Bounds for distances and geodesic dimension in Liouville first passage percolation

Gwynne, Ewain; Pfeffer, Joshua

doi:10.1214/19-ecp248

Cited by 27 publications

(28 citation statements)

References 18 publications

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“…Since this paper was posted to the arXiv, new bounds for dγ have been obtained in[GP19a] which improve on our bounds in some regimes. As in the case of our bounds, the new bounds in[GP19a] are based on Theorem 1.5, the fact that d √ 8/3 = 4, and a certain monotonicity statement for LFPP.…”

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confidence: 64%

The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds

Ding

Gwynne

2019

Commun. Math. Phys.

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105

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We prove that for each γ ∈ (0, 2), there is an exponent d γ > 2, the "fractal dimension of γ-Liouville quantum gravity (LQG)", which describes the ball volume growth exponent for certain random planar maps in the γ-LQG universality class, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of γ-LQG distances such as Liouville graph distance and Liouville first passage percolation. We also show that d γ is a continuous, strictly increasing function of γ and prove upper and lower bounds for d γ which in some cases greatly improve on previously known bounds for the aforementioned exponents. For example, for γ = √ 2 (which corresponds to spanning-tree weighted planar maps) our bounds give 3.4641 ≤ d √ 2 ≤ 3.63299 and in the limiting case we get 4.77485 ≤ lim γ→2 − d γ ≤ 4.89898.

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confidence: 64%

The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds

Ding

Gwynne

2019

Commun. Math. Phys.

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105

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“…For ξ ∈ (0.267, 0.707), stronger lower bounds were proved in [DG18,GP19], and for ξ ∈ (0, ξ 0 ), [DG16] gives λ(ξ) ≥ Ω(ξ 4/3 / log(1/ξ)).…”

Section: For Any Setmentioning

confidence: 99%

“…As before, Theorem 1.9 is the best known lower bound for the regime γ ∈ (γ 0 , 0.576), where γ 0 is small and non-explicit. The best known lower bounds for γ ∈ (0.577, 2) are proved in [DG18,GP19], and the best bound for γ ∈ (0, γ 0 ) is shown in [DG16].…”

Section: For Any Setmentioning

confidence: 99%

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Comparison of discrete and continuum Liouville first passage percolation

Ang¹

2019

Electron. Commun. Probab.

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Discrete and continuum Liouville first passage percolation (DLFPP, LFPP) are two approximations of the conjectural γ-Liouville quantum gravity (LQG) metric, obtained by exponentiating the discrete Gaussian free field (GFF) and the circle average regularization of the continuum GFF respectively. We show that these two models can be coupled so that with high probability distances in these models agree up to o(1) errors in the exponent, and thus have the same distance exponent.Ding and Gwynne (2018) give a formula for the continuum LFPP distance exponent in terms of the γ-LQG dimension exponent d γ . Using results of Ding and Li (2018) on the level set percolation of the discrete GFF, we bound the DLFPP distance exponent and hence obtain a new lower bound d γ ≥ 2 + γ 2 2 . This improves on previous lower bounds for d γ for the regime γ ∈ (γ 0 , 0.576), for some small nonexplicit γ 0 > 0.

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“…For other values of γ , we do not know the value of d γ hence we do not know these dimensions explicitly. However, we get upper and lower bounds for ess sup dim 0 H ∂B s and ess sup dim γ H ∂B s by plugging in the known bounds for d γ from [DG18,GP19a,Ang19]; see Fig. 1.…”

Section: Recall That Formentioning

confidence: 99%

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

Gwynne

2020

Commun. Math. Phys.

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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 -LQG metric ball with respect to the Euclidean (resp. $$\gamma $$ γ -LQG) metric is $$2 - \frac{\gamma }{d_\gamma }\left( \frac{2}{\gamma } + \frac{\gamma }{2} \right) + \frac{\gamma ^2}{2d_\gamma ^2}$$ 2 - γ d γ 2 γ + γ 2 + γ 2 2 d γ 2 (resp. $$d_\gamma -1$$ d γ - 1 ), where $$d_\gamma $$ d γ is the Hausdorff dimension of the whole plane with respect to the $$\gamma $$ γ -LQG metric. For $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 , in which case $$d_{\sqrt{8/3}}=4$$ d 8 / 3 = 4 , we get that the essential supremum of Euclidean (resp. $$\sqrt{8/3}$$ 8 / 3 -LQG) dimension of a $$\sqrt{8/3}$$ 8 / 3 -LQG ball boundary is 5/4 (resp. 3). We also compute the essential suprema of the Euclidean and $$\gamma $$ γ -LQG Hausdorff dimensions of the intersection of a $$\gamma $$ γ -LQG ball boundary with the set of metric $$\alpha $$ α -thick points of the field h for each $$\alpha \in \mathbb R$$ α ∈ R . Our results show that the set of $$\gamma /d_\gamma $$ γ / d γ -thick points on the ball boundary has full Euclidean dimension and the set of $$\gamma $$ γ -thick points on the ball boundary has full $$\gamma $$ γ -LQG dimension.

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Bounds for distances and geodesic dimension in Liouville first passage percolation

Cited by 27 publications

References 18 publications

The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds

The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds

Comparison of discrete and continuum Liouville first passage percolation

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

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