Abstract
We prove that the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian. A similar statement is proven for the $\lambda \phi^4$ fields over $\mathbb{R}^4$ with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models’ random current representation, in which the correlation functions’ deviation from Wick’s law is expressed in terms of intersection probabilities of random currents with sources at distances that are large on the model’s lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.
-
[Aiz82]
M. Aizenman, "Geometric analysis of $\varphi ^{4}$ fields and Ising models. I, II," Comm. Math. Phys., vol. 86, iss. 1, pp. 1-48, 1982.
@ARTICLE{Aiz82,
author = {Aizenman, Michael},
title = {Geometric analysis of {$\varphi \sp{4}$} fields and {I}sing models. {I},
{II}},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {86},
year = {1982},
number = {1},
pages = {1--48},
issn = {0010-3616},
mrclass = {81E25 (82A68)},
mrnumber = {0678000},
mrreviewer = {C. A. Hurst},
doi = {10.1007/BF01205659},
url = {https://doi.org/10.1007/BF01205659},
zblnumber = {0533.58034},
} -
[Aiz85]
M. Aizenman, "The intersection of Brownian paths as a case study of a renormalization group method for quantum field theory," Comm. Math. Phys., vol. 97, iss. 1-2, pp. 91-110, 1985.
@ARTICLE{Aiz85,
author = {Aizenman, Michael},
title = {The intersection of {B}rownian paths as a case study of a renormalization group method for quantum field theory},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {97},
year = {1985},
number = {1-2},
pages = {91--110},
issn = {0010-3616},
mrclass = {60J65 (81E15 82A25)},
mrnumber = {0782960},
mrreviewer = {Gerhard C. Hegerfeldt},
doi = {10.1007/BF01206180},
url = {https://doi.org/10.1007/BF01206180},
zblnumber = {0573.60076},
} -
[AizGra83]
M. Aizenman and R. Graham, "On the renormalized coupling constant and the susceptibility in $\varphi ^{4}_{4}$ field theory and the Ising model in four dimensions," Nuclear Phys. B, vol. 225, iss. 2, , FS 9, pp. 261-288, 1983.
@ARTICLE{AizGra83,
author = {Aizenman, Michael and Graham, Ross},
title = {On the renormalized coupling constant and the susceptibility in {$\varphi \sp{4}\sb{4}$} field theory and the {I}sing model in four dimensions},
journal = {Nuclear Phys. B},
fjournal = {Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems},
volume = {225},
year = {1983},
number = {2, , FS 9},
pages = {261--288},
issn = {0550-3213},
mrclass = {81E08 (81E15 81E25 82A25 82A68)},
mrnumber = {0720344},
mrreviewer = {Alan D. Sokal},
doi = {10.1016/0550-3213(83)90053-6},
url = {https://doi.org/10.1016/0550-3213(83)90053-6},
zblnumber = {},
} -
[AizBarFer87]
M. Aizenman, D. J. Barsky, and R. Fernández, "The phase transition in a general class of Ising-type models is sharp," J. Statist. Phys., vol. 47, iss. 3-4, pp. 343-374, 1987.
@ARTICLE{AizBarFer87,
author = {Aizenman, Michael and Barsky, D. J. and Fern\'{a}ndez, R.},
title = {The phase transition in a general class of {I}sing-type models is sharp},
journal = {J. Statist. Phys.},
fjournal = {Journal of Statistical Physics},
volume = {47},
year = {1987},
number = {3-4},
pages = {343--374},
issn = {0022-4715},
mrclass = {82A25 (81E25 82A43 82A68)},
mrnumber = {0894398},
mrreviewer = {Alan D. Sokal},
doi = {10.1007/BF01007515},
url = {https://doi.org/10.1007/BF01007515},
zblnumber = {},
} -
[AizDumSid15]
M. Aizenman, H. Duminil-Copin, and V. Sidoravicius, "Random currents and continuity of Ising model’s spontaneous magnetization," Comm. Math. Phys., vol. 334, iss. 2, pp. 719-742, 2015.
