Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-02-06T11:16:52.287Z Has data issue: false hasContentIssue false

5 - More on importance sampling Monte Carlo methods for lattice systems

Published online by Cambridge University Press:  24 November 2021

David Landau  and
Kurt Binder
David Landau
Affiliation:
University of Georgia
Kurt Binder
Affiliation:
Johannes Gutenberg Universität Mainz, Germany
Get access

Summary

Advances in simulational methods sometimes have their origin in unusual places; such is the case with an entire class of methods which attempt to beat critical slowing down in spin models on lattices by flipping correlated clusters of spins in an intelligent way instead of simply attempting single spin-flips. The first steps were taken by Fortuin and Kasteleyn (Kasteleyn and Fortuin, 1969; Fortuin and Kasteleyn, 1972), who showed that it was possible to map a ferromagnetic Potts model onto a corresponding percolation model. The reason that this observation is so important is that in the percolation problem states are produced by throwing down particles, or bonds, in an uncorrelated fashion; hence there is no critical slowing down. In contrast, as we have already mentioned, the q-state Potts model when treated using standard Monte Carlo methods suffers from slowing down. (Even for large q where the transition is first order, the time scales can become quite long.) The Fortuin–Kasteleyn transformation thus allows us to map a problem with slow critical relaxation into one where such effects are largely absent. (As we shall see, not all slowing down is eliminated, but the problem is reduced quite dramatically.)

Type
Chapter
Information
A Guide to Monte Carlo Simulations in Statistical Physics , pp. 161 - 242
Publisher: Cambridge University Press
Print publication year: 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abraham, D. B. (1986), in Phase Transitions and Critical Phenomena, eds. Domb, C. and Lebowitz, J. L. (Academic Press, London), vol. 10, p. 1.Google Scholar
Adam, E., Billard, L., and Lancon, F. (1999), Phys. Rev. E 59, 1212.CrossRefGoogle Scholar
Albano, E. V. and Binder, K. (2012), Phys. Rev. E 85, 061601.Google Scholar
Alexandrowicz, Z. (1975), J. Stat. Phys. 13, 231.Google Scholar
Almarza, N. G. and Lomba, E. (2003), Phys. Rev. E 68, 011202.CrossRefGoogle Scholar
Angles D’Auriac, J.-C., and Sourlas, N. (1997), Europhys. Lett. 39, 473.Google Scholar
Angst, S., Hucht, A., and Wolf, D. E. (2012), Phys. Rev. E 85, 051120.Google Scholar
Baity-Jesi, M., et al. (Janus collaboration). (2013), Phys. Rev. B 88, 224416. [Note: “et al.” stands for 23 names]Google Scholar
Ballesteros, H. G., Cruz, A., Fernandez, L. A., Martin-Mayor, V., Pech, J., Ruiz-Lorenzo, J. J., Tarancon, A., Tellez, P., Ullod, E. L., and Ungil, C. (2000), Phys. Rev. B 62, 14237.Google Scholar
Barash, L. and Shchur, L. N. (2006), Phys. Rev. E 73, 036701.Google Scholar
Barash, L. Y. and Shchur, L. (2013), Comput. Phys. Commun. 184, 2367.Google Scholar
Barash, L. Y. and Shchur, L. (2014), Comput. Phys. Commun. 185, 1343.CrossRefGoogle Scholar
