Some Recent Rigorous Results in the Theory of Phase Transitions and Critical Phenomena
Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Volume 30 (1982), Talk no. 2, 42 p.
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     publisher = {Institut de Recherche Math\'ematique Avanc\'ee - Universit\'e Louis Pasteur},
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     url = {http://www.numdam.org/item/RCP25_1982__30__42_0/}
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Fröhlich, Jürg; Spencer, Thomas. Some Recent Rigorous Results in the Theory of Phase Transitions and Critical Phenomena. Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Volume 30 (1982), Talk no. 2, 42 p. http://www.numdam.org/item/RCP25_1982__30__42_0/

[1] K.G. Wilson, Phys. Rev. B4, 3174 and 3184 (1971).

K.G. Wilson and M.E. Fisher, Phys. Rev. Letters 28, 240 (1972).

[2] M. Feigenbaum, J. Stat. Phys. 19, 25 (1978) | MR | Zbl

M. Feigenbaum J. Stat. Phys. 21, 669 (1979). | MR | Zbl

O.E. Lanford Iii Séminaire Bourbaki n° 563, November 1980.

P. Collet and J.-P. Eckmann, "Iterated Maps of the Interval as Dynamical Systems", Progress in Physics, vol. 1, Boston-Basel-Stuttgart, Birkhäuser, 1980. | MR | Zbl

[3] K.G. Wilson and J. Kogut, Phys. Rep. 12C, 75 (1974) refs. given there.

L. Kadanoff, Physics 2, 263 (1966).

L. Kadanoff, A. Haughton and M. Yalabik, J. Stat. Phys. 14, N° 2, 171 (1976). | MR

G. Jona-Lasinio, Nuovo Cimento B 26, 9 (1975). | MR

M. Cassandro and G. Jona-Lasinio, Adv. Physics 27, 913 (1978).

M.E. Fisher, Rev. Mod. Phys. 46, 597 (1974).

E. Brézin, J.C. Le Guillou and J. Zinn-Justin, in "Phase Transitions and Critical Phenomena", vol. 6, C. Domb and M.S. Green (eds.) New-York, Academic Press, 1976. | Zbl

E. Brézin and J. Zinn-Justin, Phys. Rev. B14, 3110 (1976). etc.

[4] "Phase Transitions and Critical Phenomena", vol. 1-6, C. Domb and M.S. Green (eds.), New York, Academic Press. | Zbl

S.K. Ma, "Modern Theory of Critical Phenomena", London-Amsterdam-DonMills, Ontario, Benjamin 1976.

G. Toulouse and P. Pfeuty, "Introduction au Groupe de Renormalisation et à Ses Applications", Presses Universitaires de Grenoble, 1975.

C.J. Thompson, "Mathematical Statistical Mechanics", Princeton, Princeton University Press, 1980. | MR | Zbl

[5] N.D. Mermin, Rev. Mod. Phys. 51, 591 (1979). | MR

L. Michel, Rev. Mod. Phys. 52, 617 (1980). | MR

[6] R. Streater and A.S. Wightman, "PCT, Spin and Statistics and All That", New York, Benjamin, 1964. | MR | Zbl

R. Jost, "The General Theory of Quantized Fields", Providence, R.I., Amer. Math. Soc. Publ., 1965. | MR | Zbl

[7] J. Fröhlich, K. Osterwalder and E. Seiler, article to appear.

[8] K. Osterwalder and R. Schrader, Commun. Math. Phys. 31, 83 (1973) | MR | Zbl

K. Osterwalder and R. Schrader, Commun. Math. Phys. 42, 281 (1975). | MR | Zbl

[9] J.-P. Eckmann and H. Epstein, Commun. Math. Phys. 64, 95-130 (1979). | MR

[10] J. Fröhlich, Ann. Inst. H. Poincaré, Sect. A 21, 271 (1974). | MR

W. Driessler and J. Fröhlich, Ann. Inst. H. Poincaré, Sect. A 27, 221 (1977). | Numdam | MR | Zbl

[11] 1) J. Fröhlich and T. Spencer, in "New Developments in Quantum Field Theory and Statistical Mechanics", M. Lévy and P. Mitter (eds.), New York-London, Plenum, 1977. | MR

