Liouville quantum gravity and the Brownian map I: the $$\mathrm{QLE}(8/3,0)$$ metric

Miller, Jason; Sheffield⋆, Scott

doi:10.1007/s00222-019-00905-1

Cited by 110 publications

(169 citation statements)

References 38 publications

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“…The limiting object here can be understood as a random generalized function which is formally a Gaussian process whose correlation kernel is

- \frac{1}{2} log false| z - w false|

for

z, w

in the unit disk. Such random generalized functions have recently been discovered to be closely related to conformally invariant Schramm‐Loewner evolution‐type random curves as well as the scaling limits of random planar maps — see, for example, .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the moments of the characteristic polynomial of a Ginibre random matrix

Webb

Wong

2018

Proc. London Math. Soc.

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In this article, we study the large N asymptotics of complex moments of the absolute value of the characteristic polynomial of an N×N complex Ginibre random matrix with the characteristic polynomial evaluated at a point in the unit disk. More precisely, we calculate the large N asymptotics of double-struckEfalse|trueprefixdetfalse(GN−zfalse)|γ, where GN is an N×N matrix whose entries are i.i.d. and distributed as N−1/2Z, Z being a standard complex Gaussian, Re (γ)>−2, and |z|<1. This expectation is proportional to the determinant of a complex moment matrix with a symbol which is supported in the whole complex plane and has a Fisher–Hartwig type of singularity: 0truedet(∫Cwiw¯jfalse|w−zfalse|γe−Nfalse|wfalse|2d2wfalse)i,j=0N−1. We study the asymptotics of this determinant using recent results due to Lee and Yang concerning the asymptotics of orthogonal polynomials with respect to the weight |w−z|γe−N|w|2d2w along with differential identities familiar from the study of asymptotics of Toeplitz and Hankel determinants with Fisher–Hartwig singularities. To our knowledge, even in the case of one singularity, the asymptotics of the determinant of such a moment matrix whose symbol has support in a two‐dimensional set and a Fisher–Hartwig singularity have been previously unknown.

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“…The limiting object here can be understood as a random generalized function which is formally a Gaussian process whose correlation kernel is

- \frac{1}{2} log false| z - w false|

for

z, w

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the moments of the characteristic polynomial of a Ginibre random matrix

Webb

Wong

2018

Proc. London Math. Soc.

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“…Recently, a huge amount of effort has been devoted to understanding the random metric associated with LQG. Building on [30,16], in [31,28,29] the authors constructed in the continuum a random metric which is presumably the scaling limit of the LQG distance for the specific choice γ = 8/3, and proved deep connections with the Brownian map [22,23,27]. Note that in these works there is no mention of a discrete approximation of the metric.…”

Section: Background and Motivationmentioning

confidence: 99%

Subsequential Scaling Limits for Liouville Graph Distance

Ding

Dunlap

2020

Commun. Math. Phys.

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For 0 < γ < 2 and δ > 0, we consider the Liouville graph distance, which is the minimal number of Euclidean balls of γ-Liouville quantum gravity measure at most δ whose union contains a continuous path between two endpoints. In this paper, we show that the renormalized distance is tight and thus has subsequential scaling limits at δ → 0. In particular, we show that for all δ > 0 the diameter with respect to the Liouville graph distance has the same order as the typical distance between two endpoints.

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“…So far, the random planar map approach has met with much success in the special case when c M = 0 (i.e., γ = 8/3), in which case we are dealing with uniform random planar maps. Indeed, it is known that uniform random planar maps converge to 8/3-LQG surfaces both in the Gromov-Hausdorff sense [MS15,MS16a,MS16b,Le 13,Mie13] and under certain embeddings into the plane [HS19]. There are also some weaker convergence results for general c M ∈ (−∞, 1) based on the paper [DMS14]; see [GHS19] for a recent survey.…”

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.

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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.

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Liouville quantum gravity and the Brownian map I: the $$\mathrm{QLE}(8/3,0)$$ metric

Cited by 110 publications

References 38 publications

On the moments of the characteristic polynomial of a Ginibre random matrix

On the moments of the characteristic polynomial of a Ginibre random matrix

Subsequential Scaling Limits for Liouville Graph Distance

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

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