Bathe, M. and Rutledge, G. C. (2003), J. Comput. Chem. 24, 876.Google Scholar
Berg, B. (2004), Markov Chain Monte Carlo Simulations and Their Statistical Analysis (World Scientific, Singapore).Google Scholar
Berg, B. A. and Neuhaus, T. (1991), Phys. Lett. B 267, 241.Google Scholar
Berg, B. A. and Neuhaus, T. (1992), Phys. Rev. Lett. 68, 9.Google Scholar
Bhatt, R. N. and Young, A. P. (1985), Phys. Rev. Lett. 54, 924.Google Scholar
Binder, K. (1977), Z. Phys. B 26, 339.Google Scholar
Binder, K. (1983), in Phase Transitions and Critical Phenomena, Vol. VIII, eds. Domb, C. and Lebowitz, J. L. (Academic Press, London), p. 1.Google Scholar
Binder, K. and Kob, W. (2011), Glussy Materials and Disordered Solids, 2nd edn. (World Scientific, Singapore).CrossRefGoogle Scholar
Binder, K. and Landau, D. P. (1988), Phys. Rev. B 37, 1745.Google Scholar
Binder, K. and Landau, D. P. (1992), J. Chem. Phys. 96, 1444.Google Scholar
Binder, K. and Schröder, K. (1976), Phys. Rev. B 14, 2142.CrossRefGoogle Scholar
Binder, K. and Virnau, P. (2016), J. Chem. Phys. 145, 211701.Google Scholar
Binder, K. and Wang, J.-S. (1989), J. Stat. Phys. 55, 87.Google Scholar
Binder, K. and Young, A. P. (1986), Rev. Mod. Phys. 58, 801.Google Scholar
Binder, K., Evans, R., Landau, D. P., and Ferrenberg, A. M. (1996), Phys. Rev. E 53, 5023.Google Scholar
Binder, K., Landau, D. P., and Wansleben, S. (1989), Phys. Rev. B 40, 6971.Google Scholar
Bortz, A. B., Kalos, M. H., and Lebowitz, J. L. (1975), J. Comput. Phys. 17, 10.CrossRefGoogle Scholar
Boulter, C. J. and Parry, A. O. (1995), Phys. Rev. Lett. 74, 3403.Google Scholar
Brown, F. R. and Woch, T. J. (1987), Phys. Rev. Lett. 58, 2394.Google Scholar
Bryk, P., and Binder, K. (2013), Phys. Rev. E 88, 030401(R).Google Scholar
Campostrini, M., Hasenbusch, M., Pelisetto, A., Rossi, P., and Vicari, E. (2002), Phys. Rev. E 65, 144520.Google Scholar
Cardy, J. L. and Jacobsen, J. L. (1997), Phys. Rev. Lett. 79, 4063.CrossRefGoogle Scholar
Challa, M. S. S. and Hetherington, J. H. (1988), Phys. Rev. Lett. 60, 77.Google Scholar
Chen, K., and Landau, D.P. (1992), Phys. Rev. B 46, 937.Google Scholar
Chen, K., Ferrenberg, A. M., and Landau, D. P. (1993), Phys. Rev. B 48, 3249.CrossRefGoogle Scholar
Chen, S., Ferrenberg, A. M., and Landau, D. P. (1995), Phys. Rev. E 52, 1377.Google Scholar
Compagner, A. (1991), J. Stat. Phys. 63, 883.Google Scholar
Creutz, M. (1980), Phys. Rev. D 21, 2308.CrossRefGoogle Scholar
Creutz, M. (1987), Phys. Rev. D 36, 515.CrossRefGoogle Scholar
Crisanti, A. and Ritort, F. (2003), J. Phys. A 36, R181.Google Scholar
de Miguel, E., Marguta, R. G., and del Rio, E. M. (2007), J. Chem. Phys. 127, 154512.Google Scholar
De Meo, M., D’Onorio, M., Heermann, D., and Binder, K. (1990), J. Stat. Phys. 60, 585.Google Scholar
Dietrich, S. (1988), in Phase Transitions and Critical Phenomena, Vol. XII, eds. Domb, C. and Lebowitz, J. L. (Academic Press, London), p. 1.Google Scholar
Duane, S., Kennedy, A. D., Pendleton, B. J., and Roweth, D. (1987), Phys. Lett. B 195, 216.CrossRefGoogle Scholar
Dukovski, I., Machta, J., and Chayes, L. V. (2002), Phys. Rev. E 65, 026702.Google Scholar
