## The Annals of Probability

### Liouville first-passage percolation: Subsequential scaling limits at high temperature

#### Abstract

Let $\{Y_{\mathfrak{B}}(x):x\in\mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which each vertex $x$ is given a weight of $e^{\gamma Y_{\mathfrak{B}}(x)}$ for some $\gamma>0$. We show that for sufficiently small but fixed $\gamma>0$, for any sequence of scales $\{S_{k}\}$ there exists a subsequence along which the appropriately scaled and interpolated Liouville FPP metric converges in the Gromov–Hausdorff sense to a random metric on the unit square in $\mathbf{R}^{2}$. In addition, all possible (conjecturally unique) scaling limits are homeomorphic by bi-Hölder-continuous homeomorphisms to the unit square with the Euclidean metric.

#### Article information

Source
Ann. Probab., Volume 47, Number 2 (2019), 690-742.

Dates
Revised: November 2017
First available in Project Euclid: 26 February 2019

https://projecteuclid.org/euclid.aop/1551171635

Digital Object Identifier
doi:10.1214/18-AOP1267

Mathematical Reviews number (MathSciNet)
MR3916932

Zentralblatt MATH identifier
07053554

#### Citation

Ding, Jian; Dunlap, Alexander. Liouville first-passage percolation: Subsequential scaling limits at high temperature. Ann. Probab. 47 (2019), no. 2, 690--742. doi:10.1214/18-AOP1267. https://projecteuclid.org/euclid.aop/1551171635

