×

zbMATH — the first resource for mathematics

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
Keywords:
scaling limit
PDF BibTeX XML Cite
Full Text: DOI arXiv
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.