Dünweg, B. and Landau, D. P. (1993), Phys. Rev. B 48, 14182.Google Scholar
Edwards, S. F. and Anderson, P. W. (1975), J. Phys. F 5, 965.Google Scholar
Eisenbach, M., Zhou, C.-G., Nicholson, D. M., Brown, G., Larkin, J., and Schulthess, T. C. (2009), in SC ’09: Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis, ed. ACM (Portland, Oregon), p. 1.Google Scholar
Evertz, H. G. and Landau, D. P. (1996), Phys. Rev. B 54, 12, 302.Google Scholar
Fisher, M. E., and Privman, V. (1985), Phys. Rev. B 32, 447.Google Scholar
Fortuin, C. M. and Kasteleyn, P. W. (1972), Physica 57, 536.Google Scholar
Frenkel, D. and Ladd, A. J. C. (1984), J. Chem. Phys. 81, 3188.Google Scholar
Fukui, K. and Todo, S. J. (2009), Comput. Phys. 228, 2629.Google Scholar
Fytas, N. G. and Martin-Mayor, V. (2013), Phys. Rev. Lett. 110, 227201.Google Scholar
Fytas, N. G., Martin-Mayor, V., Parisi, G., Picco, M., and Sourlas, N. (2019), Phys. Rev. Lett. 122, 240603.Google Scholar
Fytas, N. G., Martin-Mayor, V., Picco, M., and Sourlas, N. (2016), Phys. Rev. Lett. 116, 227201.CrossRefGoogle Scholar
Gerold, V. and Kern, J. (1987), Acta. Metall. 35, 393.Google Scholar
Geyer, C. (1991), in Proceedings of the 23rd Symposium on the Interface, ed. Keramidas, E. M. (Interface Foundation, Fairfax, VA), p. 156.Google Scholar
Goodman, J. and Sokal, A. (1986), Phys. Rev. Lett. 56, 1015.Google Scholar
Griffiths, R. B. (1969), Phys. Rev. Lett. 23, 17.Google Scholar
Hartmann, A. and Rieger, H. (2002), Optimization Algorithms in Physics (Wiley-VCH, Weinheim).Google Scholar
Hartmann, A. K., and Young, A. P. (2001), Phys. Rev. B 64, 214419.Google Scholar
Hasenbusch, M. (1995), Nucl. Phys. B 42, 764.Google Scholar
Hasenbusch, M. and Meyer, S. (1990), Phys. Lett. B 241, 238.Google Scholar
Hasenbusch, M. and Meyer, S. (1991), Phys. Rev. Lett. 66, 530.CrossRefGoogle Scholar
Hatano, N. and Gubernatis, J. E. (1999) AIP Conf. Proc. 469, 565.Google Scholar
Heermann, D. W. and Burkitt, A. N. (1990), in Computer Simulation Studies in Condensed Matter Physics II, eds. Landau, D. P., Mon, K. K., and Schüttler, H.-B. (Springer Verlag, Heidelberg), p. 16.Google Scholar
Heffelfinger, G. S. (2000), Comput. Phys. Commun. 128, 219.Google Scholar
Herrmann, H. J. (1990), in Computer Simulation Studies in Condensed Matter Physics II, eds. Landau, D. P., Mon, K. K., and Schüttler, H.-B. (Springer Verlag, Heidelberg), p. 56.Google Scholar
Heuer, A., Dünweg, B., and Ferrenberg, A. M. (1997), Comput. Phys. Commun. 103, 1.Google Scholar
Holm, C. and Janke, W. (1993), Phys. Lett. A 173, 8.Google Scholar
Hornreich, R. M., Luban, M., and Shtrikman, S. (1975), Phys. Rev. Lett. 35, 1678.Google Scholar
Hucht, A. (2009), Phys. Rev. E 80, 061138.Google Scholar
Hui, K. and Berker, A. N. (1989), Phys. Rev. Lett. 62, 2507.Google Scholar
Hukushima, K. and Kawamura, H. (2000), Phys. Rev. E 61, R1008.Google Scholar
Hukushima, K. and Nemoto, K. (1996), J. Phys. Soc. Japan 65, 1604.Google Scholar
Imry, Y. and Ma, S. (1975), Phys. Rev. Lett. 35, 1399.Google Scholar
Kandel, D., Domany, E., and Brandt, A. (1989), Phys. Rev. B 40, 330.Google Scholar
