Abstract
The relation between level lines of Gaussian free fields (GFF) and SLE4-type curves was discovered by O. Schramm and S. Sheffield. A weak interpretation of this relation is the existence of a coupling of the GFF and a random curve, in which the curve behaves like a level line of the field. In the present paper we study these couplings for the free field with different boundary conditions. We provide a unified way to determine the law of the curve (i.e. to compute the driving process of the Loewner chain) given boundary conditions of the field and to prove existence of the coupling. The proof is reduced to the verification of two simple properties of the mean and covariance of the field, which always relies on Hadamard’s formula and properties of harmonic functions. Examples include combinations of Dirichlet, Neumann and Riemann–Hilbert boundary conditions. In doubly connected domains, the standard annulus SLE4 is coupled with a compactified GFF obeying Neumann boundary conditions on the inner boundary. We also consider variants of annulus SLE coupled with free fields having other natural boundary conditions. These include boundary conditions leading to curves connecting two points on different boundary components with prescribed winding as well as those recently proposed by C. Hagendorf, M. Bauer and D. Bernard.
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Bauer, M., Bernard, D.: SLE, CFT and zig-zag probabilities. In: Proceedings of the conference ‘Conformal Invariance and Random Spatial Processes’, Edinburgh, 2003
Bauer M., Bernard D.: 2D growth processes: SLE and Loewner chains. Phys. Rep. 432(3–4), 115–222 (2006) arXiv:math-ph/0602049
Bauer, M., Bernard, D., Cantini, L.: Off-critical SLE(2) and SLE(4): a field theory approach. J. Stat. Mech. P07037 (2009). arXiv:0903.1023
Bauer, M., Bernard, D., Houdayer, J.: Dipolar stochastic Loewner evolutions. J. Stat. Mech. (3), P03001, 18 pp (2005, electronic)
Cardy, J.: SLE(kappa,rho) and Conformal Field Theory (2004). arXiv:math-ph/0412033
Dubédat J.: SLE and the free field: Partition functions and couplings. J. Am. Math. Soc. 22, 995–1054 (2009) arXiv:0712.3018
Hagendorf C., Bauer M., Bernard D.: The Gaussian free field and SLE(4) on doubly connected domains. J. Stat. Phys. 140, 1–26 (2010) arXiv:1001.4501
Kang, N.-G., Makarov, N.: Gaussian free field and conformal field theory. arXiv:1101.1024
Kytölä K.: On conformal field theory of SLE(kappa, rho). J. Stat. Phys. 123(6), 1169–1181 (2006) arXiv:math-ph/0504057
Lawler, G.F.: Conformally invariant processes in the plane. In: Mathematical Surveys and Monographs, vol. 114. American Mathematical Society, Providence, RI, 2005
Makarov, N., Smirnov, S.: Off-critical lattice models and massive SLEs. In: Proceedings of ICMP, 2009 (to appear)
Makarov, N., Zhan, D.: in preparation
Schramm O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000) arXiv:math/9904022
Schramm O., Sheffield S.: Harmonic explorer and its convergence to SLE 4. Ann. Probab. 33(6), 2127–2148 (2005) arXiv:math.PR/0310210
Sheffield, S.: Local sets of the Gaussian Free Field. Presentation at “Percolation, SLE, and related topics Workshop”, Fields Institute, Toronto, 2005. http://www.fields.utoronto.ca/audio/05-06/#percolation_SLE
Schramm O., Sheffield S.: Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202, 21–137 (2009) arXiv:math.PR/0605337
Schramm, O., Wilson, D.B.: SLE coordinate changes. New York J. Math. 11, 659–669 (2005, electronic). arXiv:math.PR/0505368
Werner, W.: Random planar curves and Schramm-Loewner evolutions. In: Lectures on probability theory and statistics. Lecture Notes in Math., vol. 1840, pp. 107–195. Springer, Berlin (2004)
Zhan D.: Stochastic Loewner evolution in doubly connected domains. Probab. Theory Relat. Fields 129((3), 340–380 (2004) arXiv:math/0310350
Zhan, D.: Some properties of annulus SLE. Electron. J. Probab. 11, Paper no. 41, 10691093 (2006). arXiv:math/0610304
Zhan D.: The scaling limits of planar LERW in finitely connected domains. Ann. Probab. 36(2), 467–529 (2008) arXiv:math/0610304
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Izyurov, K., Kytölä, K. Hadamard’s formula and couplings of SLEs with free field. Probab. Theory Relat. Fields 155, 35–69 (2013). https://doi.org/10.1007/s00440-011-0391-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-011-0391-2