Skip to main content Accessibility help
×
Home

UNIQUENESS OF THE WELDING PROBLEM FOR SLE AND LIOUVILLE QUANTUM GRAVITY

  • Oliver McEnteggart (a1), Jason Miller (a1) and Wei Qian (a1)

Abstract

We give a simple set of geometric conditions on curves $\unicode[STIX]{x1D702}$ , $\widetilde{\unicode[STIX]{x1D702}}$ in $\mathbf{H}$ from $0$ to $\infty$ so that if $\unicode[STIX]{x1D711}:\mathbf{H}\rightarrow \mathbf{H}$ is a homeomorphism which is conformal off $\unicode[STIX]{x1D702}$ with $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702})=\widetilde{\unicode[STIX]{x1D702}}$ then $\unicode[STIX]{x1D711}$ is a conformal automorphism of $\mathbf{H}$ . Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm–Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if $\unicode[STIX]{x1D702}$ is a non-space-filling $\text{SLE}_{\unicode[STIX]{x1D705}}$ curve in $\mathbf{H}$ from $0$ to $\infty$ , and $\unicode[STIX]{x1D711}$ is a homeomorphism which is conformal on $\mathbf{H}\setminus \unicode[STIX]{x1D702}$ , and $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702})$ , $\unicode[STIX]{x1D702}$ are equal in distribution, then $\unicode[STIX]{x1D711}$ is a conformal automorphism of $\mathbf{H}$ . Applying this result for $\unicode[STIX]{x1D705}=4$ establishes that the welding operation for critical ( $\unicode[STIX]{x1D6FE}=2$ ) LQG is well defined. Applying it for $\unicode[STIX]{x1D705}\in (4,8)$ gives a new proof that the welding of two independent $\unicode[STIX]{x1D705}/4$ -stable looptrees of quantum disks to produce an $\text{SLE}_{\unicode[STIX]{x1D705}}$ on top of an independent $4/\sqrt{\unicode[STIX]{x1D705}}$ -LQG surface is well defined.

