In order to study the quantum geometry of random surfaces in Liouville gravity, we propose a definition of geodesic distance associated to a Gaussian free field on a regular lattice. This geodesic distance is used to numerically determine the Hausdorff dimension associated to shortest cycles of 2d quantum gravity on the torus coupled to conformal matter fields, showing agreement with a conjectured formula by Y. Watabiki. Finally, the numerical tools are put to test by quantitatively comparing the distribution of lengths of shortest cycles to the corresponding distribution in large random triangulations.

Type

Publication

Nucl. Phys. B 889 (2014) 676-691

Date

May, 2014

Links

- Borot, Gaëtan, Jérémie Bouttier, and Bertrand Duplantier. “Nesting statistics in the O(n) loop model on random planar maps.” arXiv preprint arXiv:1605.02239 (2016).
- Sheffield, Scott. “Conformal weldings of random surfaces: SLE and the quantum gravity zipper.” The Annals of Probability 44.5 (2016): 3474-3545.
- Ding, Jian, and Fuxi Zhang. “Non-universality for first passage percolation on the exponential of log-correlated Gaussian fields.” Probability Theory and Related Fields (2015): 1-32.
- Ding, Jian, and Subhajit Goswami. “Upper bounds on Liouville first passage percolation and Watabiki’s prediction.” arXiv preprint arXiv:1610.09998 (2016).
- Ding, Jian, and Subhajit Goswami. “Liouville first passage percolation: the weight exponent is strictly less than 1 at high temperatures.” arXiv preprint arXiv:1605.08392 (2016).
- Gwynne, Ewain, Nina Holden, and Xin Sun. “A distance exponent for Liouville quantum gravity.” Probability Theory and Related Fields (2016): 1-67.
- Cooperman, Joshua H. “Scale-dependent homogeneity measures for causal dynamical triangulations.” Physical Review D 90.12 (2014): 124053.
- Ding, Jian, and Ewain Gwynne. “The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds.” arXiv preprint arXiv:1807.01072 (2018).
- Gwynne, Ewain, Nina Holden, and Xin Sun. “A mating-of-trees approach to graph distances in random planar maps.” arXiv preprint arXiv:1711.00723 (2017).
- Alevy, Ian M. “Regular Polygon Surfaces.” arXiv preprint arXiv:1804.05452 (2018).
- Goswami, Subhajit. “Some metric properties of planar Gaussian free field.” (2017).
- Gwynne, E., Holden, N., Pfeffer, J., & Remy, G. “Liouville quantum gravity with central charge in (1, 25) : a probabilistic approach”. arXiv preprint arXiv:1903.09111 (2019).
- J. Barkley and T. Budd, “Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity”, arXiv preprint arXiv:1908.09469 (2019).
- E. Gwynne, “The dimension of the boundary of a Liouville quantum gravity metric ball”, arXiv preprint arXiv::1909.08588 (2019).
- Pfeffer, Joshua William. “Frontiers of Liouville quantum gravity.” PhD diss., Massachusetts Institute of Technology, 2020.