Some recent rigorous results in the theory of phase transitions and critical phenomena

 Access to full text

 Full (PDF)

1. [1] K.G. Wilson, Phys. Rev.B4, 3174 and 3184 (1971). K.G. Wilson and M.E. Fisher, Phys. Rev. Letters28, 240 (1972). 
2. [2] M. Feigenbaum, J. Stat. Phys.19, 25 (1978) and 21, 669 (1979). 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. MR501179
3. [3] K.G. Wilson and J. Kogut, Phys. Rep.12C, 75 (1974), and refs. given there. 
4. L. Kadanoff, Physics2, 263 (1966). 
5. L. Kadanoff, A. Haughton and M. Yalabik, J. Stat. Phys.14, N° 2, 171 (1976). MR456229
6. G. Jona-Lasinio, Nuovo CimentoB26, 9 (1975). MR368667
7. M. Cassandro and G. Jona-Lasinio, Adv. Physics27, 913 (1978). 
8. M.E. Fisher, Rev. Mod. Phys.46, 597 (1974). 
9. E. Br Ézin, 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. 
10. E. Brézin and J. Zinn-Justin, Phys. Rev.B14, 3110 (1976). etc. 
11. [4] "Phase Transitions and Critical Phenomena", vol. 1-6, C. Domb and M.S. Green (eds.), New York, Academic Press. Zbl1276.82001
12. S.K. Ma, "Modern Theory of Critical Phenomena", London-Amsterdam-DonMills, Ontario, Benjamin1976. 
13. G. Toulouse and P. Pfeuty, "Introduction au Groupe de Renormalisation et à Ses Applications", Presses Universitaires de Grenoble, 1975. 
14. C.J. Thompson, "Mathematical Statistical Mechanics", Princeton, Princeton University Press, 1980. Zbl0417.60096MR548873
15. [5] N.D. Mermin, Rev. Mod. Phys.51, 591 (1979). L. Michel, Rev. Mod. Phys.52, 617 (1980). MR541885
16. [6] R. Streater and A.S. Wightman, "PCT, Spin and Statistics and All That", New York, Benjamin, 1964. Zbl0135.44305MR161603
17. R. Jost, "The General Theory of Quantized Fields", Providence, R.I., Amer. Math. Soc. Publ., 1965. Zbl0127.19105MR177667
18. [7] J. Fröhlich, K. Osterwalder and E. Seiler, article to appear. 
19. [8] K. Osterwalder and R. Schrader, Commun. Math. Phys.31, 83 (1973), and 42, 281 (1975). 
20. [9] J.-P. Eckmann and H. Epstein, Commun. Math. Phys.64, 95-130 (1979). MR519920
21. [10] J. Fröhlich, Ann. Inst. H. Poincaré, Sect. A 21, 271 (1974). W. Driessler and J. Fröhlich, Ann. Inst. H. Poincaré, Sect. A 27, 221 (1977). MR373494
22. [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. MR508189
23. 2) J. Fröhlich and E.H. Lieb, Commun. Math. Phys.60, 233 (1978). MR523697
24. 3) J. Fröhlich, R. Israel, E.H. Lieb and B. Simon, Commun. Math. Phys.62, 1 (1978). MR506363
25. [12] 1) "Constructive Quantum Field Theory", G. Velo and A.S. Wightman (eds.), Lecture Notes in Physics25, Berlin-Heidelberg-New York, Springer-Verlag, 1973. (See, in particular, the contribution by E. Nelson). Zbl0325.00006MR395513
26. 2) F. Guerra, L. Rosen and B. Simon, Ann. Math.101, 111 (1975). MR378670
27. 3) B. Simon, "The P(φ)2 Euclidean (Quantum) Field Theory", Princeton, Princeton University Press1974. 
28. [13] D. Ruelle, "Statistical Mechanics", London-Amsterdam-DonMills, Ontario, Benjamin, 1969. Zbl0177.57301MR1747792
29. [14] R. Israel, "Convexity in the Theory of Lattice Gases", Princeton, PrincetonUniversity Press, 1979. Zbl0399.46055MR517873
30. [15] J. Fröhlich and T. Spencer, Phys. Rev. Letters46, 1006 (1981), and Commun. Math. Phys.81, 277 (1981). MR607429
31. [16] R. Kotecky and S. Shlosman, Preprint 1981, to appear in Commun. Math. Phys. 
32. [17] "Phase Transitions and Critical Phenomena", vol. 6 : See ref. [4]. "New Developments in Quantum Field Theory..." : See ref. [11], 1). 
