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

TL;DR

The paper develops a discrete gravity framework for edge-length dynamics on graphs by defining a graph Ricci curvature and a discrete Einstein–Hilbert–like action with a boundary term, applicable when the bulk graph is a tree or has long cycles. In the p-adic AdS/CFT setting provided by the Bruhat–Tits tree, linearized fluctuations of edge lengths obey a massless equation controlled by an edge Laplacian, and holographic correlators of the dual operator T are computed, revealing a boundary-dimension match and scalar-like two- and three-point structures. The two-point function scales as $\langle T(z_1) T(z_2) \rangle \propto |z_{12}|^{-2n}$ with $n$ the boundary dimension, while the mixed three-point function $\langle T O O \rangle$ is explicitly determined and the purely geometric three-point function $\langle T T T \rangle$ vanishes for separated points under specific curvature choices. The work also demonstrates exact non-constant edge-length solutions preserving negative curvature and discusses implications for non-archimedean holography, potential entanglement connections, and extensions to broader graph geometries.

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.

Figures (6)

• Figure 1: A regular graph in black, and its line graph in green.
• Figure 2: Left: Small spherical neighborhoods of nearby points in a smooth manifold provide a starting point for defining Ricci curvature without first defining the Riemann tensor. Right: A similar construction on graphs hinges on replacing the small spherical neighborhood around a point $x_0$ with a probability distribution $\psi_{x_0}(t)$ which for small $t$ is concentrated at $x_0$ with a little bit of weight on neighboring vertices.
• Figure 3: Part of a graph which may qualify as "almost a tree." The important criterion is that the alternate route from $x_1$ to $y_1$, passing through the top four edges, must be longer than the path from $x_1$ to $y_1$ through the edge $xy$.
• Figure 4: Left: A fat subgraph $\Sigma$ of a regular tree. The dashed line passes through the points on the boundary $\partial\Sigma$ of $\Sigma$. Any point $x$ on the boundary has a unique neighbor $x'$ in the interior of $\Sigma$. Right: A subgraph of the same regular tree which is not fat.
• Figure 5: The "local solution circle" for edge $xy$. The physical solution subspace lies inside the interval $\theta_{xy} \in (-\pi/4, 3\pi/4)$ (the solid blue semi-circle).