Kandel, D., Domany, E., Ron, D., Brandt, A., and Loh, E. Jr. (1988), Phys. Rev. Lett. 60, 1591.Google Scholar
Kasteleyn, P. W. and Fortuin, C. M. (1969), J. Phys. Soc. Japan Suppl. 26s, 11.Google Scholar
Katzgraber, H. G., Palassini, M., and Young, A. P. (2001), Phys. Rev. B 63, 184422.Google Scholar
Kawashima, N. and Young, A. P. (1996), Phys. Rev. B 53, R484.Google Scholar
Keen, D. A. and Pusztai, L., eds. (2007), Proceedings of the 3rd Workshop on Reverse Monte Carlo Methods, J. Phys. Condens. Matter 19, issue 33.Google Scholar
Khoshbakht, H. and Weigel, M. (2018), Phys. Rev. B 97, 064410.Google Scholar
Kim, J.-K. (1993), Phys. Rev. Lett. 70, 1735.Google Scholar
Kim, J.-K., de Souza, A. J. F., and Landau, D. P. (1996), Phys. Rev. E 54, 2291.CrossRefGoogle Scholar
Kinzel, W. and Kanter, I. (2003), J. Phys. A: Math. Gen. 36, 11173.Google Scholar
Kirkpatrick, S., Gelatt, S. C. Jr., and Vecchi, M. P. (1983), Science 220, 671.Google Scholar
Kolesik, M., Novotny, M. A., and Rikvold, P. A. (1998), Phys. Rev. Lett. 80, 3384.Google Scholar
Krech, M. (1994), The Casimir Effect in Critical Systems (World Scientific, Singapore).Google Scholar
Krech, M. and Landau, D. P. (1996), Phys. Rev. E 53, 4414.Google Scholar
Landau, D. P. (1992), in The Monte Carlo Method in Condensed Matter Physics, ed. Binder, K. (Springer, Berlin), p. 23.CrossRefGoogle Scholar
Landau, D. P. (1994), Physica A 205, 41.Google Scholar
Landau, D. P. (1996), in Monte Carlo and Molecular Dynamics of Condensed Matter Systems, eds. Binder, K. and Ciccotti, G. (Società Italiana de Fisica, Bologna), p. 181.Google Scholar
Landau, D. P. and Binder, K. (1981), Phys. Rev. B 24, 1391.Google Scholar
Landau, D. P. and Krech, M. (1999), J. Phys. Cond. Mat. 11, 179.Google Scholar
Laradji, M., Landau, D. P., and Dünweg, B. (1995), Phys. Rev. B 51, 4894.Google Scholar
Lee, J. and Kosterlitz, J. M. (1990), Phys. Rev. Lett. 65, 137.Google Scholar
Lee, L. W. and Young, A. P. (2003), Phys. Rev. Lett. 90, 227203.Google Scholar
Lees, A. W. and Edwards, S. F. (1972), J. Phys. C 5, 1921.Google Scholar
Luijten, E. and Blöte, H. W. J. (1995), Int. J. Mod. Phys. C 6, 359.CrossRefGoogle Scholar
Lüscher, M. (1994), Comput. Phys. Commun. 79, 100.Google Scholar
Lyubartsev, A. P., Martsinnovski, A. A., Shevkunov, S. V., and Vorontsov-Velyaminov, P. N. (1992), J. Chem. Phys. 96, 1776.CrossRefGoogle Scholar
Machta, J., Choi, Y. S., Lucke, A., and Schweizer, T. (1995), Phys. Rev. Lett. 75, 2792.Google Scholar
Machta, J., Choi, Y. S., Lucke, A., and Schweizer, T. (1996), Phys. Rev. E 54, 1332.CrossRefGoogle Scholar
Manssen, M., Weigel, M., and Hartmann, A. K. (2012), Eur. Phys. J. Special Topics 210, 53.CrossRefGoogle Scholar
Marinari, E. and Parisi, G. (1992), Europhys. Lett. 19, 451.Google Scholar
Marsaglia, G. (1972), Ann. Math. Stat. 43, 645.CrossRefGoogle Scholar
Matsumoto, M. and Nishimura, T. (1998), ACM Trans. Model. Comput. Simul. 8, 3.CrossRefGoogle Scholar
McGreevy, R. L. (2001), J. Phys.: Condens. Matter 13, R877.Google Scholar
Meirovitch, H. and Alexandrowicz, Z. (1977), Mol. Phys. 34, 1027.Google Scholar
Mertens, S. and Bauke, H. (2004), Phys. Rev. E 69, 055702(R).Google Scholar
