Annals of Probability

Harmonic explorer and its convergence to SLE4

Oded Schramm and Scott Sheffield

Full-text: Open access


The harmonic explorer is a random grid path. Very roughly, at each step the harmonic explorer takes a turn to the right with probability equal to the discrete harmonic measure of the left-hand side of the path from a point near the end of the current path. We prove that the harmonic explorer converges to SLE4 as the grid gets finer.

Article information

Ann. Probab., Volume 33, Number 6 (2005), 2127-2148.

First available in Project Euclid: 7 December 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 82B43: Percolation [See also 60K35]

SLE SLE_4 scaling limit harmonic explorer


Schramm, Oded; Sheffield, Scott. Harmonic explorer and its convergence to SLE 4. Ann. Probab. 33 (2005), no. 6, 2127--2148. doi:10.1214/009117905000000477.

Export citation


  • Ahlfors, L. V. (1973). Conformal Invariants: Topics in Geometric Function Theory. McGraw–Hill, New York.
  • Benjamini, I. and Schramm, O. (1996). Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab. 24 1219–1238.
  • Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA.
  • Fernández, J. L., Heinonen, J. and Martio, O. (1989). Quasilines and conformal mappings. J. Anal. Math. 52 117–132.
  • He, Z.-X. and Schramm, O. (1995). Hyperbolic and parabolic packings. Discrete Comput. Geom. 14 123–149.
  • Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Boston.
  • Lawler, G. F. (2004). An introduction to the stochastic Loewner evolution. In Random Walks and Geometry 261–293. de Gruyter, Berlin.
  • Lawler, G. F., Schramm, O. and Werner, W. (2003). Conformal restriction: The chordal case. J. Amer. Math. Soc. 16 917–955.
  • Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 939–995.
  • Lawler, G. F. and Werner, W. (2000). Universality for conformally invariant intersection exponents. J. Eur. Math. Soc. 2 291–328.
  • McCaughan, G. (1998). A recurrence/transience result for circle packings. Proc. Amer. Math. Soc. 126 3647–3656.
  • Pommerenke, Ch. (1992). Boundary Behaviour of Conformal Maps. Springer, Berlin.
  • Rohde, S. and Schramm, O. (2001). Basic properties of SLE. Ann. of Math. 161 879–920.
  • Schramm, O. (1995). Transboundary extremal length. J. Anal. Math. 66 307–329.
  • Schramm, O. (2001). A percolation formula. Electron. Comm. Probab. 6 115–120.
  • Schramm, O. (2001). Scaling limits of random processes and the outer boundary of planar Brownian motion. In Current Developments in Mathematics 2000 233–253. International Press, Somerville, MA.
  • Schramm, O. and Sheffield, S. (2003). The 2D discrete Gaussian free field interface. In preparation.
  • Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy's formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239–244.
  • Werner, W. (2004). Random planar curves and Schramm–Loewner evolutions. Lecture Notes in Math. 1840 107–195. Springer, Berlin.