2) J. Fröhlich and E.H. Lieb, Commun. Math. Phys. 60, 233 (1978). | MR

3) J. Fröhlich, R. Israel, E.H. Lieb and B. Simon, Commun. Math. Phys. 62, 1 (1978). | MR

[12] 1) "Constructive Quantum Field Theory", G. Velo and A.S. Wightman (eds.), Lecture Notes in Physics 25, Berlin-Heidelberg-New York, Springer-Verlag, 1973. (See, in particular, the contribution by E. Nelson). | MR | Zbl

2) F. Guerra, L. Rosen and B. Simon, Ann. Math. 101, 111 (1975). | MR

3) B. Simon, "The P(φ) 2 Euclidean (Quantum) Field Theory", Princeton, Princeton University Press 1974. | MR | Zbl

[13] D. Ruelle, "Statistical Mechanics", London-Amsterdam-DonMills, Ontario, Benjamin, 1969. | MR

[14] R. Israel, "Convexity in the Theory of Lattice Gases", Princeton, Princeton University Press, 1979. | MR | Zbl

[15] J. Fröhlich and T. Spencer, Phys. Rev. Letters 46, 1006 (1981) | MR

J. Fröhlich and T. Spencer, Commun. Math. Phys. 81, 277 (1981). | MR

[16] R. Kotecky and S. Shlosman, Preprint 1981, to appear in Commun. Math. Phys.

[17] "Phase Transitions and Critical Phenomena", vol. 6 : See ref. [4]. | Zbl

"New Developments in Quantum Field Theory..." : See ref. [11], 1)

[18] E.H. Lieb and A. Sokal, Commun. Math. Phys. 80, 153 (1981). | MR

[19] F. Wegner, Phys. Rev. B5, 4529 (1972).

E. Brézin, J.C. Le Guillou and J. Zinn-Justin : See ref. [3].

[20] J. Glimm and A. Jaffe, in "Quantum Dynamics : Models and Mathematics", L. Streit (ed.), Berlin - Heidelberg - New York, Springer-Verlag, 1976 and refs. given there. | MR

J. Glimm and A. Jaffe, Commun. Math. Phys. 51, 1-14 (1976) | MR

J. Glimm and A. Jaffe, Commun. Math. Phys. 52, 203 (1977). | MR

[21] A. Sokal, More Inequalities for Critical Exponents, J. Stat, Phys., to appear ; Phys. Letters 71A, 451 (1979). | MR

[22] M. Aizenman, Phys. Rev. Letters 47, 1 (1981) ; article in preparation, (to appear in Commun. Math. Phys.). | MR

[23] J. Fröhlich, Nuclear Physics B200[FS4], 281 (1982)

D. Brydges, J. Fröhlich and A. Sokal, article in preparation.

[24] M. Jimbo, T. Miwa and M. Sato, in "Mathematical Problems in Theoretical Physics" K. Osterwalder (ed.), Lecture Notes in Physics 116, Berlin-Heidelberg-New York, Springer-Verlag, 1980 ; and refs. given there. | MR | Zbl

[25] E.K. Sklyanin, L.A. Takhtajan and L.D. Faddeev, Theor. Math. Phys. 40, 688 (1979).

V.E. Korepin, Commun. Math. Phys. 76, 165 (1980). | MR

L.D. Faddeev, Physica Scripta 24, 832 (1981). | MR | Zbl

[26] H.B. Thacker, Rev. Mod. Phys. 53, N° 2, 253 (1981), and refs. given there. | MR

N. Andrej and J.H. Lowenstein, Phys. Rev. Letters 43, 1698 (1979),

N. Andrej and J.H. Lowenstein Phys. Letters 90B, 106 (1980) | MR

N. Andrej and J.H. Lowenstein Phys. Letters 91B, 401 (1980). | MR

[27] E.H. Lieb, Physica 73, 226-236 (1974).

