The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

Gwynne, Ewain

doi:10.1007/s00220-020-03783-4

“…Another natural random fractal associated with the LQG metric is the boundary of a (non-filled) LQG metric ball (note that this boundary is typically not connected). It is shown in [44,48] that a.s. the Hausdorff dimension of the LQG metric ball boundary w.r.t. the Euclidean (resp.…”

Section: Dimension Calculationsmentioning

confidence: 99%

Existence and uniqueness of the Liouville quantum gravity metric for $$\gamma \in (0,2)$$

Gwynne

¹

,

Miller

²

2020

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We show that for each $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , there is a unique metric (i.e., distance function) associated with $$\gamma $$ γ -Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) h, there is a unique random metric $$D_h$$ D h associated with the Riemannian metric tensor “$$e^{\gamma h} (dx^2 + dy^2)$$ e γ h ( d x 2 + d y 2 ) ” on $${\mathbb {C}}$$ C which is characterized by a certain list of axioms: it is locally determined by h and it transforms appropriately when either adding a continuous function to h or applying a conformal automorphism of $$\mathbb {C}$$ C (i.e., a complex affine transformation). Metrics associated with other variants of the GFF can be constructed using local absolute continuity. The $$\gamma $$ γ -LQG metric can be constructed explicitly as the scaling limit of Liouville first passage percolation (LFPP), the random metric obtained by exponentiating a mollified version of the GFF. Earlier work by Ding et al. (Tightness of Liouville first passage percolation for $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , 2019. arXiv:1904.08021) showed that LFPP admits non-trivial subsequential limits. This paper shows that the subsequential limit is unique and satisfies our list of axioms. In the case when $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 , our metric coincides with the $$\sqrt{8/3}$$ 8 / 3 -LQG metric constructed in previous work by Miller and Sheffield, which in turn is equivalent to the Brownian map for a certain variant of the GFF. For general $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , we conjecture that our metric is the Gromov–Hausdorff limit of appropriate weighted random planar map models, equipped with their graph distance. We include a substantial list of open problems.

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“…The Hausdorff dimension d c M of the resulting metric space is unknown except in the case c M = 0, where it is equal to 4. There are several exponents defined in terms of various approximations of LQG or in terms of the continuum LQG distance function itself which can be expressed in terms of d c M [GHS17, DZZ18, DG18, DFG + 19, GP19b,Gwy19]. See [DG18,GHS17,Ang19] for upper and lower bounds for d c M .…”

Section: Resultsmentioning

confidence: 99%

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

Gwynne

¹

,

Holden

²

,

Pfeffer

³

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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“…The problem of quantifying the source of the singularity in Theorem 3, analogous to the result in [17] stating that a typical point sampled from the LQG measure is a γ−thick point, will be taken up in future research. In [20,28], related themes of the Hausdorff dimension of the infinite geodesic Γ, as well as the boundary of metric balls in LQG were explored.…”

Section: Theoremmentioning

confidence: 99%

“…We now come to the case when h 2 is any GFF plus a continuous function. Though a GFF plus a continuous function on D was defined to be a zero boundary GFF on D plus a random continuous function, we note that it also be written as h| D + f for a random continuous function f on D coupled with h; to see this, note that by the Markov property for h (see [20,Lemma A.1]), h| D can be decomposed as a sum of a zero boundary GFF and a random harmonic function. Since f is continuous, it is bounded on compact sets and we locally use f ∞ to denote the finite quantity sup z∈D ) for all x, y ∈ C (0,1) simultaneously.…”

Section: Proof Ofmentioning

confidence: 99%

Environment seen from infinite geodesics in Liouville Quantum Gravity

Basu,

Bhatia,

Ganguly

2021

Preprint

0

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First passage percolation (FPP) on Z d or R d is a canonical model of a random metric space where the standard Euclidean geometry is distorted by random noise. Of central interest is the length and the geometry of the geodesic, the shortest path between points. Since the latter, owing to its length minimization, traverses through atypically low values of the underlying noise variables, it is an important problem to quantify the disparity between the environment rooted at a point on the geodesic and the typical one. We investigate this in the context of γ-Liouville Quantum Gravity (LQG) (where γ ∈ (0, 2) is a parameter) -a random Riemannian surface induced on the complex plane by the random metric tensor e 2γh/dγ (dx 2 + dy 2 ), where h is the whole plane, properly centered, Gaussian Free Field (GFF), and dγ is the associated dimension. We consider the unique infinite geodesic Γ from the origin, parametrized by the logarithm of its chemical length, and show that, for an almost sure realization of h, the distributions of the appropriately scaled field and the induced metric on a ball, rooted at a point "uniformly" sampled on Γ, converge to deterministic measures on the space of generalized functions and continuous metrics on the unit disk respectively. Moreover, towards a better understanding of the limiting objects living on the unit disk, we show that they are singular with respect to their typical counterparts, but become absolutely continuous away from the origin. Our arguments rely on unearthing a regeneration structure with fast decay of correlation in the geodesic owing to coalescence and the domain Markov property of the GFF. While there have been significant recent advances around this question for stochastic planar growth models in the KPZ class, the present work initiates this research program in the context of LQG.

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The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

Abstract: Let $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , let h be the planar Gaussian free field, and consider the $$\gamma $$ γ -Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundary of a $$\gamma $$ γ… Show more

Cited by 15 publications

References 45 publications

Existence and uniqueness of the Liouville quantum gravity metric for $$\gamma \in (0,2)$$

Existence and uniqueness of the Liouville quantum gravity metric for $$\gamma \in (0,2)$$

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

Environment seen from infinite geodesics in Liouville Quantum Gravity

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