A note on Loewner energy, conformal restriction and Werner’s measure on self-avoiding loops
[Note sur l’énergie de Loewner, la restriction conforme et la mesure de Werner sur les boucles auto-évitantes]
Annales de l'Institut Fourier, Online first, 15 p.

Nous établissons une expression de l’énergie de Loewner d’une courbe de Jordan en termes de la mesure de Werner sur les boucles auto-évitantes du type SLE 8/3 . La preuve est basée sur la variation de l’énergie de Loewner sous l’effet des transformations conformes. Cette formule rappelle la propriété de la restriction conforme de SLE.

We establish an expression of the Loewner energy of a Jordan curve in terms of Werner’s measure on simple loops of SLE 8/3 type. The proof is based on a formula for the change of the Loewner energy under a conformal map that is reminiscent of SLE processes’ conformal restriction property.

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DOI : https://doi.org/10.5802/aif.3427
Classification : 30C55,  60J67
Mots clés : énergie de Loewner, restriction conforme, mesure de Werner, evolution Schramm–Loewner
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Wang, Yilin. A note on Loewner energy, conformal restriction and Werner’s measure on self-avoiding loops. Annales de l'Institut Fourier, Online first, 15 p.

[1] de Branges, Louis A proof of the Bieberbach conjecture, Acta Math., Volume 154 (1985) no. 1-2, pp. 137-152 | Article | MR 772434 | Zbl 0573.30014

[2] Dubédat, Julien Commutation relations for Schramm-Loewner evolutions, Commun. Pure Appl. Math., Volume 60 (2007) no. 12, pp. 1792-1847 | Article | MR 2358649 | Zbl 1137.82009

[3] Dubédat, Julien SLE and the free field: partition functions and couplings, J. Am. Math. Soc., Volume 22 (2009) no. 4, pp. 995-1054 | Article | MR 2525778 | Zbl 1204.60079

[4] Friz, Peter K.; Shekhar, Atul On the existence of SLE trace: finite energy drivers and non-constant κ, Probab. Theory Relat. Fields, Volume 169 (2017) no. 1-2, pp. 353-376 | Article | MR 3704771 | Zbl 1407.60113

[5] Kemppainen, Antti; Werner, Wendelin The nested simple conformal loop ensembles in the Riemann sphere, Probab. Theory Relat. Fields, Volume 165 (2016) no. 3-4, pp. 835-866 | Article | MR 3520020 | Zbl 1352.60117

[6] Lawler, Gregory; Schramm, Oded; Werner, Wendelin Conformal restriction: the chordal case, J. Am. Math. Soc., Volume 16 (2003) no. 4, pp. 917-955 | Article | MR 1992830 | Zbl 1030.60096

[7] Lawler, Gregory; Werner, Wendelin The Brownian loop soup, Probab. Theory Relat. Fields, Volume 128 (2004) no. 4, pp. 565-588 | Article | MR 2045953 | Zbl 1049.60072

[8] Le Jan, Yves Markov loops, determinants and Gaussian fields (2006) (https://arxiv.org/abs/math/0612112)

[9] Löwner, Karl Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I, Math. Ann., Volume 89 (1923) no. 1-2, pp. 103-121 | Article | MR 1512136 | Zbl 49.0714.01

[10] Nacu, Şerban; Werner, Wendelin Random soups, carpets and fractal dimensions, J. Lond. Math. Soc., Volume 83 (2011) no. 3, pp. 789-809 | Article | MR 2802511 | Zbl 1223.28012

[11] Osgood, Brad; Phillips, Ralph; Sarnak, Peter Extremals of determinants of Laplacians, J. Funct. Anal., Volume 80 (1988) no. 1, pp. 148-211 | Article | MR 960228 | Zbl 0653.53022

[12] Ray, Daniel B.; Singer, Isadore M. R-torsion and the Laplacian on Riemannian manifolds, Adv. Math., Volume 7 (1971), pp. 145-210 | Article | MR 295381

[13] Rohde, Steffen; Wang, Yilin The Loewner energy of loops and regularity of driving functions, Volume 2021 (2021) no. 10, pp. 7433-7469 (Int. Math. Res. Not.) | Article | MR 4259153 | Zbl 07398529

[14] Schramm, Oded Scaling limits of loop-erased random walks and uniform spanning trees, Isr. J. Math., Volume 118 (2000), pp. 221-288 | Article | MR 1776084 | Zbl 0968.60093

[15] Takhtajan, Leon A.; Teo, Lee-Peng Weil-Petersson metric on the universal Teichmüller space, Mem. Am. Math. Soc., Volume 183 (2006) no. 861, p. viii+119 | Article | MR 2251887 | Zbl 1243.32010

[16] Wang, Yilin The energy of a deterministic Loewner chain: reversibility and interpretation via SLE 0+ , J. Eur. Math. Soc., Volume 21 (2019) no. 7, pp. 1915-1941 | Article | MR 3959854 | Zbl 1422.30031

[17] Wang, Yilin Equivalent descriptions of the Loewner energy, Invent. Math., Volume 218 (2019) no. 2, pp. 573-621 | Article | MR 4011706 | Zbl 1435.30074

[18] Werner, Wendelin The conformally invariant measure on self-avoiding loops, J. Am. Math. Soc., Volume 21 (2008) no. 1, pp. 137-169 | Article | MR 2350053 | Zbl 1130.60016

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