[28] R. Peierls, Proc. Cambridge Phil. Soc. 32, 477 (1936). | Zbl

[29] R.B. Griffiths, Phys. Rev. A 136, 437 (1964). | MR

R.L. Dobrushin, Soviet Phys. Doklady 10, 111 (1965). | MR

[30] R.A. Minlos and Ya. G. Sinai, Theor. Math. Phys. 2, 167 (1970).

S.A. Pirogov and Ya. G. Sinai, Funct. Anal. Appl. 8, 21 (1974) ;

S.A. Pirogov and Ya. G. Sinai, Theor. Math. Phys. 25, 1185 (1975)

S.A. Pirogov and Ya. G. Sinai, Theor. Math. Phys. 26, 61 (1976).

[31] J. Glimm, A. Jaffe and T. Spencer, Commun. Math. Phys. 45, 203 (1975). | MR | Zbl

[32] J. Glimm, A. Jaffe and T. Spencer, Ann. Phys. (NY) 101, 610 (1976), | MR

J. Glimm, A. Jaffe and T. Spencer, Ann. Phys. (NY) 101, 631 (1976) | MR

[33] J. Fröhlich, R. Israel, E.H. Lieb and B. Simon, J. Stat. Phys. 22, 297 (1980). | MR

[34] D. Brydges, Commun. Math. Phys. 58, 313 (1978). | MR

D. Brydges and P. Federbush, Commun. Math. Phys. 73, 197 (1980). | MR

J. Imbrie, Commun. Math. Phys. 82, 261 (1981), | MR

J. Imbrie 82, 305 (1981). | MR

[35] J. Fröhlich and T. Spencer, "Massless Phases and Symmetry Restoration in Abelian Gauge Theories and Spin Systems", Commun. Math. Phys. 83, 411 (1982). | MR

[36] J. Fröhlich and T. Spencer, "The Phase Transition in the One-Dimensional Ising Model with 1/r2 Interaction Energy", to appear in Commun. Math. Phys. | MR | Zbl

[37] J.V. José, L.P. Kadanoff, S. Kirkpatrick, D.R. Nelson, Phys. Rev. B16, 1217 (1977)

[38] N.D. Mermin and H. Wagner, Phys. Rev. Letters 17, 1132 (1966).

N.D. Mermin, J. Math. Phys. 8, 1061 (1967).

O. Mcbryan and T. Spencer, Commun. Math. Phys. 53, 99 (1977). | MR

[39] D. Brydges : See ref. [34].

D. Brydges and P. Federbush : See ref. [34].

[40] J. Fröhlich, B. Simon and T. Spencer, Commun. Math. Phys. 50, 79 (1976). | MR

[41] J. Fröhlich, Bull. Amer. Math. Soc. 84, 165 (1978). | MR

[42] M. Reed and B. Simon, "Methods of Modern Mathematical Physics", vol. 2, pg. 70, New York-San Francisco-London, Academic Press, 1975. | MR | Zbl

[43] J. Fröhlich and C. Pfister, Commun. Math. Phys. 81, 277 (1981). | MR

[44] J.L. Lebowitz and A. Martin-Löf, Commun. Math. Phys. 25, 276 (1972). | MR

[45] A. Messager, S. Miracle-Sole and C. Pfister, Commun. Math. Phys. 58, 19 (1978). | MR

[46] J.L. Lebowitz, J. Stat. Phys. 16, 463 (1977). | MR

[47] M. Aizenman, Commun. Math. Phys. 73, 83 (1980). | MR

[48] J. Fröhlich and C. Pfister, article to appear.

