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We study the proposal that a $d+1$ dimensional induced metric is constructedfrom a $d$ dimensional field theory using gradient flow. Applying the idea tothe O($N$) $\varphi^4$ model and normalizing the flow field, we have shown inthe large $N$ limit that the induced metric is finite and universal in thesense that it does not depend on the details of the flow equation and theoriginal field theory except for the renormalized mass, which is the onlyrelevant quantity in this limit. We have found that the induced metricdescribes Euclidean Anti-de-Sitter (AdS) space in both ultra-violet (UV) andinfra-red (IR) limits of the flow direction, where the radius of the AdS isbigger in the IR than in the UV.

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