Mézard, M. and Montanari, A. (2009), Information, Physics and Computation (Oxford University Press, Oxford).Google Scholar
Milchev, A., Müller, M., Binder, K., and Landau, D. P. (2003a), Phys. Rev. Lett. 90, 136101.Google Scholar
Milchev, A., Müller, M., Binder, K., and Landau, D. P. (2003b), Phys. Rev. E 68, 031601.Google Scholar
Miller, R. G. (1974), Biometrika 61, 1.Google Scholar
Mon, K. K. (1985), Phys. Rev. Lett. 54, 2671.Google Scholar
Mon, K. K., Wansleben, S., Landau, D. P., and Binder, K. (1988), Phys. Rev. Lett. 60, 708.Google Scholar
Mon, K. K., Landau, D. P., and Stauffer, D. (1990), Phys. Rev. B 42, 545.Google Scholar
Nadler, W. and Hansmann, U. H. E. (2007a), Phys. Rev. E 75, 026109.Google Scholar
Nadler, W. and Hansmann, U. H. E. (2007b), Phys. Rev. E 76, 609701.Google Scholar
Nadler, W. and Hansmann, U. H. E. (2008), J. Phys. Chem. B 112, 10386.Google Scholar
Nattermann, T. (1998), In Spin Glasses and Random Fields, ed. Young, A. P., (World Scientific, Singapore), p. 277.Google Scholar
Nishimori, H. (2001), Statistical Physics of Spin Glasses and Information Processing: An Introduction (Oxford University Press, Oxford).Google Scholar
Novotny, M. A. (1995a), Phys. Rev. Lett. 74, 1.CrossRefGoogle Scholar
Novotny, M. A. (1995b), Computers in Physics 9, 46.Google Scholar
Ogielski, A. T. (1985), Phys. Rev. B 32, 7384.Google Scholar
Ogielski, T. (1986), Phys. Rev. Lett. 57, 1251.CrossRefGoogle Scholar
Onuki, A. (1997), J. Phys.: Condens. Matter 9, 6119.Google Scholar
Pang, L., Landau, D. P., and Binder, K. (2011), Phys. Rev. Lett. 106, 236102.Google Scholar
Pang, L., Landau, D. P., and Binder, K. (2019), Phys. Rev. E 100, 023303.Google Scholar
Parisi, G. and Sourlas, N. (1979), Phys. Rev. Lett. 43, 744.Google Scholar
Parry, A. D., Rascón, C., Bernadino, N. R., and Romero-Enrique, J. M. (2008), Phys. Rev. Lett. 100, 136105.Google Scholar
Polson, J. M., Trizac, E., Pronk, S., and Frenkel, D. (2000), J. Chem. Phys. 112, 5339.Google Scholar
Plascak, J.-A., Ferrenberg, A. M., and Landau, D. P. (2002), Phys. Rev. E 65, 066702.CrossRefGoogle Scholar
Rampf, F., Paul, W., and Binder, K. (2005), Europhys. Lett. 70, 628.CrossRefGoogle Scholar
Reuter, K., Stampfl, C., and Scheffler, M. (2005), Handbook of Materials Modeling, ed. Yip, S. (Springer, Berlin), vol. 1, p. 149.Google Scholar
Rosta, E. and Hummer, G. (2009), J. Chem. Phys. 131, 165102.Google Scholar
Rosta, E. and Hummer, G. (2010), J. Chem. Phys. 132, 034102.Google Scholar
Saracco, G. P. and Gonella, G. (2009), Phys. Rev. E 80, 051126.Google Scholar
Sasaki, M. (2009), Phys. Rev. E 82, 031118.Google Scholar
Sasaki, M. and Matsubara, F. (2008), J. Phys. Soc. Japan 77, 024004.Google Scholar
Savvidy, K. G. (2015), Comput. Phys. Commun. 196, 161.Google Scholar
Schilling, T. and Schmid, F. (2009), J. Chem. Phys. 131, 231102.Google Scholar
Schmid, F. and Wilding, N. B. (1995), Int. J. Mod. Phys. C 6, 781.Google Scholar
Schmitz, F., Virnau, P., and Binder, K. (2013), Phys. Rev. E 87, 053302.Google Scholar
Schneider, J. J. and Kirkpatrick, S. (2006), Stochastic Optimization (Springer, Berlin).Google Scholar