33. [18] E.H. Lieb and A. Sokal, Commun. Math. Phys.80, 153 (1981). MR623156
34. [19] F. Wegner, Phys. Rev.B5, 4529 (1972). E. Brézin, J.C. Le Guillou and J. Zinn-Justin : See ref. [3]. 
35. [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. MR673762
36. J. Glimm and A. Jaffe, Commun. Math. Phys.51, 1-14 (1976), and 52, 203 (1977). MR432054
37. [21] A. Sokal, More Inequalities for Critical Exponents, J. Stat. Phys., to appear ; Phys. Letters71A, 451 (1979). MR610690
38. [22] M. Aizenman, Phys. Rev. Letters47, 1 (1981) ; article in preparation, (to appear in Commun. Math. Phys.). MR620135
39. [23] J. Fröhlich, Nuclear PhysicsB200[FS4], 281 (1982) ; and MR643591
40. D. Brydges, J. Fröhlich and A. Sokal, article in preparation. 
41. [24] M. Jimbo, T. Miwa and M. Sato, in "Mathematical Problems in Theoretical Physics", K. Osterwalder (ed.), Lecture Notes in Physics116, Berlin-Heidelberg- New York, Springer-Verlag, 1980 ; and refs. given there. Zbl0425.00027MR579762
42. [25] E.K. Sklyanin, L.A. Takhtajan and L.D. Faddeev, Theor. Math. Phys.40, 688 (1979). 
43. V. E. Korepin, Commun. Math. Phys.76, 165 (1980). MR587509
44. L.D. Faddeev, Physica Scripta24, 832 (1981). Zbl1063.81643MR639411
45. [26] H.B. Thacker, Rev. Mod. Phys.53, N° 2, 253 (1981), and refs. given there. MR611318
46. N. Andrej and J.H. Lowenstein, Phys. Rev. Letters43, 1698 (1979), Phys. Letters90B, 106 (1980) and 91B, 401 (1980). 
47. [27] E.H. Lieb, Physica73, 226-236 (1974). 
48. [28] R. Peierls, Proc. Cambridge Phil. Soc.32, 477 (1936). Zbl0014.33604
49. [29] R.B. Griffiths, Phys. Rev. A136, 437 (1964). Zbl0129.23205MR189681
50. R.L. Dobrushin, Soviet Phys. Doklady10, 111 (1965). MR182405
51. [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) ; Theor. Math. Phys.25, 1185 (1975) ; Theor. Math. Phys.26, 61 (1976). 
52. [31] J. Glimm, A. Jaffe and T. Spencer, Commun. Math. Phys.45, 203 (1975). Zbl0956.82501MR391797
53. [32] J. Glimm, A. Jaffe and T. Spencer, Ann. Phys. (NY)101, 610 (1976), 101, 631 (1976). 
54. [33] J. Fröhlich, R. Israel, E.H. Lieb and B. Simon, J. Stat. Phys.22, 297 (1980). MR570370
55. [34] D. Brydges, Commun. Math. Phys.58, 313 (1978). MR475434
56. D. Brydges and P. Federbush, Commun. Math. Phys.73, 197 (1980). MR574172
57. J. Imbrie, Commun. Math. Phys.82, 261 (1981), 82, 305 (1981). 
58. [35] J. Fröhlich and T. Spencer, "Massless Phases and Symmetry Restoration in Abelian Gauge Theories and Spin Systems", Commun. Math. Phys.83, 411 (1982). MR649811
59. [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.. Zbl1110.82302
60. [37] J.V. José, L.P. Kadanoff, S. Kirkpatrick, D.R. Nelson, Phys. Rev.B16, 1217 (1977). MR589901
61. [38] N.D. Mermin and H. Wagner, Phys. Rev. Letters17, 1132 (1966). 
62. N.D. Mermin, J. Math. Phys.8, 1061 (1967). 
63. O. McBryan and T. Spencer, Commun. Math. Phys.53, 99 (1977). MR441179
64. [39] D. Brydges : 
65. } See ref. [34]. D. Brydges and P. Federbush : 
66. [40] J. Fröhlich, B. Simon and T. Spencer, Commun. Math. Phys.50, 79 (1976). MR421531
67. [41] J. Fröhlich, Bull. Amer. Math. Soc. 84, 165 (1978). MR475121
68. [42] M. Reed and B. Simon, "Methods of Modern Mathematical Physics", vol. 2, pg. 70, New York-San Francisco-London, Academic Press, 1975. Zbl0308.47002MR493420
69. [43] J. Fröhlich and C. Pfister, Commun. Math. Phys.81, 277 (1981). MR632763
70. [44] J.L. Lebowitz and A. Martin-Löf, Commun. Math. Phys.25, 276 (1972). MR312854
71. [45] A. Messager, S. Miracle-Sole and C. Pfister, Commun. Math. Phys.58, 19 (1978). MR475552
72. [46] J.L. Lebowitz, J. Stat. Phys.16, 463 (1977). MR456178
73. [47] M. Aizenman, Commun. Math. Phys.73, 83 (1980). MR573615
74. [48] J. Fröhlich and C. Pfister, article to appear. 
