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Contour lines of the two-dimensional discrete Gaussian free field. (English) Zbl 1210.60051
The paper deals with the 2-dimensional massless Gaussian free field (GFF) which is a 2-dimensional-time analog of the Wiener process. The GFF is a scaling limit of several discrete models for random surfaces just as the Wiener process is the hydrodynamic limit of random evolutions. The authors prove that the chordal contour lines of the GFF converge to forms of SLE(4), where SLE(4) is the scaling limit of a random interface called the harmonic explorer. Specifically, there is a constant $$c>0$$ such that h is an interpolation of the discrete GFF on a Jordan domain with values $$-c$$ on one boundary arc and $$c$$ on the complementary arc, the zero level line of $$c$$ joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are $$-a<0$$ on the first arc and $$b>0$$ on the complementary arc, then the convergence is to SLE$$(4;a/c-1,b/c-1)$$, a variant of SLE(4).

##### MSC:
 60G60 Random fields 60G15 Gaussian processes
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##### References:
 [1] Ahlfors, L. V., Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill Series in Higher Mathematics. McGraw-Hill, New York, 1973. · Zbl 0272.30012 [2] Beffara, V., The dimension of the SLE curves. Ann. Probab., 36:4 (2008), 1421–1452. · Zbl 1165.60007 [3] Belavin, A. A., Polyakov, A. M. & Zamolodchikov, A. B., Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B, 241 (1984), 333–380. · Zbl 0661.17013 [4] Bricmont, J., El Mellouki, A. & Fröhlich, J., Random surfaces in statistical mechanics: roughening, rounding, wetting,.... J. Stat. Phys., 42:5–6 (1986), 743–798. [5] Camia, F. & Newman, C. M., Two-dimensional critical percolation: the full scaling limit. Comm. Math. Phys., 268 (2006), 1–38. · Zbl 1117.60086 [6] Cardy, J., SLE for theoretical physicists. Ann. Physics, 318 (2005), 81–118. · Zbl 1073.81068 [7] Coniglio, A., Fractal structure of Ising and Potts clusters: exact results. Phys. Rev. Lett., 62:26 (1989), 3054–3057. [8] Di Francesco, P., Mathieu, P. & Sénéchal, D., Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, New York, 1997. [9] Doyle, P. G. & Snell, J. L., Random Walks and Electric Networks. Carus Mathematical Monographs, 22. Mathematical Association of America, Washington, DC, 1984. · Zbl 0583.60065 [10] Duplantier, B., Two-dimensional fractal geometry, critical phenomena and conformal invariance. Phys. Rep., 184 (1989), 229–257. [11] Duplantier, B. & Saleur, H., Exact critical properties of two-dimensional dense self-avoiding walks. Nuclear Phys. B, 290 (1987), 291–326. [12] – Winding-angle distributions of two-dimensional self-avoiding walks from conformal invariance. Phys. Rev. Lett., 60:23 (1988), 2343–2346. [13] Foltin, G., An alternative field theory for the Kosterlitz–Thouless transition. J. Phys. A, 34:26 (2001), 5327–5333. [14] Fröhlich, J. & Spencer, T., The Kosterlitz–Thouless transition in two-dimensional abelian spin systems and the Coulomb gas. Comm. Math. Phys., 81 (1981), 527–602. [15] Gedzki, K., Lectures on conformal field theory, in Quantum Fields and Strings: a Course for Mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), pp. 727–805. Amer. Math. Soc., Providence, RI, 1999. [16] Giacomin, G., Limit theorems for random interface models of Ginzburg–Landau type, in Stochastic Partial Differential Equations and Applications (Trento, 2002), Lecture Notes in Pure and Appl. Math., 227, pp. 235–253. Dekker, New York, 2002. · Zbl 0996.82044 [17] Glimm, J. & Jaffe, A., Quantum Physics. Springer, New York, 1987. · Zbl 0461.46051 [18] Huber, G. & Kondev, J., Passive-scalar turbulence and the geometry of loops. Bull. Amer. Phys. Soc., DCOMP Meeting 2001, Q2.008. [19] Isichenko, M. B., Percolation, statistical topography, and transport in random media. Rev. Modern Phys., 64:4 (1992), 961–1043. [20] Kadanoff, L. P., Lattice Coulomb gas representations of two-dimensional problems. J. Phys. A, 11:7 (1978), 1399–1417. [21] Kager, W. & Nienhuis, B., A guide to stochastic Löwner evolution and its applications. J. Stat. Phys., 