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.
Highlights
Xy indicates a sum over edges, and axy is the length of the edge xy, while V is a potential for the bulk scalar field φx
We formulate a Euclidean theory of edge length dynamics based on a notion of Ricci curvature on graphs with variable edge lengths
A key idea that will lead us to a more interesting class of edge length actions is a notion of Ricci curvature on graphs with variable edge lengths
Summary
We briefly review two well-known mathematical concepts. In subsection 2.1 we explain the Bruhat-Tits tree, a regular tree whose boundary is the p-adic numbers. We describe this privileged path as the “trunk” of the tree We consider another path ( with no back-tracking) starting from the point ∞ and leading to some other boundary point that we are going to associate with the p-adic number x. This new path must run along the trunk for a while, and the location where it diverges from the trunk can be labeled by the valuation v of x (as it appears in (2.2)). We have to choose among p possible directions, and each such choice can be labeled by an element ci ∈ Fp. In short, we see that the data required to select the new path is in precise correspondence with the information required to specify a non-zero p-adic number. It is possible to consider more general extensions of Qp than the unramified extension Qq, but we leave an explicit account along such lines for future work
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