75. [49] P.M. Bleher and Ya.G. Sinai, Commun. Math. Phys.33, 23 (1973), 45, 347 (1975). MR1552598
76. See also : P. Collet and J.-P. Eckmann, "A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics", Lecture Notes in Physics74, Berlin-Heidelberg-New York, Springer-Verlag, 1978. MR503070
77. [50] J. Glimm and A. Jaffe, "Quantum Physics", Berlin-Heidelberg-New York, Springer-Verlag, 1981. Zbl0461.46051MR628000
78. [51] K. Gawedzki and A. Kupiainen, Commun. Math. Phys.77, 31 (1980), 82, 407 (1981); articles to appear in Commun. Math. Phys. (1982). MR588686
79. [52] J. Magnen and R. Sénéor, article to appear. 
80. [53] A. Sokal, "An Alternate Constructive Approach to the φ43 Quantum Field Theory,..", Princeton Ph D thesis1981, to appear in Ann. Inst. H. Poincaré. 
81. [54] Ya.G. Sinai, in "Mathematical Problems in Theoretical Physics", G. Dell'Antonio, S. Doplicher and G. Jona-Lasinio (eds.), Lecture Notes in Physics80, Springer-Verlag1978. Ya.G. Sinai, Theor. Prob. Appl.21, 64 (1976). Zbl0416.46054MR504505
82. [55] G. Jona-Lasinio, in "New Developments in Quantum Field Theory..." : See ref. [111, 1). Zbl0903.58039
83. [56] H. van Beyeren, G. Gallavotti and H. Knops, Physica78, 541 (1974). G. Gallavotti and H. Knops, Nuovo Cimento5, 341 (1975). 
84. [57] G. Jona-Lasinio, Nuovo CimentoB26, 9 (1975). G. Gallavotti and G. Jona-Lasinio, Commun. Math. Phys.41, 301 (1975). Zbl0343.60043
85. [58] G. Benfatto, M. Cassandro, G. Gallavotti et al., Commun. Math. Phys.59, 143 (1978), 71, 95 (1980). 
86. [59] J. Glimm and A. Jaffe, Fortschritte der Physik, 21, 327 (1973). MR408581
87. [60] T. Balaban, in "Mathematical Problems in Theoretical Physics", Lecture Notes in Physics116 (see ref. [24]) ; Aarhus preprint, Jan. 1982. 
88. [61] K. Gawedzki and A. Kupiainen, Commun. Math. Phys.77, 31 (1980). MR588686
89. [62] K. Gawedzki and A. Kupiainen, ref. [51]. 
90. [63] J. Magnen and R. Sénéor, ref. [52]. 
91. [64] K. Symanzik, in "Local Quantum Theory", R. Jost (ed.), New York-London, Academic Press, 1969. 
92. [65] D. Brydges, J. Fröhlich and T. Spencer, Commun. Math. Phys.83, 123 (1982) ; MR648362
93. [66] J. Fröhlich, Nuclear Physics B200[FS4], 281 (1982). MR643591
94. [67] R.B. Griffiths, J. Math. Phys.8, 478 (1967), 8, 484 (1967), Commun. Math. Phys.6, 121 (1967). MR1552524
95. J. Ginibre, Commun. Math. Phys.16, 310 (1970). MR269252
96. P. Cartier, Séminaire Bourbaki n° 431 (1972/73). 
97. [68] J. Lebowitz, Commun. Math. Phys.35, 87 (1974). MR339738
98. [69] J. Glimm and A. Jaffe, Ann. Inst. H. Poincaré, Sect. A, 22, 97 (1975). MR384012
99. [70] G. Lawler, Duke Math. Journal47, 655 (1980) ; Preprint, Courant Institute1982. Zbl0445.60058MR587173
100. [71] A. Dvoretzky, P. Erdös and S. Kakutani, Acta Sci. Math. Szeged12B, 75 (1950). MR34972