## Harmonic explorer and its convergence to $$\text{SLE}_4$$.(English)Zbl 1095.60007

The harmonic explorer is a new model similar in spirit to the loop erased random walk and diffusion limited aggregation. It is a random grid path in the planar honeycomb lattice that at each step of generation takes a turn to the right with probability equal to a discrete harmonic measure. The authors prove that the harmonic explorer converges to the chordal stochastic Loewner evolution with parameter 4 $$\text{(SLE}_4)$$ as the hexagonal mesh gets finer.

### MSC:

 60D05 Geometric probability and stochastic geometry 82B43 Percolation

scaling limit
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### References:

 [1] Ahlfors, L. V. (1973). Conformal Invariants : Topics in Geometric Function Theory . McGraw–Hill, New York. · Zbl 0272.30012 [2] Benjamini, I. and Schramm, O. (1996). Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab. 24 1219–1238. · Zbl 0862.60053 [3] Dudley, R. M. (1989). Real Analysis and Probability . Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA. · Zbl 0686.60001 [4] Fernández, J. L., Heinonen, J. and Martio, O. (1989). Quasilines and conformal mappings. J. Anal. Math. 52 117–132. · Zbl 0677.30012 [5] He, Z.-X. and Schramm, O. (1995). Hyperbolic and parabolic packings. Discrete Comput. Geom. 14 123–149. · Zbl 0830.52010 [6] Lawler, G. F. (1991). Intersections of Random Walks . Birkhäuser, Boston. · Zbl 1228.60004 [7] Lawler, G. F. (2004). An introduction to the stochastic Loewner evolution. In Random Walks and Geometry 261–293. de Gruyter, Berlin. · Zbl 1061.60107 [8] Lawler, G. F., Schramm, O. and Werner, W. (2003). Conformal restriction: The chordal case. J. Amer. Math. Soc. 16 917–955. · Zbl 1030.60096 [9] 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. · Zbl 1126.82011 [10] Lawler, G. F. and Werner, W. (2000). Universality for conformally invariant intersection exponents. J. Eur. Math. Soc. 2 291–328. · Zbl 1098.60081 [11] McCaughan, G. (1998). A recurrence/transience result for circle packings. Proc. Amer. Math. Soc. 126 3647–3656. · Zbl 0912.30002 [12] Pommerenke, Ch. (1992). Boundary Behaviour of Conformal Maps . Springer, Berlin. · Zbl 0762.30001 [13] Rohde, S. and Schramm, O. (2001). Basic properties of SLE. Ann. of Math. 161 879–920. · Zbl 1081.60069 [14] Schramm, O. (1995). Transboundary extremal length. J. Anal. Math. 66 307–329. · Zbl 0842.30006 [15] Schramm, O. (2001). A percolation formula. Electron. Comm. Probab. 6 115–120. · Zbl 1008.60100 [16] 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. [17] Schramm, O. and Sheffield, S. (2003). The 2D discrete Gaussian free field interface. In preparation. · Zbl 1210.60051 [18] 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. · Zbl 0985.60090 [19] Werner, W. (2004). Random planar curves and Schramm–Loewner evolutions. Lecture Notes in Math. 1840 107–195. Springer, Berlin. · Zbl 1057.60078
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