Summary
In this article, I give a pedagogical introduction and overview of percolation theory. Special emphasis will be put on the review of some of the most prominent algorithms devised to study percolation numerically. The real-space renormalization group treatment of the percolation problem is then discussed. As a rather novel application of this approach to percolation, I will review recent results using similar real-space renormalization ideas that have been applied to the quantum Hall transition.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
We note that the WEH-Ferienkurs, following which this article was prepared, took place during early fall.
S.R. Broadbent, J.M. Hammersley: Proc. Camb Philos. Soc. 53, 629 (1957)
J.M. Hammersley: Proc. Camb. Philos. Soc. 53, 642 (1957)
J.M. Hammersley: Ann. Math. Statist. 28, 790 (1957)
D. Stauffer, A. Aharony: Perkolationstheorie (VCH, Weinheim 1995 )
K. Binder: Rep. Prog. Phys. 60, 487 (1997)
G. Grimmett: Percolation ( Springer, Berlin 1989 )
A. Bunde, S. Havlin (Eds.): Percolation and Disordered Systems: Theory and Applications ( North-Holland, Amsterdam 1999 )
J. Voit: The Statistical Mechanics of Capital Markets ( Springer, Heidelberg 2001 )
J. Goldenberg, B. Libai, S. Solomon, N. Jan, D. Stauffer: Marketing Percolation (2000). Cond-mat/9905426
http://de.arXiv.org/, 1993–2000
J. Hoshen, R. Kopelman: Phys. Rev. B 14, 3438 (1976)
C. Adami: (1997), http://www.krl.caltech.edu/“adami/CD1/Percolation/percolation.html, likely to change without prior notice
P. Leath: Phys. Rev. B 14, 5056 (1976)
W. Kinzel, G. Reents: Physics by Computer ( Springer, Berlin 1998 )
W. Kinzel, G. Reents: (1999), http://wptxl5.physik.uni-wuerzburg.de/TP3/applet_java/percgr.html, likely to change without prior notice
R.F. Voss: J. Phys. A: Math. Gen. 17 (7), L373 (1984)
R.M. Ziff, P.T. Cummings, G. Stell: J. Phys. A Math. Gen. 17, 3009 (1984)
M. Rosso, J.F. Gouyet, B. Sapoval: Phys. Rev. B 32, 6053 (1985)
R.M. Ziff, B. Sapoval: J. Phys. A Math. Gen. 19, L1169 (1986)
P.N. Suding, R.M. Ziff: Phys. Rev. E 60, 275 (1999)
M.F. Sykes, J.W. Essam: Phys. Rev. Lett. 10, 3 (1963)
R.M. Ziff, P.N. Suding: J. Phys. A Math. Gen. 30, 5351 (1997)
C.D. Lorenz, R.M. Ziff: Phys. Rev. B 57, 230 (1998)
P. Kleban, R.M. Ziff: Phys. Rev. B 57, R8075 (1998)
M.E.J. Newman, R.M. Ziff: Phys. Rev. Lett. 85, 4104 (2000). Cond-mat/0005264
A.B. Harris, T.C. Lubensky, W.K. Holcomb, C. Dasgupta: Phys. Rev. Lett. 35, 327 (1975)
P.J. Reynolds, W. Klein, H.E. Stanley: J. Phys. C Solid State Phys. 10, L167 (1977)
J. Bernasconi: Phys. Rev. B 18, 2185 (1978)
P.J. Reynolds, H.E. Stanley, W. Klein: Phys. Rev. B 21, 1223 (1980)
P.D. Eschbach, D. Stauffer, H. Herrmann: Phys. Rev. B 23, 422 (1981)
K. von Klitzing, G. Dorda, M. Pepper: Phys. Rev. Lett. 45, 494 (1980)
http://www.tu-chemnitz.de/physik/HERAEUS/2000/Springer.html
M. Janssen, O. Viehweger, U. Fastenrath, J. Hajdu: Introduction to the Theory of the Integer Quantum Hall effect ( VCH, Weinheim 1994 )
T. Chakraborty, P. Pietilänen: The Quantum Hall Effects ( Springer, Berlin 1995 )
R.B. Laughlin: Phys. Rev. B 23, 5632 (1981)
D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs: Phys. Rev. Lett. 49, 405 (1982)
R.E. Prange: Phys. Rev. B 23, 4802 (1981)
A.M.M. Pruisken: Nucl. Phys. B 235, 277 (1984)
D.L. Landau: Z. Phys. 64, 629 (1930)
S.V. Iordanskii: Solid State Commun. 43, 1 (1982)
J.T. Chalker, P.D. Coddington: J. Phys. Condens. Matter 21, 2665 (1988)
D.H. Lee, Z. Wang, S. Kivelson: Phys. Rev. Lett. 70, 4130 (1993)
S. Koch, R.J. Haug, K. von Klitzing, K. Ploog: Phys. Rev. B 43, 6828 (1991)
R.T.F. van Schaijk, A. de Visser, S.M. Olsthoorn, H.P. Wei, A.M.M. Pruisken: Phys. Rev. Lett. 84, 1567 (2000)
A.G. Galstyan, M.E. Raikh: Phys. Rev. B 56, 1422 (1997)
M. Büttiker, Y. Imry, R. Landauer, S. Pinhas• Phys. Rev. B 31, 6207 (1985)
W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling: Numerical Recipes in FORTRAN, 2nd edn. ( Cambridge University Press, Cambridge 1992 )
Z. Wang, B. Jovanovic, D.H. Lee: Phys. Rev. Lett. 77, 4426 (1996)
A. Weymer, M. Janssen: Ann. Phys. (Leipzig) 7, 159 (1998). Cond-mat/9805063
Y. Avishai, Y. Band, D. Brown: Phys. Rev. B 60, 8992 (1999)
D.H. Cobden, E. Kogan: Phys. Rev. B 54, R17 316 (1996)
P. Cain, R.A. Römer, M.E. Raikh, M. Schreiber: Integer Quantum Hall Transition in the Presence of a Quenched Disorder,submitted to Phys. Rev. B (2001). Condmat/0104045
A. Aho, J.E. Hoperoft, J.D. Ullman: Data Structures and Algorithms ( Addison-Wesley, New York 1983 )
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Römer, R.A. (2002). Percolation, Renormalization and Quantum Hall Transition. In: Computational Statistical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04804-7_17
Download citation
DOI: https://doi.org/10.1007/978-3-662-04804-7_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07571-1
Online ISBN: 978-3-662-04804-7
eBook Packages: Springer Book Archive