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Flow equation for the large $N$ scalar model and induced geometries

We study the proposal that a $d+1$ dimensional induced metric is constructed from a $d$ dimensional field theory using gradient flow. Applying the idea to the O($N$) $φ^4$ model and normalizing the flow field, we have shown in the large $N$ limit that the induced metric is finite and universal in the sense that it does not depend on the details of the flow equation and the original field theory except for the renormalized mass, which is the only relevant quantity in this limit. We have found that the induced metric describes Euclidean Anti-de-Sitter (AdS) space in both ultra-violet (UV) and infra-red (IR) limits of the flow direction, where the radius of the AdS is bigger in the IR than in the UV.