Edge length dynamics on graphs with applications to $p$-adic AdS/CFT

TL;DR

This paper develops a discrete Ricci curvature-based theory of edge length dynamics on graphs, connecting it to p-adic AdS/CFT and computing correlators of dual operators.

Contribution

It introduces a Euclidean edge length dynamics framework on graphs with a discrete Einstein-Hilbert action, linking graph curvature to p-adic AdS/CFT.

Findings

Infinite regular tree models constant negative curvature.

Explicit correlators of edge fluctuation dual operators computed.

Edge fluctuation operator shares features with stress tensor.

Abstract

We formulate a Euclidean theory of edge length dynamics based on a notion of Ricci curvature on graphs with variable edge lengths. In order to write an explicit form for the discrete analog of the Einstein-Hilbert action, we require that the graph should either be a tree or that all its cycles should be sufficiently long. The infinite regular tree with all edge lengths equal is an example of a graph with constant negative curvature, providing a connection with $p$-adic AdS/CFT, where such a tree takes the place of anti-de Sitter space. We compute simple correlators of the operator holographically dual to edge length fluctuations. This operator has dimension equal to the dimension of the boundary, and it has some features in common with the stress tensor.