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References

  • Aizenman, M.; Chayes, J. T.; Chayes, L.; Newman, C. M. Discontinuity of the magnetization in one-dimensional $1/\vert x-y\vert ^ 2$ Ising and Potts models. J. Statist. Phys. 50 (1988), no. 1-2, 1--40. MR0939480
  • Aizenman, Michael; Grimmett, Geoffrey. Strict monotonicity for critical points in percolation and ferromagnetic models. J. Statist. Phys. 63 (1991), no. 5-6, 817--835. MR1116036
  • Bezuidenhout, C. E.; Grimmett, G. R.; Kesten, H. Strict inequality for critical values of Potts models and random-cluster processes. Comm. Math. Phys. 158 (1993), no. 1, 1--16. MR1243713
  • Bricmont, J.; Kupiainen, A. Phase transition in the $3$d random field Ising model. Comm. Math. Phys. 116 (1988), no. 4, 539--572. MR0943702
  • Campanino, M. Strict inequality for critical percolation values in frustrated random-cluster models. Markov Process. Related Fields 4 (1998), no. 3, 395--410. MR1670039
  • E. De Santis, Strict inequalities in phase transition between ferromagnetic and frustrated systems. Ph.D. thesis, Rome, 1998.
  • De Santis, E.; Gandolfi, A. Bond percolation in frustrated systems. Ann. Probab. 27 (1999), no. 4, 1781--1808. MR1742888
  • Diestel, Reinhard. Graph theory. Translated from the 1996 German original. Graduate Texts in Mathematics, 173. Springer-Verlag, New York, 1997. xiv+289 pp. ISBN: 0-387-98210-8 MR1448665
  • R. Dobrushin, Gibbs state describing coexistence of phases for a three dimensional Ising model. Theory Prob. Appl., 13% , 1972, pp. 582-601.
  • S. Edwards, P. Anderson, Theory of spin glasses. J. Phys. F 5, 1975, pp. 965-974.
  • Edwards, Robert G.; Sokal, Alan D. Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D (3) 38 (1988), no. 6, 2009--2012. MR0965465
  • Fortuin, C. M.; Kasteleyn, P. W.; Ginibre, J. Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22 (1971), 89--103. MR0309498
  • Fortuin, C. M.; Kasteleyn, P. W. On the random-cluster model. I. Introduction and relation to other models. Physica 57 (1972), 536--564. MR0359655
  • Fortuin, C. M. On the random-cluster model. II. The percolation model. Physica 58 (1972), 393--418. MR0378660
  • Fortuin, C. M. On the random-cluster model. III. The simple random-cluster model. Physica 59 (1972), 545--570. MR0432137
  • A. Gandolfi, Inequalities for critical points in disordered ferromagnets. preprint 1998.
  • Gandolfi, A.; Keane, M. S.; Newman, C. M. Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probab. Theory Related Fields 92 (1992), no. 4, 511--527. MR1169017
  • Grimmett, Geoffrey. Percolation. Springer-Verlag, New York, 1989. xii+296 pp. ISBN: 0-387-96843-1 MR0995460
  • Grimmett, Geoffrey. Potts models and random-cluster processes with many-body interactions. J. Statist. Phys. 75 (1994), no. 1-2, 67--121. MR1273054
  • Grimmett, Geoffrey. The stochastic random-cluster process and the uniqueness of random-cluster measures. Ann. Probab. 23 (1995), no. 4, 1461--1510. MR1379156
  • Grimmett, Geoffrey R. Inequalities and entanglements for percolation and random-cluster models. Perplexing problems in probability, 91--105, Progr. Probab., 44, Birkhäuser Boston, Boston, MA, 1999. MR1703126
  • E. Ising, Beitrag zur Theorie des Ferromagnetismus. Z. Phys., 31, 1925, pp. 253-258. omment
  • Y. Kasai, A. Okji, Percolation problem describing pm J Ising spin glass system. Progress in Theoretical Physics 79, 1988, pp. 1080-1094.
  • P. Kasteleyn, C. Fortuin, Phase transitions in lattice systems with random local properties. J. Phys. Soc. Japan 26 (Suppl.) 1969, pp. 11-14.
  • Lebowitz, Joel L.; Martin-Löf, Anders. On the uniqueness of the equilibrium state for Ising spin systems. Comm. Math. Phys. 25 (1972), 276--282. MR0312854
  • Liggett, Thomas M. Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4 MR0776231
  • Lindvall, Torgny. Lectures on the coupling method. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1992. xiv+257 pp. ISBN: 0-471-54025-0 MR1180522
  • Newman, Charles M. Disordered Ising systems and random cluster representations. Probability and phase transition (Cambridge, 1993), 247--260, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 420, Kluwer Acad. Publ., Dordrecht, 1994. MR1283186
  • Newman, Charles M. Topics in disordered systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1997. viii+88 pp. ISBN: 3-7643-5777-0 MR1480664
  • E. Olivieri, J.F. Perez, S. Goulart Rosa Jr., Some rigorous results on the phase diagram of the dilute Ising model. Phys. lett., 94A, 1983, pp. 309-311.
  • Strassen, V. The existence of probability measures with given marginals. Ann. Math. Statist. 36 1965 423--439. MR0177430
  • R. Swendsen, J. Wang, Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett., 58 , 1987, pp. 86-88.


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