Annals of Probability

Confluence of geodesics in Liouville quantum gravity for $\gamma \in (0,2)$

Ewain Gwynne and Jason Miller

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Abstract

We prove that for any metric, which one can associate with a Liouville quantum gravity (LQG) surface for $\gamma \in (0,2)$ satisfying certain natural axioms, its geodesics exhibit the following confluence property. For any fixed point $z$, a.s. any two $\gamma $-LQG geodesics started from distinct points other than $z$ must merge into each other and subsequently coincide until they reach $z$. This is analogous to the confluence of geodesics property for the Brownian map proven by Le Gall (Acta Math. 205 (2010) 287–360). Our results apply for the subsequential limits of Liouville first passage percolation and are an important input in the proof of the existence and uniqueness of the LQG metric for all $\gamma \in (0,2)$.

Article information

Source
Ann. Probab., Volume 48, Number 4 (2020), 1861-1901.

Dates
Received: July 2019
Revised: October 2019
First available in Project Euclid: 20 July 2020

Permanent link to this document
https://projecteuclid.org/euclid.aop/1595232097

Digital Object Identifier
doi:10.1214/19-AOP1409

Mathematical Reviews number (MathSciNet)
MR4124527

Zentralblatt MATH identifier
07224962

Subjects
Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 60G52: Stable processes

Keywords
Liouville quantum gravity Gaussian free field LQG metric Liouville first passage percolation confluence of geodesics

Citation

Gwynne, Ewain; Miller, Jason. Confluence of geodesics in Liouville quantum gravity for $\gamma \in (0,2)$. Ann. Probab. 48 (2020), no. 4, 1861--1901. doi:10.1214/19-AOP1409. https://projecteuclid.org/euclid.aop/1595232097


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References

  • [1] Bettinelli, J. and Miermont, G. (2017). Compact Brownian surfaces I: Brownian disks. Probab. Theory Related Fields 167 555–614.
  • [2] Burago, D., Burago, Y. and Ivanov, S. (2001). A Course in Metric Geometry. Graduate Studies in Mathematics 33. Amer. Math. Soc., Providence, RI.
  • [3] Ding, J., Dubédat, J., Dunlap, A. and Falconet, H. (2019). Tightness of Liouville first passage percolation for $\gamma \in (0,2)$. ArXiv e-prints.
  • [4] Ding, J. and Gwynne, E. (2018). The fractal dimension of Liouville quantum gravity: Universality, monotonicity, and bounds. Comm. Math. Phys. To appear.
  • [5] Dubédat, J., Falconet, H., Gwynne, E., Pfeffer, J. and Sun, X. (2019). Weak LQG metrics and Liouville first passage percolation. ArXiv e-prints.
  • [6] Duplantier, B. and Sheffield, S. (2011). Liouville quantum gravity and KPZ. Invent. Math. 185 333–393.
  • [7] Gwynne, E. and Miller, J. (2019). Local metrics of the Gaussian free field. ArXiv e-prints.
  • [8] Gwynne, E. and Miller, J. (2019). Existence and uniqueness of the Liouville quantum gravity metric for $\gamma \in (0,2)$. ArXiv e-prints.
  • [9] Gwynne, E., Miller, J. and Sheffield, S. (2018). The Tutte embedding of the Poisson–Voronoi tessellation of the Brownian disk converges to $\sqrt{8/3}$-Liouville quantum gravity. ArXiv e-prints.
  • [10] Gwynne, E. and Pfeffer, J. (2019). KPZ formulas for the Liouville quantum gravity metric. ArXiv e-prints.
  • [11] Kahane, J.-P. (1985). Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9 105–150.
  • [12] Le Gall, J.-F. (2010). Geodesics in large planar maps and in the Brownian map. Acta Math. 205 287–360.
  • [13] Le Gall, J.-F. (2013). Uniqueness and universality of the Brownian map. Ann. Probab. 41 2880–2960.
  • [14] Miermont, G. (2013). The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 319–401.
  • [15] Miller, J. and Qian, W. (2018). The geodesics in Liouville quantum gravity are not Schramm–Loewner evolutions. ArXiv e-prints.
  • [16] Miller, J. and Sheffield, S. (2015). Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric. Invent. Math. To appear.
  • [17] Miller, J. and Sheffield, S. (2015). An axiomatic characterization of the Brownian map. ArXiv e-prints.
  • [18] Miller, J. and Sheffield, S. (2016). Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding. ArXiv e-prints.
  • [19] Miller, J. and Sheffield, S. (2016). Liouville quantum gravity and the Brownian map III: The conformal structure is determined. ArXiv e-prints.
  • [20] Miller, J. and Sheffield, S. (2016). Imaginary geometry I: Interacting SLEs. Probab. Theory Related Fields 164 553–705.
  • [21] Miller, J. and Sheffield, S. (2016). Quantum Loewner evolution. Duke Math. J. 165 3241–3378.
  • [22] Miller, J. and Sheffield, S. (2017). Imaginary geometry IV: Interior rays, whole-plane reversibility, and space-filling trees. Probab. Theory Related Fields 169 729–869.
  • [23] Pitt, L. D. (1982). Positively correlated normal variables are associated. Ann. Probab. 10 496–499.
  • [24] Polyakov, A. M. (1981). Quantum geometry of bosonic strings. Phys. Lett. B 103 207–210.
  • [25] Rhodes, R. and Vargas, V. (2014). Gaussian multiplicative chaos and applications: A review. Probab. Surv. 11 315–392.
  • [26] Schramm, O. and Sheffield, S. (2013). A contour line of the continuum Gaussian free field. Probab. Theory Related Fields 157 47–80.
  • [27] Sheffield, S. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 521–541.