Liouville quantum gravity on the unit disk

Huang, Yichao; Rhodes, Rémi; Vargas, Vincent

doi:10.1214/17-aihp852

Cited by 59 publications

(99 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The additional condition on p, p < (1 + 4 γ 2 (1 + a)) ∧ (1 + 4 γ 2 (1 + b)), comes from the presence of the insertions. A proof of the bounds (1.5) can be found in [29,14]. Now the goal of our paper is simply to prove the following exact formula for M (γ, p, a, b):…”

mentioning

confidence: 99%

The distribution of Gaussian multiplicative chaos on the unit interval

Rémy¹,

Zhu²

2020

Ann. Probab.

View full text Add to dashboard Cite

We consider a sub-critical Gaussian multiplicative chaos (GMC) measure defined on the unit interval [0, 1] and prove an exact formula for the fractional moments of the total mass of this measure. Our formula includes the case where log-singularities (also called insertion points) are added in 0 and 1, the most general case predicted by the Selberg integral. The idea to perform this computation is to introduce certain auxiliary functions resembling holomorphic observables of conformal field theory that will be solutions of hypergeometric equations. Solving these equations then provides non-trivial relations that completely determine the moments we wish to compute. We also include a detailed discussion of the so-called reflection coefficients appearing in tail expansions of GMC measures and in Liouville theory. Our theorem provides an exact value for one of these coefficients. Lastly we mention some additional applications to small deviations for GMC measures, to the behavior of the maximum of the log-correlated field on the interval and to random hermitian matrices. †, ‡ Research supported in part by the ANR grant Liouville (ANR-15-CE40-0013). MSC 2010 subject classifications: Primary 60G57; secondary 60G15, 60G60, 60G70, 82B23.1 1 Our normalization differs from the ln 1 |x−y| usually found in the literature.

show abstract

mentioning

confidence: 99%

The distribution of Gaussian multiplicative chaos on the unit interval

Rémy¹,

Zhu²

2020

Ann. Probab.

View full text Add to dashboard Cite

show abstract

“…Our paper also clarifies a few points that were not addressed in [4,13]. First, we justify for any background metric g the expression of the covariance of our field X g by diagonalizing an operator T defined using Neumann boundary conditions.…”

Section: Introductionmentioning

confidence: 67%

“…The probabilistic framework used throughout this paper was introduced in [4] where the authors provide a construction of LQG for the Riemann sphere. Following the same approach, the theory has been defined on the unit disk in [13], on the complex tori in [5] and on compact Riemann surfaces of higher genus in [12]. The Riemann sphere is the simplest case as it corresponds to a simply connected compact surface without boundary.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Liouville quantum gravity on the annulus

Rémy

2018

Journal of Mathematical Physics

View full text Add to dashboard Cite

In this work we construct Liouville quantum gravity on an annulus in the complex plane. This construction is aimed at providing a rigorous mathematical framework to the work of theoretical physicists initiated by Polyakov in 1981 [22]. It is also a very important example of a conformal field theory (CFT). Results have already been obtained on the Riemann sphere [4] and on the unit disk [13] so this paper will follow the same approach. The case of the annulus contains two difficulties: it is a surface with two boundaries and it has a non-trivial moduli space. We recover the Weyl anomaly -a formula verified by all CFT -and deduce from it the KPZ formula. We also show that the full partition function of Liouville quantum gravity integrated over the moduli space is finite. This allows us to give the joint law of the Liouville measures and of the random modulus and to write the conjectured link with random planar maps.

show abstract

“…We also record the formula c L = 1 + 6Q 2 , which is obtained by combining (1.1) and (1.3). It was shown rigorously in [DKRV16] that the partition function of LCFT transforms under conformal rescaling as the partition function of a CFT with central charge c L for c L ≥ 25, which corresponds to c M ∈ (−∞, 1], in the case of the sphere topology (see [HRV18,DRV16,GRV16] for other topologies). From the point of view of constructive quantum field theory, this means that LCFT is a CFT with central charge c L .…”

Section: Introduction 1overviewmentioning

confidence: 99%

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

Gwynne

Holden

Pfeffer

et al. 2020

Commun. Math. Phys.

View full text Add to dashboard Cite

There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge c M ∈ (−∞, 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 c M = 25 − 6(2/γ + γ/2) 2 . Physics considerations suggest that LQG should also make sense in the regime when c M > 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 c M ∈ (−∞, 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 c M ∈ (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 c M ∈ (1, 25), which gives a finite quantum dimension if and only if the Euclidean dimension is at most (25 − c 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 c M ∈ (1, 25) is infinite).We include a substantial list of open problems.

show abstract

Liouville quantum gravity on the unit disk

Cited by 59 publications

References 36 publications

The distribution of Gaussian multiplicative chaos on the unit interval

The distribution of Gaussian multiplicative chaos on the unit interval

Liouville quantum gravity on the annulus

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

Contact Info

Product

Resources

About