry from Functional Renormalization Group in the Massless Critical Limit

We present a holographic dual description for the O(N) vector model in $d$-dimensional Euclidean space within the functional renormalization group (FRG) framework. By iterating Wilsonian renormalization group transformations, the extra-dimensional scale coordinate is identified as the radial direction of an emergent $(d+1)$-dimensional bulk spacetime. We construct a bidirectional holographic dictionary that maps non-perturbative fluctuations to the emergent bulk metric warping factors. Under the massless critical configuration, the emergent gravitational vacuum reduces to an Anti-de Sitter (${\text{AdS}}_{d+1}$) geometry, satisfying the local energy conditions.