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


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.

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  1. [1]

    S.S. Gubser, J. Knaute, S. Parikh, A. Samberg and P. Witaszczyk, p-adic AdS/CFT, Commun. Math. Phys. 352 (2017) 1019 [arXiv:1605.01061] [INSPIRE].

  2. [2]

    M. Heydeman, M. Marcolli, I. Saberi and B. Stoica, Tensor networks, p-adic fields and algebraic curves: arithmetic and the AdS 3 /CFT 2 correspondence, arXiv:1605.07639 [INSPIRE].

  3. [3]

    Y.I. Manin and M. Marcolli, Holography principle and arithmetic of algebraic curves, Adv. Theor. Math. Phys. 5 (2002) 617 [hep-th/0201036] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    A.V. Zabrodin, Nonarchimedean Strings and Bruhat-Tits Trees, Commun. Math. Phys. 123 (1989) 463 [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  5. [5]

    D. Harlow, S.H. Shenker, D. Stanford and L. Susskind, Tree-like structure of eternal inflation: A solvable model, Phys. Rev. D 85 (2012) 063516 [arXiv:1110.0496] [INSPIRE].

    ADS  Google Scholar 

  6. [6]

    D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de probabilités de Strasbourg 19 (1985) 177.

    MathSciNet  MATH  Google Scholar 

  7. [7]

    Y. Ollivier, Ricci curvature of Markov chains on metric spaces, J. Funct. Anal. 256 (2009) 810.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    Y. Lin, L. Lu and S.-T. Yau, Ricci curvature of graphs, Tohoku Math. J. 63 (2011) 605.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    L. Brekke, P.G.O. Freund, M. Olson and E. Witten, Nonarchimedean String Dynamics, Nucl. Phys. B 302 (1988) 365 [INSPIRE].

    ADS  Article  Google Scholar 

  10. [10]

    M.-K. von Renesse and K.-T. Sturm, Entropic measure and Wasserstein diffusion, Ann. Probab. 37 (2009) 1114.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    F. Chung, Y. Lin and S.-T. Yau, Harnack inequalities for graphs with non-negative Ricci curvature, J. Math. Anal. Appl. 415 (2014) 25.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    F. Bauer, F. Chung, Y. Lin and Y. Liu, Curvature Aspects of Graphs, Proc. Amer. Math. Soc. 145 (2017) 2033.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    C. Cao, S.M. Carroll and S. Michalakis, Space from Hilbert Space: Recovering Geometry from Bulk Entanglement, Phys. Rev. D 95 (2017) 024031 [arXiv:1606.08444] [INSPIRE].

    ADS  Google Scholar 

  14. [14]

    W. Donnelly, B. Michel, D. Marolf and J. Wien, Living on the Edge: A Toy Model for Holographic Reconstruction of Algebras with Centers, JHEP 04 (2017) 093 [arXiv:1611.05841] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  15. [15]

    L. Brekke and P.G.O. Freund, p-adic numbers in physics, Phys. Rept. 233 (1993) 1 [INSPIRE].

  16. [16]

    B. Casselman, The Bruhat-Tits tree of SL(2), https://www.math.ubc.ca/~cass/research/pdf/Tree.pdf.

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Correspondence to Sarthak Parikh.

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ArXiv ePrint: 1612.09580

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Gubser, S.S., Heydeman, M., Jepsen, C. et al. Edge length dynamics on graphs with applications to p-adic AdS/CFT. J. High Energ. Phys. 2017, 157 (2017). https://doi.org/10.1007/JHEP06(2017)157

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  • Lattice Models of Gravity
  • AdS-CFT Correspondence
  • Classical Theories of Gravity