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

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Abstract

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.

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Notes

  1. 1.

    See [DRV16, Rem18, GRV16] for works concerning LQG with non-simply connected topologies.

  2. 2.

    In the physics literature it is common to use \(b = \frac{\gamma }{2}\) instead of \(\gamma \).

  3. 3.

    Here we mean distance function in the sense of a metric which gives the distance between any pair of points. We use the phrase “distance function” rather than “metric” to avoid confusion with the Riemannian metric tensor \(e^{\gamma h}(dx^2+dy^2)\).

  4. 4.

    The original KPZ formula in [KPZ88] described what the primary fields of the matter field CFT become when they are coupled to quantum gravity. This question seems to be mathematically out of reach so far but mathematicians have proved a weaker formulation [DS11, RV11] which relates the fractal dimension of a set sampled independently from the GFF as measured with the Euclidean metric to the fractal dimension of the same set as measured by the random distance function corresponding to (1.2). This weaker formulation also goes under the name KPZ formula and is sometimes called the geometric KPZ formula. In this paper we use the terms KPZ formula and geometric KPZ formula interchangeably and we are only concerned with the formula that relates the two notions of dimension for random fractals.

  5. 5.

    One reason why it is natural for this ball volume exponent to coincide with the dimension of LQG is that the Minkowski dimension d of a metric space can be defined by the condition that the number of metric balls of radius \(\delta >0\) needed to cover a metric ball of radius 1 is of order \(\delta ^{-d}\). For M, the number of graph distance balls of radius 1 (i.e., singleton sets of vertices) needed to cover the ball of radius r is its cardinality.

  6. 6.

    To be more precise, [DZZ18, Section 5] shows that (1.19) holds for the variant of \(D_h^\epsilon \) where we replace the circle average by the truncated white-noise decomposition of the field (here we note that \(\chi = 2/d_{{{\mathbf {c}}_{\mathrm M}}}\) in the notation of [DZZ18], see [DG18]; and that the parameter \(\epsilon \) in [DZZ18] corresponds to \(\epsilon ^{2/\gamma }\) in our setting). One can compare this variant of \(D_h^\epsilon \) to \(D_h^\epsilon \) itself using Lemma 4.2 below.

  7. 7.

    Due to lattice effects of the dyadic tiling we do not expect the approximation (2.3) to converge exactly to the LQG measure of X as defined earlier in this subsection for all fixed choices of X. For example, the line segment \([0,1] \times \{0\}\) typically intersects approximately twice as many squares as the line segment \([0,1]\times \{r\}\) for \(r\in \mathbb {R}\setminus \mathbb {Q}\) close to 1, but these two line segments should have approximately the same LQG measure. However, we believe that certain variants of (2.3) do converge to the LQG measure of X, e.g. we can consider versions of the square subdivision where the set of possible boxes are translated by \(z\in [0,1]^2\) and average over z.

  8. 8.

    Here at the classical level the exponential term \(e^{\gamma \varphi }\) transforms correctly under the coordinate change provided that \(Q = \frac{2}{\gamma }\), but in the quantum case when \(\varphi \) is a GFF type distribution the correct value of Q is \( \frac{\gamma }{2} + \frac{2}{\gamma }\). See the coordinate change formula (2.2).

  9. 9.

    This object may only make sense as a distribution, not a (complex) measure; see [JSW18a] for the case of purely imaginary \(\gamma \).

  10. 10.

    For technical reasons, [DDDF19, GM19b] define LFPP using the convolution of h with the heat kernel rather than the circle average approximation but the scaling limit of both versions of LFPP should be the same.

  11. 11.

    Here we are using the fact that, if \(f: \mathbb {R}^+ \rightarrow \mathbb {R}\) is a continuous function that is increasing on (0, a) and decreasing on \((a,\infty )\), then the sum \(\sum _{n=1}^{\infty } f(n)\) is a lower Riemann sum for the integral of the function \(f^*\) that equals f on (0, a), the constant f(a) on \((a,a+1)\), and \(f(x-1)\) on \((a+1,\infty )\). Hence, \(\sum _{n=1}^{\infty } f(n) \le \int _0^{\infty } f^*(x) dx = \int _0^{\infty } f(x) dx + f(a)\).

  12. 12.

    The truncated white noise decomposition is called \(\eta \) in [DZZ18] and has a slightly more complicated definition than \(\widehat{h}^{\mathrm {tr}}\), but the same (in fact, a slightly easier) argument works in the case of \(\widehat{h}^{\mathrm {tr}}\).

  13. 13.

    The reason for considering \([1,2n-1]\times [1,n-1]\) instead of \(\mathcal R_n \) is so that if \(S\in \mathcal S(\mathcal R_n)\), then each of the four \(1\times 2\) or \(2\times 1\) rectangles with corners in \(\mathbb {Z}^2\) which contain S are contained in \(\mathcal R_n \).

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Acknowledgements

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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Gwynne, E., Holden, N., Pfeffer, J. et al. Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach. Commun. Math. Phys. (2020). https://doi.org/10.1007/s00220-019-03663-6

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