Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach

  • Ewain Gwynne
  • Nina HoldenEmail author
  • Joshua Pfeffer
  • Guillaume Remy


There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge \({{\mathbf {c}}}_{\mathrm M} \in (-\infty ,1]\). Via the DDK ansatz, LQG can equivalently be described as the random geometry obtained by exponentiating \(\gamma \) times a variant of the planar Gaussian free field, where \(\gamma \in (0,2]\) satisfies \({\mathbf {c}}_{\mathrm M} = 25 - 6(2/\gamma + \gamma /2)^2\). Physics considerations suggest that LQG should also make sense in the regime when \({\mathbf {c}}_{\mathrm M} > 1\). However, the behavior in this regime is rather mysterious in part because the corresponding value of \(\gamma \) is complex, so analytic continuations of various formulas give complex answers which are difficult to interpret in a probabilistic setting. We introduce and study a discretization of LQG which makes sense for all values of \({\mathbf {c}}_{\mathrm M} \in (-\infty ,25)\). Our discretization consists of a random planar map, defined as the adjacency graph of a tiling of the plane by dyadic squares which all have approximately the same “LQG size" with respect to the Gaussian free field. We prove that several formulas for dimension-related quantities are still valid for \(\mathbf{c}_{\mathrm M} \in (1,25)\), with the caveat that the dimension is infinite when the formulas give a complex answer. In particular, we prove an extension of the (geometric) KPZ formula for \(\mathbf{c}_{\mathrm M} \in (1,25)\), which gives a finite quantum dimension if and only if the Euclidean dimension is at most \((25-\mathbf{c}_{\mathrm M} )/12\). We also show that the graph distance between typical points with respect to our discrete model grows polynomially whereas the cardinality of a graph distance ball of radius r grows faster than any power of r (which suggests that the Hausdorff dimension of LQG in the case when \({\mathbf {c}}_{\mathrm M} \in (1,25)\) is infinite). We include a substantial list of open problems.



We are grateful to several individuals for helpful discussions, including Timothy Budd, Jian Ding, Bertrand Duplantier, Antti Kupiainen, Greg Lawler, Eveliina Peltola, Rémi Rhodes, Scott Sheffield, Xin Sun, and Vincent Vargas. We thank Scott Sheffield for suggesting the idea of using square subdivisions to approximate LQG for \({{\mathbf {c}}_{\mathrm M}}\in (1,25)\). We also thank the anonymous referee for numerous helpful suggestions and comments. E.G. was partially supported by a Herchel Smith fellowship and a Trinity College junior research fellowship. N.H. was supported by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation. G.R. was partially supported by a National Science Foundation mathematical sciences postdoctoral research fellowship. J.P. was partially supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1122374.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  1. 1.University of CambridgeCambridgeUK
  2. 2.ETH ZürichZürichSwitzerland
  3. 3.MITCambridgeUSA
  4. 4.Columbia UniversityNew York CityUSA

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