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2016

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

Gubser, Steven S., Heydeman, Matthew, Jepsen, Christian et al.

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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.

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

Built on

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L. Brekke and P. G. O. Freund, “ p p -adic numbers in physics,” Phys. Rept
1993
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S. S. Gubser, J. Knaute, S. Parikh, A. Samberg, and P. Witaszczyk, “ p p -adic AdS/CFT,” 1605.01061
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M. Heydeman, M. Marcolli, I. Saberi, and B. Stoica, “Tensor networks, p p -adic fields, and algebraic curves: arithmetic and the AdS 3 /CFT 2 correspondence,” 1605.07639
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F. Bauer, F. Chung, Y. Lin, and Y. Liu, “Curvature aspects of graphs.” http://www.math.ucsd.edu/~fan/wp/CLLcurvature3.pdf
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W. Donnelly, B. Michel, D. Marolf, and J. Wien, “Living on the Edge: A Toy Model for Holographic Reconstruction of Algebras with Centers,” 1611.05841
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B. Casselman, “The bruhat–tits tree of sl ⁡ ( 2 ) {\rm sl}(2) .” https://www.math.ubc.ca/~cass/research/pdf/Tree.pdf
Cited in the paper.

Then

Y. Lin, L. Lu, and S.-T. Yau, “Ricci curvature of graphs,” Tohoku Math. J. (2)
2011
Later among the works it cites.
D. Harlow, S. H. Shenker, D. Stanford, and L. Susskind, “Tree-like structure of eternal inflation: A solvable model,” Phys. Rev
Original
2012
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F. Chung, Y. Lin, and S.-T. Yau, “Harnack inequalities for graphs with non-negative Ricci curvature,” Journal of Mathematical Analysis and Applications
2014
Later among the works it cites.

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