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A contour line of the continuum Gaussian free field. (English) Zbl 1331.60090
The paper deals with the two-dimensional Gaussian free field on a simply connected domain $$D$$ with boundary condition $$-\lambda$$ on one boundary arc and boundary condition $$\lambda$$ on the complementary arc. In previous papers, the authors studied the discrete Gaussian free field which is a random function on a graph that (when defined on increasingly fine lattices) has the Gaussian free field as a scaling limit. They showed that there is a special constant $$\lambda>0$$ such that, if the boundary conditions are set to $$\pm\lambda$$ on the two boundary arcs, then the zero chordal contour line connecting the two endpoints of these arcs converges in law to the Schramm-Loewner evolution as the lattice size tends to zero, but $$\lambda$$ was not determined. In this paper, the exact value of $$\lambda$$ is determined and the Gaussian free field is considered for $$\lambda=\sqrt{\pi/8}$$. The authors construct the zero level line in two ways: as the limit of the chordal zero contour lines of the projections of the Gaussian free field onto certain spaces of piecewise linear functions, and as the only path-valued function on the space of distributions with a natural Markov property. They also show that, as a function of the two-dimensional Gaussian free field, the zero level line does not change when the Gaussian free field is modified away from the zero level line and derive some general properties of local sets.

##### MSC:
 60G60 Random fields 60G15 Gaussian processes 60J67 Stochastic (Schramm-)Loewner evolution (SLE)
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##### References:
 [1] Duplantier, B., Sheffield, S.: Liouville Quantum Gravity and KPZ. ArXiv e-prints, August 2008, 0808.1560 · Zbl 1226.81241 [2] Dubédat, J, SLE and the free field: partition functions and couplings, J. Am. Math. Soc., 22, 995-1054, (2009) · Zbl 1204.60079 [3] Dudley, R.M.: Real Analysis and Probability, volume 74 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2002). Revised reprint of the 1989 original [4] Gross, L.: Abstract Wiener spaces. In: Proceedings of Fifth Berkeley Symposium. Mathematical Statistical and Probability, Berkeley, California (1965/1966), vol. II: Contributions to Probability Theory, Part 1, pp. 31-42. University of California Press, Berkeley, California (1967) [5] Hagendorf, C., Bernard, D., Bauer, M.: The Gaussian free field and SLE$$\_4$$ on doubly connected domains. J. Stat. Phys. 140, 1-26 (2010), 1001.4501 · Zbl 1193.82027 [6] Izyurov, K., Kytölä, K.: Hadamard’s formula and couplings of SLEs with free field. ArXiv e-prints, (2010), 1006.1853 · Zbl 1204.60079 [7] Svante, J.: Gaussian Hilbert Spaces, volume 129 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1997) · Zbl 0887.60009 [8] Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939-995 (2004), arXiv:math.PR/0112234 · Zbl 1126.82011 [9] Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Springer-Verlag, Berlin (1992) · Zbl 0762.30001 [10] Protter, P.: Stochastic Integration and Differential Equations, volume 21 of Applications of Mathematics (New York): A new approach. Springer-Verlag, Berlin (1990) · Zbl 0694.60047 [11] Rohde, S; Schramm, O, Basic properties of SLE, Ann. Math. (2), 161, 883-924, (2005) · Zbl 1081.60069 [12] Rider, B., Virág, B.: The noise in the circular law and the Gaussian free field. Int. Math. Res. Not. IMRN, (2):Art. ID rnm006, 33 (2007) · Zbl 1130.60030 [13] Sheffield, S.: Local sets of the Gaussian free field: Slides and audio. www.fields.utoronto.ca/audio/05-06/percolation_SLE/sheffield1, www.fields.utoronto.ca/audio/05-06/percolation_SLE/sheffield2, www.fields.utoronto.ca/audio/05-06/percolation_SLE/sheffield3 (2005) [14] Scott, S.: Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139(3-4), 521-541 (2007) · Zbl 1132.60072 [15] Sheffield, S., Sun, N.: Strong path convergence from Loewner driving function convergence. Ann. Probab. 40(2), 578-610 (2012). doi:10.1214/10-AOP627 · Zbl 1255.60148 [16] Oded, S; Scott, S, Contour lines of the two-dimensional discrete Gaussian free field, Acta Math., 202, 21-137, (2009) · Zbl 1210.60051 [17] Schramm, O., Wilson, D.B.: SLE coordinate changes. New York J. Math. 11, 659-669, (2005). http://nyjm.albany.edu:8000/j/2005/11-31.html · Zbl 1094.82007 [18] Vaillant, N.: Probability tutorials: Tutorial 13, Regular Measure (2012). www.probability.net · Zbl 1210.60051
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