TY - JOUR

T1 - Liouville Quantum Gravity with Matter Central Charge in (1, 25)

T2 - A Probabilistic Approach

AU - Gwynne, Ewain

AU - Holden, Nina

AU - Pfeffer, Joshua

AU - Remy, Guillaume

N1 - Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/6/1

Y1 - 2020/6/1

N2 - There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge cM∈ (- ∞, 1]. Via the DDK ansatz, LQG can equivalently be described as the random geometry obtained by exponentiating γ times a variant of the planar Gaussian free field, where γ∈ (0 , 2] satisfies cM= 25 - 6 (2 / γ+ γ/ 2) 2. Physics considerations suggest that LQG should also make sense in the regime when cM> 1. However, the behavior in this regime is rather mysterious in part because the corresponding value of γ 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 cM∈ (- ∞, 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 cM∈ (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 cM∈ (1 , 25) , which gives a finite quantum dimension if and only if the Euclidean dimension is at most (25 - cM) / 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 cM∈ (1 , 25) is infinite). We include a substantial list of open problems.

AB - There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge cM∈ (- ∞, 1]. Via the DDK ansatz, LQG can equivalently be described as the random geometry obtained by exponentiating γ times a variant of the planar Gaussian free field, where γ∈ (0 , 2] satisfies cM= 25 - 6 (2 / γ+ γ/ 2) 2. Physics considerations suggest that LQG should also make sense in the regime when cM> 1. However, the behavior in this regime is rather mysterious in part because the corresponding value of γ 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 cM∈ (- ∞, 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 cM∈ (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 cM∈ (1 , 25) , which gives a finite quantum dimension if and only if the Euclidean dimension is at most (25 - cM) / 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 cM∈ (1 , 25) is infinite). We include a substantial list of open problems.

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U2 - 10.1007/s00220-019-03663-6

DO - 10.1007/s00220-019-03663-6

M3 - Article

AN - SCOPUS:85078607191

SN - 0010-3616

VL - 376

SP - 1573

EP - 1625

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 2

ER -