[49] P.M. Bleher and Ya. G. Sinai, Commun. Math. Phys. 33, 23 (1973), | MR

P.M. Bleher and Ya. G. Sinai, Commun. Math. Phys. 45, 347 (1975). | MR

See also : P. Collet and J.-P. Eckmann, "A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics", Lecture Notes in Physics 74, Berlin-Heidelberg-New York, Springer-Verlag, 1978. | MR

[50] J. Glimm and A. Jaffe, "Quantum Physics", Berlin-Heidelberg-New York, Springer-Verlag, 1981. | MR | Zbl

[51] K. Gawedzki and A. Kupiainen, Commun. Math. Phys. 77, 31 (1980), | MR

K. Gawedzki and A. Kupiainen, Commun. Math. Phys. 82, 407 (1981) ; | MR

K. Gawedzki and A. Kupiainen, articles to appear in Commun. Math. Phys. (1982). | MR

[52] J. Magnen and R. Sénéor, article to appear.

[53] A. Sokal, "An Alternate Constructive Approach to the ϕ 3 4 Quantum Field Theory,..", Princeton Ph D thesis 1981, to appear in Ann. Inst. H. Poincaré. | MR

[54] Ya. G. Sinai, in "Mathematical Problems in Theoretical Physics", G. Dell'Antonio, S. Doplicher and G. Jona-Lasinio (eds.), Lecture Notes in Physics 80, Springer-Verlag 1978. | MR | Zbl

Ya. G. Sinai, Theor. Prob. Appl. 21, 64 (1976). | Zbl

[55] G. Jona-Lasinio, in "New Developments in Quantum Field Theory..." : See ref. [11], 1)

[56] H. Van Beyeren, G. Gallavotti and H. Knops, Physica 78, 541 (1974).

G. Gallavotti and H. Knops, Nuovo Cimento 5, 341 (1975).

[57] G. Jona-Lasinio, Nuovo Cimento B26, 9 (1975). | MR

G. Gallavotti and G. Jona-Lasinio, Commun. Math. Phys. 41, 301 (1975). | MR | Zbl

[58] G. Benfatto, M. Cassandro, G. Gallavotti et al., Commun. Math. Phys. 59, 143 (1978), | MR | Zbl

G. Benfatto, M. Cassandro, G. Gallavotti et al., Commun. Math. Phys. 71, 95 (1980). | MR | Zbl

[59] J. Glimm and A. Jaffe, Fortschritte der Physik, 21, 327 (1973). | MR

[60] T. Balaban, in "Mathematical Problems in Theoretical Physics", Lecture Notes in Physics 116 (see ref. [24]) ; Aarhus preprint, Jan. 1982.

[61] K. Gawedzki and A. Kupiainen, Commun. Math. Phys. 77, 31 (1980). | MR

[62] K. Gawedzki and A. Kupiainen, ref. [51].

[63] J. Magnen and R. Sénéor, ref. [52].

[64] K. Symanzik, in "Local Quantum Theory", R. Jost (ed.), New York-London, Academic Press, 1969.

[65] D. Brydges, J. Fröhlich and T. Spencer, Commun. Math. Phys. 83, 123 (1982) ; | MR

[66] J. Fröhlich, Nuclear Physics B200[FS4], 281 (1982). | MR

[67] R.B. Griffiths, J. Math. Phys. 8, 478 (1967),

R.B. Griffiths, J. Math. Phys. 8, 484 (1967),

R.B. Griffiths Commun. Math. Phys. 6, 121 (1967). | MR

J. Ginibre, Commun. Math. Phys. 16, 310 (1970). | MR

P. Cartier, Séminaire Bourbaki n° 431 (1972/73). | MR

[68] J. Lebowitz, Commun. Math. Phys. 35, 87 (1974). | MR

[69] J. Glimm and A. Jaffe, Ann. Inst. H. Poincaré, Sect. A, 22, 97 (1975). | Numdam | MR

[70] G. Lawler, Duke Math. Journal 47, 655 (1980) ; | MR | Zbl

G. Lawler, Duke Math. Journal Preprint, Courant Institute 1982.

[71] A. Dvoretzky, P. Erdös and S. Kakutani, Acta Sci. Math. Szeged 12B, 75 (1950). | MR