Schwartz, M. (1991), Europhys. Lett. 15, 777.Google Scholar
Selke, W. (1988), Phys. Repts. 170, 213.Google Scholar
Selke, W., Shchur, L. N., and Talapov, A. L. (1994), in Annual Reviews of Computer Science I, ed. Stauffer, D. (World Scientific, Singapore) p. 17.Google Scholar
Shchur, L. N. and Butera, P. (1998), Int. J. Mod. Phys. C 9, 607.Google Scholar
Sweeny, M. (1983), Phys. Rev. B 27, 4445.Google Scholar
Swendsen, R. H. and Wang, J.-S. (1986), Phys. Rev. Lett. 57, 2607.Google Scholar
Swendsen, R. H. and Wang, J.-S. (1987), Phys. Rev. Lett. 58, 86.CrossRefGoogle Scholar
Tipton, W. W. and Hennig, R. G. (2013), J. Phys.: Condens. Matter 25, 495401.Google Scholar
Tomita, Y. and Okabe, Y. (2001), Phys. Rev. Lett. 86, 572.Google Scholar
Trobo, M. L., Albano, E. V., and Binder, K. (2018), J. Chem. Phys. 148, 114701.Google Scholar
Tröster, A. (2007), Phys. Rev. B 76, 012402.Google Scholar
Tröster, A. (2008a), Phys. Rev. Lett. 100, 140602.Google Scholar
Tröster, A. (2008b), Comput. Phys. Comm. 179, 30.Google Scholar
Tröster, A. (2013), Phys. Rev. B 87, 104112.CrossRefGoogle Scholar
Tröster, A., Schmitz, F., Virnau, P., and Binder, K. (2018), J. Phys. Chem. B 122, 3407.Google Scholar
Uhlherr, A. (2003), Comput. Phys. Commun. 155, 31.Google Scholar
Urbach, C. (2018), Comput. Phys. Commun. 224, 44.Google Scholar
Vink, R. L. C., Binder, K., and Löwen, H. (2006), Phys. Rev. Lett. 97, 230603.Google Scholar
Vink, R. L. C., Binder, K., and Löwen, H. (2008), J. Phys.: Condens. Matter 20, 404222.Google Scholar
Vink, R. L. C., Fischer, T., and Binder, K. (2010), Phys. Rev. E 82, 051134.Google Scholar
Vogel, T., and Perez, D. (2015), Phys. Rev. Lett. 115, 190602.CrossRefGoogle Scholar
Vollmayr, K., Reger, J. D., Scheucher, M., and Binder, K. (1993), Z. Physik B: Condens. Matter 91, 113.Google Scholar
Wang, J.-S., Swendsen, R. H., and Kotecký, R. (1989), Phys. Rev. Lett. 63, 109.Google Scholar
Wang, J.-S. (1989), Physica A 161, 249Google Scholar
Wang, J.-S. (1990), Physica A 164, 240.Google Scholar
Wansleben, S. (1987), Comput. Phys. Commun. 43, 315.Google Scholar
Winter, D., Virnau, P., Horbach, J., and Binder, K. (2010), Europhys. Lett. 91, 60002.Google Scholar
Wiseman, S. and Domany, E. (1995), Phys. Rev. E 52, 3469.CrossRefGoogle Scholar
Wiseman, S. and Domany, E. (1998), Phys. Rev. Lett. 81, 22.Google Scholar
Wolff, U. (1988), Nucl. Phys. B 300, 501.Google Scholar
Wolff, U. (1989a), Phys. Rev. Lett. 62, 361.Google Scholar
Wolff, U. (1989b), Nucl. Phys. B 322, 759.Google Scholar
Wolff, U. (1990), Nucl. Phys. B 334, 581.Google Scholar
Wu, Y., and Machta, J. (2006), Phys. Rev. B 74, 064418.Google Scholar
Xu, J., Tsai, S.-H., Landau, D. P., and Binder, K. (2018), Phys. Rev. E 99, 023309.Google Scholar
Young, A. P. (1996), in Monte Carlo and Molecular Dynamics of Condensed Matter Systems, eds. Binder, K. and Ciccotti, G. (Società Italiana di Fisica, Bologna), p. 285.Google Scholar
Young, A. P. (ed.) (1998), Spin Glasses and Random Fields (World Scientific, Singapore).Google Scholar
Zorn, R., Herrmann, H. J., and Rebbi, C. (1981), Comput. Phys. Commun. 23, 337.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×