Copyright

References

Hide All
1. Ahlfors, L. V., Lectures on Quasiconformal Mappings, second edition, University Lecture Series, Volume 38 (American Mathematical Society, Providence, RI, 2006). With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard.
2. Aru, J., Huang, Y. and Sun, X., Two perspectives of the 2D unit area quantum sphere and their equivalence, Commun. Math. Phys. 356(1) (2017), 261283.
3. Astala, K., Jones, P., Kupiainen, A. and Saksman, E., Random conformal weldings, Acta Math. 207(2) (2011), 203254.
4. Bass, R. F., Probabilistic Techniques in Analysis, Probability and its Applications (New York), (Springer, New York, 1995).
5. Berestycki, N., An elementary approach to Gaussian multiplicative chaos, Electron. Commun. Probab. 22(Paper No. 27) (2017), 112.
6. David, F., Kupiainen, A., Rhodes, R. and Vargas, V., Liouville quantum gravity on the Riemann sphere, Commun. Math. Phys. 342(3) (2016), 869907.
7. Ding, J., Dubédat, J., Dunlap, A. and Falconet, H., Tightness of Liouville first passage percolation for $\unicode[STIX]{x1D6FE}\in (0,2)$ , Preprint, April 2019, arXiv:1904.08021.
8. Dubédat, J., SLE and the free field: partition functions and couplings, J. Amer. Math. Soc. 22(4) (2009), 9951054.
9. Dubédat, J., Falconet, H., Gwynne, E., Pfeffer, J. and Sun, X., Weak LQG metrics and Liouville first passage percolation, Preprint, May 2019, arXiv:1905.00380.
10. Duplantier, B., Miller, J. and Sheffield, S., Liouville quantum gravity as a mating of trees, Preprint, September 2014, ArXiv e-prints.
11. Duplantier, B., Rhodes, R., Sheffield, S. and Vargas, V., Critical Gaussian multiplicative chaos: convergence of the derivative martingale, Ann. Probab. 42(5) (2014), 17691808.
12. Duplantier, B., Rhodes, R., Sheffield, S. and Vargas, V., Renormalization of critical Gaussian multiplicative chaos and KPZ relation, Commun. Math. Phys. 330(1) (2014), 283330.
13. Duplantier, B. and Sheffield, S., Liouville quantum gravity and KPZ, Invent. Math. 185(2) (2011), 333393.
14. Gwynne, E., Kassel, A., Miller, J. and Wilson, D. B., Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for 𝜅 > 8, Commun. Math. Phys. 358(3) (2018), 10651115.
15. Gwynne, E. and Miller, J., Convergence of the self-avoiding walk on random quadrangulations to SLE $_{8/3}$ on $\sqrt{8/3}$ -Liouville quantum gravity, Preprint, August 2016, ArXiv e-prints.
16. Gwynne, E. and Miller, J., Convergence of percolation on uniform quadrangulations with boundary to SLE $_{6}$ on $\sqrt{8/3}$ -Liouville quantum gravity, Preprint, January 2017, ArXiv e-prints.
17. Gwynne, E. and Miller, J., Confluence of geodesics in Liouville quantum gravity for $\unicode[STIX]{x1D6FE}\in (0,2)$ , Preprint, May 2019, arXiv:1905.00381.
18. Gwynne, E. and Miller, J., Conformal covariance of the Liouville quantum gravity metric for $\unicode[STIX]{x1D6FE}\in (0,2)$ , Preprint, May 2019, arXiv:1905.00384.
19. Gwynne, E. and Miller, J., Existence and uniqueness of the Liouville quantum gravity metric for $\unicode[STIX]{x1D6FE}\in (0,2)$ , Preprint, May 2019, arXiv:1905.00383.
20. Gwynne, E. and Miller, J., Local metrics of the Gaussian free field, Preprint, May 2019, arXiv:1905.00379.
21. Gwynne, E., Miller, J. and Sun, X., Almost sure multifractal spectrum of Schramm–Loewner evolution, Duke Math. J. 167(6) (2018), 10991237.
22. Høegh Krohn, R., A general class of quantum fields without cut-offs in two space–time dimensions, Commun. Math. Phys. 21 (1971), 244255.
23. Holden, N. and Powell, E., Conformal welding for critical Liouville quantum gravity, Preprint, December 2018, arXiv:1812.11808.
24. Huang, Y., Rhodes, R. and Vargas, V., Liouville quantum gravity on the unit disk, Ann. Inst. Henri Poincaré Probab. Stat. 54(3) (2018), 16941730.
25. Jones, P. W., On removable sets for Sobolev spaces in the plane, in Essays on Fourier Analysis in Honor of Elias M. Stein (Princeton, NJ, 1991), Princeton Mathematical Series, Volume 42, pp. 250267 (Princeton University Press, Princeton, NJ, 1995).
26. Jones, P. W. and Smirnov, S. K., Removability theorems for Sobolev functions and quasiconformal maps, Ark. Mat. 38(2) (2000), 263279.
27. Junnila, J. and Saksman, E., Uniqueness of critical Gaussian chaos, Electron. J. Probab. 22(Paper No. 11) (2017), 131.