115:5–6 (2004), 1149–1229. · Zbl 1157.82327 [22] Karatzas, I. & Shreve, S. E., Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, 113. Springer, New York, 1988. · Zbl 0638.60065 [23] Kenyon, R., Dominos and the Gaussian free field. Ann. Probab., 29:3 (2001), 1128–1137. · Zbl 1034.82021 [24] Kondev, J., Du, S. & Huber, G., Two-dimensional passive-scalar turbulence and the geometry of loops. Bull. Amer. Phys. Soc., March Meeting 2002, U4.003. [25] Kondev, J. & Henley, C. L., Geometrical exponents of contour loops on random Gaussian surfaces. Phys. Rev. Lett., 74:23 (1995), 4580–4583. [26] Kondev, J., Henley, C. L. & Salinas, D. G., Nonlinear measures for characterizing rough surface morphologies. Phys. Rev. E, 61 (2000), 104–125. [27] Kosterlitz, J. M., The d-dimensional Coulomb gas and the roughening transition. J. Phys. C, 10:19 (1977), 3753–3760. [28] Kosterlitz, J. M. & Thouless, D. J., Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C, 6:7 (1973), 1181–1203. [29] Lawler, G. F., Intersections of Random Walks. Probability and its Applications. Birkhäuser, Boston, MA, 1991. · Zbl 1228.60004 [30] – Strict concavity of the intersection exponent for Brownian motion in two and three dimensions. Math. Phys. Electron. J., 4 (1998), Paper 5, 67 pp. · Zbl 0909.60065 [31] – An introduction to the stochastic Loewner evolution, in Random Walks and Geometry, pp. 261–293. de Gruyter, Berlin, 2004. · Zbl 1061.60107 [32] – Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs, 114. Amer. Math. Soc., Providence, RI, 2005. [33] Lawler, G. F., Schramm, O. & Werner, W., Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math., 187 (2001), 237–273. · Zbl 1005.60097 [34] – Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist., 38 (2002), 109–123. · Zbl 1006.60075 [35] – Conformal restriction: the chordal case. J. Amer. Math. Soc., 16 (2003), 917–955. · Zbl 1030.60096 [36] – Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab., 32:1B (2004), 939–995. · Zbl 1126.82011 [37] Mandelbrot, B., How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science, 156 (1967), 636–638. [38] Naddaf, A. & Spencer, T., On homogenization and scaling limit of some gradient perturbations of a massless free field. Comm. Math. Phys., 183 (1997), 55–84. · Zbl 0871.35010 [39] Nienhuis, B., Exact critical point and critical exponents of O(n) models in two dimensions. Phys. Rev. Lett., 49:15 (1982), 1062–1065. [40] – Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Stat. Phys., 34:5–6 (1984), 731–761. · Zbl 0595.76071 [41] den Nijs, M., Extended scaling relations for the magnetic critical exponents of the Potts model. Phys. Rev. B, 27:3 (1983), 1674–1679. [42] Pommerenke, C., Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften, 299. Springer, Berlin–Heidelberg, 1992. · Zbl 0762.30001 [43] Revuz, D. & Yor, M., Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften, 293. Springer, Berlin–Heidelberg, 1999. · Zbl 0917.60006 [44] Rohde, S. & Schramm, O., Basic properties of SLE. Ann. of Math., 161 (2005), 883–924. · Zbl 1081.60069 [45] Saleur, H. & Duplantier, B., Exact determination of the percolation hull exponent in two dimensions. Phys. Rev. Lett., 58:22 (1987), 2325–2328. [46] Schramm, O., Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118 (2000), 221–288. · Zbl 0968.60093 [47] Schramm, O. & Sheffield, S., Harmonic explorer and its convergence to SLE4. Ann. Probab., 33:6 (2005), 2127–2148. · Zbl 1095.60007 [48] Sheffield, S., Gaussian free fields for mathematicians. Probab. Theory Related Fields, 139 (2007), 521–541. · Zbl 1132.60072 [49] Smirnov, S., Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 239–244. · Zbl 0985.60090 [50] Spencer, T., Scaling, the free field and statistical mechanics, in The Legacy of Norbert Wiener: A Centennial Symposium (Cambridge, MA, 1994), Proc. Sympos. Pure Math., 60, pp. 373–389. Amer. Math. Soc., Providence, RI, 1997. [51] Werner, W., Random planar curves and Schramm–Loewner evolutions, in Lectures on Probability Theory and Statistics, Lecture Notes in Math., 1840, pp. 107–195. Springer, Berlin–Heidelberg, 2004.
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