# The Smart Kinetic Self-Avoiding Walk and Schramm Loewner Evolution

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## Abstract

The smart kinetic self-avoiding walk (SKSAW) is a random walk which never intersects itself and grows forever when run in the full-plane. At each time step the walk chooses the next step uniformly from among the allowable nearest neighbors of the current endpoint of the walk. In the full-plane a nearest neighbor is allowable if it has not been visited before and there is a path from the nearest neighbor to infinity through sites that have not been visited before. It is well known that on the hexagonal lattice the SKSAW in a bounded domain between two boundary points is equivalent to an interface in critical percolation, and hence its scaling limit is the chordal Schramm–Loewner evolution with \(\kappa =6\) (SLE\(_6\)). Like SLE there are variants of the SKSAW depending on the domain and the initial and terminal points. On the hexagonal lattice these variants have been shown to converge to the corresponding version of SLE\(_6\). It is believed that the scaling limit of all these variants on any regular lattice is the corresponding version of SLE\(_6\). We test this conjecture for the square lattice by simulating the SKSAW in the full-plane and find excellent agreement with the predictions of full-plane SLE\(_6\).

## Keywords

Percolation explorer Laplacian random walk Smart kinetic walk Schramm–Loewner evolution## Notes

### Acknowledgments

An allocation of computer time from the UA Research Computing High Performance Computing (HPC) and High Throughput Computing (HTC) at the University of Arizona is gratefully acknowledged.

## References

- 1.Amit, D.J., Parisi, G., Peliti, L.: Asymptotic behavior of the “true” self-avoiding walk. Phys. Rev. B
**27**, 1635 (1983)MathSciNetADSCrossRefGoogle Scholar - 2.Camia, F. , Newman, C.M.: Critical percolation exploration path and SLE\(_6\) : a proof of convergence. Probab. Theory Relat. Fields
**139**, 473–519 (2007). Archived as arXiv:math/0605035 [math.PR] - 3.Coniglio, A., Jan, N., Majid, I., Stanley, H.E.: Conformation of a polymer chain at the Theta\(^\prime \) point: connection to the external perimeter of a percolation cluster. Phys. Rev. B
**35**, 3617 (1987)ADSCrossRefGoogle Scholar - 4.Duplantier, B., Saleur, H.: Exact tricritical exponents for polymers at the FTHETA point in two dimensions. Phys. Rev. Lett.
**59**, 539 (1987)MathSciNetADSCrossRefGoogle Scholar - 5.Gherardi, M.: Theta-point polymers in the plane and Schramm–Loewner evolution. Phys. Rev. E
**88**, 032128 (2013). Archived as arXiv:1306.4993 [cond-mat.stat-mech] - 6.Gunn, J.M.F., Ortuño, M.: Percolation and motion in a simple random environment. J. Phys. A
**18**, L1095 (1985)ADSCrossRefGoogle Scholar - 7.Jiang, J.: Exploration processes and SLE\(_6\). Preprint (2014). Archived as arXiv:1409.6834 [math.PR]
- 8.Kennedy, T.: Monte Carlo tests of SLE predictions for 2D self-avoiding walks. Phys. Rev. Lett.
**88**, 130601 (2002). Archived as arXiv:math/0112246v1 [math.PR] - 9.Kennedy, T.: Conformal invariance and stochastic Loewner evolution predictions for the 2D self-avoiding walk—Monte Carlo tests. J. Stat. Phys.
**114**, 51–78 (2004). Archived as arXiv:math/0207231v2 [math.PR] - 10.Kremer, K., Lyklema, J.W.: Indefinitely growing self-avoiding walk. Phys. Rev. Lett.
**54**, 267 (1985)ADSCrossRefGoogle Scholar - 11.Lawler, G.: Conformally Invariant Processes in the Plane. American Mathematical Society, Providence (2005)Google Scholar
- 12.Lawler, G.: The Laplacian-\(b\) random walk and the Schramm–Loewner evolution. Illinois J. Math.
**50**, 701–746 (2006)zbMATHMathSciNetGoogle Scholar - 13.Lawler, G.F., Schramm, O., Werner, W.: On the scaling limit of planar self-avoiding walk, Fractal Geometry and Applications: a Jubilee of Benoit Mandelbrot, Part 2, 339, Proceedings of the Symposium on Pure Mathematics 72, American Mathematical Society, Providence, RI, (2004). Archived as arXiv:math/0204277v2 [math.PR]
- 14.Lawler, G., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab.
**32**, 939–995, (2004). Archived as arXiv:math/0112234 [math.PR] - 15.Lyklema, J.W., Evertsz, C., Pietronero, L.: The Laplacian random walk. Europhys. Lett.
**2**, 77 (1986)ADSCrossRefGoogle Scholar - 16.Madras, N., Slade, G.: The Self-Avoiding Walk. Birkhäuser, Boston (1996)zbMATHCrossRefGoogle Scholar
- 17.Majid, I., Jan, N., Coniglio, A., Stanley, H.E.: Kinetic growth walk: A new model for linear polymers. Phys. Rev. Lett.
**52**, 1257 (1984)ADSCrossRefGoogle Scholar - 18.Manna, S.S., Guttmann, A.J.: Kinetic growth walks and trails on oriented square lattices: Hull percolation and percolation hulls. J. Phys. A
**22**, 3113 (1989)ADSCrossRefGoogle Scholar - 19.Smirnov, S.: Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Math. Acad. Sci. Paris
**333**, 239–244 (2001). Archived as arXiv:0909.4499 [math.PR] - 20.Weinrib, A., Trugman, S.A.: A new kinetic walk and percolation perimeters. Phys. Rev. B
**31**, 2993 (1985)ADSCrossRefGoogle Scholar - 21.Werner, W.: Lectures on Two-Dimensional Critical Percolation, Statistical Mechanics (IAS/Park City Mathemematics Series v. 16), S. Sheffield, T. Spencer (eds.) (2007). Archived as arXiv:0710.0856 [math.PR]
- 22.Wu, H.: Conformal restriction: the radial case. Stoch. Process. Appl.
**125**, 552–570 (2013). Archived as arXiv:1304.5712 [math.PR] - 23.Ziff, R.M., Cummings, P.T., Stell, G.: Generation of percolation cluster perimeters by a random walk. J. Phys. A
**17**, 3009 (1984)ADSCrossRefGoogle Scholar