28. Kahane, J.-P., Sur le chaos multiplicatif, Ann. Sci. Math. Québec 9(2) (1985), 105150.
29. Kaufman, R. and Wu, J.-M., On removable sets for quasiconformal mappings, Ark. Mat. 34(1) (1996), 141158.
30. Kenyon, R., Miller, J., Sheffield, S. and Wilson, D. B., Bipolar orientations on planar maps and SLE $_{12}$ , Ann. Probab., Preprint, November 2015, ArXiv e-prints, to appear.
31. Koskela, P. and Nieminen, T., Quasiconformal removability and the quasihyperbolic metric, Indiana Univ. Math. J. 54(1) (2005), 143151.
32. Lawler, G. F. and Rezaei, M. A., Minkowski content and natural parameterization for the Schramm–Loewner evolution, Ann. Probab. 43(3) (2015), 10821120.
33. Li, Y., Sun, X. and Watson, S. S., Schnyder woods, SLE(16), and Liouville quantum gravity, Preprint, May 2017, ArXiv e-prints.
34. Miller, J., Dimension of the SLE light cone, the SLE fan, and SLE𝜅(𝜌) for 𝜅 ∈ (0, 4) and 𝜌 ∈ [[[()[]mml:mfrac[]()]][[()[]mml:mrow []()]]𝜅[[()[]/mml:mrow[]()]] [[()[]mml:mrow []()]]2[[()[]/mml:mrow[]()]][[()[]/mml:mfrac[]()]] - 4, -2), Commun. Math. Phys. 360(3) (2018), 10831119.
35. Miller, J. and Sheffield, S., Liouville quantum gravity and the Brownian map I: the QLE(8/3,0) metric, Preprint, July 2015, ArXiv e-prints.
36. Miller, J. and Sheffield, S., Gaussian free field light cones and SLE $_{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D70C})$ , Ann. Probab., Preprint, June 2016, ArXiv e-prints, to appear.
37. Miller, J. and Sheffield, S., Imaginary geometry I: interacting SLEs, Probab. Theory Related Fields 164(3–4) (2016), 553705.
38. Miller, J. and Sheffield, S., Imaginary geometry III: reversibility of SLE𝜅 for 𝜅 ∈ (4, 8), Ann. of Math. (2) 184(2) (2016), 455486.
39. Miller, J. and Sheffield, S., Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding, Preprint, May 2016, arXiv:1605.03563.
40. Miller, J. and Sheffield, S., Liouville quantum gravity and the Brownian map III: the conformal structure is determined, Preprint, August 2016, arXiv:1608.05391.
41. Miller, J. and Sheffield, S., Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees, Probab. Theory Related Fields 169(3–4) (2017), 729869.
42. Miller, J., Sheffield, S. and Werner, W., CLE percolations, Forum Math. Pi 5(e4) (2017), 1102.
43. Miller, J., Sheffield, S. and Werner, W., (2018). In preparation.
44. Ntalampekos, D., A removability theorem for Sobolev functions and detour sets, Preprint, June 2017, arXiv:1706.07687.
45. Ntalampekos, D., Non-removability of the Sierpinski Gasket, Inventiones, Preprint, April 2018, arXiv:1804.10239, to appear.
46. Powell, E., Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation, Electron. J. Probab. 23(Paper No. 31) (2018), 126.
47. Rezaei, M. A. and Zhan, D., Green’s functions for chordal SLE curves, Probab. Theory Related Fields 171(3–4) (2018), 10931155.
48. Rezaei, M. A. and Zhan, D., Higher moments of the natural parameterization for SLE curves, Ann. Inst. Henri Poincaré Probab. Stat. 53(1) (2017), 182199.
49. Rhodes, R. and Vargas, V., Gaussian multiplicative chaos and applications: a review, Probab. Surv. 11 (2014), 315392.
50. Robert, R. and Vargas, V., Gaussian multiplicative chaos revisited, Ann. Probab. 38(2) (2010), 605631.
51. Rohde, S. and Schramm, O., Basic properties of SLE, Ann. of Math. (2) 161(2) (2005), 883924.
52. Schramm, O., Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221288.
53. Schramm, O. and Sheffield, S., A contour line of the continuum Gaussian free field, Probab. Theory Related Fields 157(1–2) (2013), 4780.
54. Sheffield, S., Exploration trees and conformal loop ensembles, Duke Math. J. 147(1) (2009), 79129.
55. Sheffield, S., Conformal weldings of random surfaces: SLE and the quantum gravity zipper, Ann. Probab. 44(5) (2016), 34743545.
56. Sheffield, S., Quantum gravity and inventory accumulation, Ann. Probab. 44(6) (2016), 38043848.
57. Sheffield, S. and Werner, W., Conformal loop ensembles: the Markovian characterization and the loop-soup construction, Ann. of Math. (2) 176(3) (2012), 18271917.
58. Werness, B. M., Regularity of Schramm–Loewner evolutions, annular crossings, and rough path theory, Electron. J. Probab. 17(81) (2012), 121.
59. Zhan, D., Reversibility of chordal SLE, Ann. Probab. 36(4) (2008), 14721494.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed