Liouville heat kernel: Regularity and bounds

Maillard, Pascal; Rhodes, Rémi; Vargas, Vincent; Zeitouni, Ofer

doi:10.1214/15-aihp676

Cited by 26 publications

(54 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will not directly use the Liouville heat kernel, so we do not say anything further about it here and instead refer the interested reader to [GRV14,MRVZ16,AK16,DZZ18a] for additional background.…”

Section: Overviewmentioning

confidence: 99%

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

Ding

Gwynne

2019

Commun. Math. Phys.

105

View full text Add to dashboard Cite

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.

show abstract

Section: Overviewmentioning

confidence: 99%

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

Ding

Gwynne

2019

Commun. Math. Phys.

105

View full text Add to dashboard Cite

show abstract

“…In [13,11], some non-universality aspects (when considering underlying logcorrelated fields other than GFF) for LQG distances were demonstrated. In [26,12], a type of equivalence between the Liouville graph distance and the heat kernel for Liouville Brownian motion was proved. In [10], it was shown that there is a single number which determines the distance exponents for a few reasonable choices of distances associated with LQG, as well as the distance exponents for random planar maps.…”

Section: Background and Motivationmentioning

confidence: 99%

Subsequential Scaling Limits for Liouville Graph Distance

Ding

Dunlap

2020

Commun. Math. Phys.

View full text Add to dashboard Cite

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.

show abstract

“…Actually, one can even show that p γ t (x, y) is a continuous function of (t, x, y) (see [23,Section 3]). In what follows, we will also consider the standard heat kernel p t (x, y) of the planar Brownian motion B on R 2 .…”

Section: Liouville Measure and Lbmmentioning

confidence: 99%

“…This elementary Euclidean argument can be formulated in the LQG context. Consider the Liouville heat kernel p γ t (x, y) of the Brownian motion associated to the metric tensor (1.1), which has been constructed in [15] (see also [23] for further properties). A moment of thought shows that finding the quantum exponent Δ of the set A consists of finding the number Δ such that…”

Section: Introductionmentioning

confidence: 99%

“…Of course, such a discussion is heuristic, as even the distance d γ (x, y) remains out of reach so far, and it is not clear that the heat kernel has the shape (1.4) (see [23] as well as [2] for a thorough discussion as well as theorems in this direction), except of course in the Euclidean case when γ = 0. However, we point out that the analogy with the Euclidean case γ = 0 has limitations and hides some more complex issues.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

KPZ formula derived from Liouville heat kernel

Berestycki

Garban²,

Rhodes³

et al. 2016

J. London Math. Soc.

View full text Add to dashboard Cite

In this paper, we establish the Knizhnik-Polyakov-Zamolodchikov (KPZ) formula of Liouville quantum gravity, using the heat kernel of Liouville Brownian motion. This derivation of the KPZ formula was first suggested by F. David and M. Bauer in order to get a geometrically more intrinsic way of measuring the dimension of sets in Liouville quantum gravity. We also provide a careful study of the (no)-doubling behaviour of the Liouville measures in the appendix, which is of independent interest.

show abstract

Liouville heat kernel: Regularity and bounds

Abstract: We initiate in this paper the study of analytic properties of the Liouville heat kernel. In particular, we establish regularity estimates on the heat kernel and derive non trivial lower and upper bounds.

Cited by 26 publications

References 38 publications

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

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

Subsequential Scaling Limits for Liouville Graph Distance

KPZ formula derived from Liouville heat kernel

Contact Info

Product

Resources

About