#### References

• [1] Adler, R. J. (1990). An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. Institute of Mathematical Statistics Lecture Notes—Monograph Series 12. IMS, Hayward, CA.
• [2] Ahlberg, D., Tassion, V. and Teixeira, A. (2017). Sharpness of the phase transition for continuum percolation in $\mathbb{R}^{2}$. Probab. Theory Related Fields 172 525–581.
• [3] Auffinger, A., Damron, M. and Hanson, J. (2017). 50 Years of First-Passage Percolation. University Lecture Series 68. Amer. Math. Soc., Providence, RI.
• [4] Beffara, V. and Duminil-Copin, H. (2012). The self-dual point of the two-dimensional random-cluster model is critical for $q\geq 1$. Probab. Theory Related Fields 153 511–542.
• [5] Benjamini, I. (2010). Random planar metrics. In Proceedings of the International Congress of Mathematicians. Volume IV 2177–2187. Hindustan Book Agency, New Delhi.
• [6] Biskup, M. (2017). Extrema of 2D discrete Gaussian free field. Lecture notes from the 2017 PIMS Summer School in Probability. Available at https://www.math.ucla.edu/~biskup/PIMS/notes.html.
• [7] Bouttier, J., Di Francesco, P. and Guitter, E. (2004). Planar maps as labeled mobiles. Electron. J. Combin. 11 Research Paper 69, 27.
• [8] Bramson, M., Ding, J. and Zeitouni, O. (2016). Convergence in law of the maximum of the two-dimensional discrete Gaussian free field. Comm. Pure Appl. Math. 69 62–123.
• [9] Cori, R. and Vauquelin, B. (1981). Planar maps are well labeled trees. Canad. J. Math. 33 1023–1042.
• [10] Ding, J. and Goswami, S. (2016). Upper bounds on Liouville first passage percolation and Watabiki’s prediction. Preprint. Available at https://arxiv.org/abs/1610.09998.
• [11] Ding, J. and Goswami, S. (2017). First passage percolation on the exponential of two-dimensional branching random walk. Electron. Commun. Probab. 22 Paper No. 69, 14.
• [12] Ding, J. and Zhang, F. (2017). Non-universality for first passage percolation on the exponential of log-correlated Gaussian fields. Probab. Theory Related Fields 171 1157–1188.
• [13] Duminil-Copin, H., Hongler, C. and Nolin, P. (2011). Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model. Comm. Pure Appl. Math. 64 1165–1198.
• [14] Duminil-Copin, H., Manolescu, I. and Tassion, V. (2018). An RSW theorem for Gaussian free field. In preparation.
• [15] Duminil-Copin, H., Raoufi, A. and Tassion, V. (2018). A new computation of the critical point for the planar random-cluster model with $q\ge1$. Ann. Inst. Henri Poincaré Probab. Stat. 54 422–436.
• [16] Duminil-Copin, H., Sidoravicius, V. and Tassion, V. (2017). Continuity of the phase transition for planar random-cluster and Potts models with $1\leq q\leq 4$. Comm. Math. Phys. 349 47–107.
• [17] Duplantier, B., Miller, J. and Sheffield, S. (2014). Liouville quantum gravity as a mating of trees. Preprint. Available at http://arxiv.org/abs/1409.7055.
• [18] Duplantier, B. and Sheffield, S. (2011). Liouville quantum gravity and KPZ. Invent. Math. 185 333–393.
• [19] Fernique, X. (1975). Regularité des trajectoires des fonctions aléatoires gaussiennes. In École d’Été de Probabilités de Saint-Flour, IV-1974. Lecture Notes in Math. 480 1–96. Springer, Berlin.
• [20] Grimmett, G. R. and Kesten, H. (2012). Percolation since Saint-Flour. In Percolation Theory at Saint-Flour. Probab. St.-Flour. Springer, Heidelberg.
• [21] Gwynne, E., Holden, N. and Sun, X. (2016). A distance exponent for Liouville quantum gravity. Preprint. Available at http://arxiv.org/abs/1606.01214.
• [22] Lawler, G. F. and Limic, V. (2010). Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge Univ. Press, Cambridge.
• [23] Le Gall, J.-F. (2007). The topological structure of scaling limits of large planar maps. Invent. Math. 169 621–670.
• [24] Le Gall, J.-F. (2010). Geodesics in large planar maps and in the Brownian map. Acta Math. 205 287–360.
• [25] Le Gall, J.-F. (2013). Uniqueness and universality of the Brownian map. Ann. Probab. 41 2880–2960.
• [26] Le Gall, J.-F. and Paulin, F. (2008). Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Geom. Funct. Anal. 18 893–918.
• [27] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI.
• [28] Lyons, R. and Peres, Y. (2016). Probability on Trees and Networks. Cambridge Series in Statistical and Probabilistic Mathematics 42. Cambridge Univ. Press, New York.
• [29] Miermont, G. (2013). The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 319–401.
• [30] Miermont, G. (2014). Aspects of random maps. Lecture Notes of the 2014 Saint-Flour Probability Summer School. Preliminary draft. Available at http://perso.ens-lyon.fr/gregory.miermont/coursSaint-Flour.pdf.
• [31] Miller, J. and Sheffield, S. (2015). Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric. Preprint. Available at http://arxiv.org/abs/1507.00719.
• [32] Miller, J. and Sheffield, S. (2016). Quantum Loewner evolution. Duke Math. J. 165 3241–3378.
• [33] Miller, J. and Sheffield, S. (2016). Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding. Preprint. Available at http://arxiv.org/abs/1605.03563.
• [34] Miller, J. and Sheffield, S. (2016). Liouville quantum gravity and the Brownian map III: the conformal structure is determined. Preprint. Available at http://arxiv.org/abs/1608.05391.
• [35] Pitt, L. D. (1982). Positively correlated normal variables are associated. Ann. Probab. 10 496–499.
• [36] Polyakov, A. M. (1981). Quantum geometry of bosonic strings. Phys. Lett. B 103 207–210.
• [37] Rhodes, R. and Vargas, V. (2014). Gaussian multiplicative chaos and applications: A review. Probab. Surv. 11 315–392.
• [38] Russo, L. (1978). A note on percolation. Z. Wahrsch. Verw. Gebiete 43 39–48.
• [39] Russo, L. (1981). On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete 56 229–237.
• [40] Schaeffer, G. (1988). Conjugaison d’arbres et cartes combinatoires aléatoires. Ph.D. thesis. Univ. Bordeaux I.
• [41] Seymour, P. D. and Welsh, D. J. A. (1978). Percolation probabilities on the square lattice. Ann. Discrete Math. 3 227–245.
• [42] Steele, J. M. (1986). An Efron–Stein inequality for nonsymmetric statistics. Ann. Statist. 14 753–758.
• [43] Tassion, V. (2016). Crossing probabilities for Voronoi percolation. Ann. Probab. 44 3385–3398.