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

## 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. Here, 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.

- Authors:

- Princeton Univ., NJ (United States). Joseph Henry Lab.
- California Inst. of Technology (CalTech), Pasadena, CA (United States). Walter Burke Inst. for Theoretical Physics
- California Inst. of Technology (CalTech), Pasadena, CA (United States). Dept. of Mathematics
- Ruprecht-Karls Univ. Heidelberg, Heidelberg (Germany). Mathematisches Inst.
- Brandeis Univ., Waltham, MA (United States). Martin A. Fisher School of Physics

- Publication Date:

- Research Org.:
- Princeton Univ., NJ (United States); California Inst. of Technology (CalTech), Pasadena, CA (United States)

- Sponsoring Org.:
- USDOE; National Science Foundation (NSF)

- OSTI Identifier:
- 1425489

- Grant/Contract Number:
- FG02-91ER40671; SC0011632; DMS-1201512; PHY-1205440; SC0009987

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 6; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Lattice Models of Gravity; AdS-CFT Correspondence; Classical Theories of Gravity

### Citation Formats

```
Gubser, Steven S., Heydeman, Matthew, Jepsen, Christian, Marcolli, Matilde, Parikh, Sarthak, Saberi, Ingmar, Stoica, Bogdan, and Trundy, Brian.
```*Edge length dynamics on graphs with applications to p-adic AdS/CFT*. United States: N. p., 2017.
Web. doi:10.1007/JHEP06(2017)157.

```
Gubser, Steven S., Heydeman, Matthew, Jepsen, Christian, Marcolli, Matilde, Parikh, Sarthak, Saberi, Ingmar, Stoica, Bogdan, & Trundy, Brian.
```*Edge length dynamics on graphs with applications to p-adic AdS/CFT*. United States. doi:10.1007/JHEP06(2017)157.

```
Gubser, Steven S., Heydeman, Matthew, Jepsen, Christian, Marcolli, Matilde, Parikh, Sarthak, Saberi, Ingmar, Stoica, Bogdan, and Trundy, Brian. Fri .
"Edge length dynamics on graphs with applications to p-adic AdS/CFT". United States.
doi:10.1007/JHEP06(2017)157. https://www.osti.gov/servlets/purl/1425489.
```

```
@article{osti_1425489,
```

title = {Edge length dynamics on graphs with applications to p-adic AdS/CFT},

author = {Gubser, Steven S. and Heydeman, Matthew and Jepsen, Christian and Marcolli, Matilde and Parikh, Sarthak and Saberi, Ingmar and Stoica, Bogdan and Trundy, Brian},

abstractNote = {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. Here, 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.},

doi = {10.1007/JHEP06(2017)157},

journal = {Journal of High Energy Physics (Online)},

number = 6,

volume = 2017,

place = {United States},

year = {Fri Jun 30 00:00:00 EDT 2017},

month = {Fri Jun 30 00:00:00 EDT 2017}

}

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