@ARTICLE{AizDumSid15,
author = {Aizenman, Michael and Duminil-Copin, Hugo and Sidoravicius, Vladas},
title = {Random currents and continuity of {I}sing model's spontaneous magnetization},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {334},
year = {2015},
number = {2},
pages = {719--742},
issn = {0010-3616},
mrclass = {82B20 (82D40)},
mrnumber = {3306602},
doi = {10.1007/s00220-014-2093-y},
url = {https://doi.org/10.1007/s00220-014-2093-y},
zblnumber = {1315.82004},
} -
[AizDumTasWar18]
M. Aizenman, H. Duminil-Copin, V. Tassion, and S. Warzel, "Emergent planarity in two-dimensional Ising models with finite-range interactions," Invent. Math., vol. 216, iss. 3, pp. 661-743, 2019.
@ARTICLE{AizDumTasWar18,
author = {Aizenman, Michael and Duminil-Copin, Hugo and Tassion, Vincent and Warzel, Simone},
title = {Emergent planarity in two-dimensional {I}sing models with finite-range interactions},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {216},
year = {2019},
number = {3},
pages = {661--743},
issn = {0020-9910},
mrclass = {82B20 (60K35)},
mrnumber = {3955708},
mrreviewer = {Rongfeng Sun},
doi = {10.1007/s00222-018-00851-4},
url = {https://doi.org/10.1007/s00222-018-00851-4},
zblnumber = {1417.82022},
} -
[AizFer86]
M. Aizenman and R. Fernández, "On the critical behavior of the magnetization in high-dimensional Ising models," J. Statist. Phys., vol. 44, iss. 3-4, pp. 393-454, 1986.
@ARTICLE{AizFer86,
author = {Aizenman, Michael and Fern\'{a}ndez, R.},
title = {On the critical behavior of the magnetization in high-dimensional {I}sing models},
journal = {J. Statist. Phys.},
fjournal = {Journal of Statistical Physics},
volume = {44},
year = {1986},
number = {3-4},
pages = {393--454},
issn = {0022-4715},
mrclass = {82A68 (82A05)},
mrnumber = {0857063},
mrreviewer = {C. A. Hurst},
doi = {10.1007/BF01011304},
url = {https://doi.org/10.1007/BF01011304},
zblnumber = {0629.60106},
} -
[ACF83]
C. Aragão de Carvalho, S. Caracciolo, and J. Fröhlich, "Polymers and $g\varphi ^{4}$ theory in four dimensions," Nuclear Phys. B, vol. 215, iss. 2, pp. 209-248, 1983.
@ARTICLE{ACF83,
author = {{Arag\~{a}o de Carvalho},
C. and Caracciolo, S. and Fröhlich, J.},
title = {Polymers and {$g\varphi \sp{4}$} theory in four dimensions},
journal = {Nuclear Phys. B},
fjournal = {Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems},
volume = {215},
year = {1983},
number = {2},
pages = {209--248},
issn = {0550-3213},
mrclass = {81E10 (81E25 82A25 82A51)},
mrnumber = {0690735},
mrreviewer = {V. Rivasseau},
doi = {10.1016/0550-3213(83)90213-4},
url = {https://doi.org/10.1016/0550-3213(83)90213-4},
zblnumber = {},
} -
[BauBrySla14]
R. Bauerschmidt, D. C. Brydges, and G. Slade, "Scaling limits and critical behaviour of the 4-dimensional $n$-component $|\phi|^4$ spin model," J. Stat. Phys., vol. 157, iss. 4-5, pp. 692-742, 2014.
@ARTICLE{BauBrySla14,
author = {Bauerschmidt, Roland and Brydges, David C. and Slade, Gordon},
title = {Scaling limits and critical behaviour of the 4-dimensional {$n$}-component {$|\phi|^4$} spin model},
journal = {J. Stat. Phys.},
fjournal = {Journal of Statistical Physics},
volume = {157},
year = {2014},
number = {4-5},
pages = {692--742},
issn = {0022-4715},
mrclass = {82B28 (60K35 82B20 82B27)},
mrnumber = {3269689},
mrreviewer = {Mei Yin},
doi = {10.1007/s10955-014-1060-5},
url = {https://doi.org/10.1007/s10955-014-1060-5},
zblnumber = {1308.82026},
} -
[BryFroSpe82]
D. Brydges, J. Fröhlich, and T. Spencer, "The random walk representation of classical spin systems and correlation inequalities," Comm. Math. Phys., vol. 83, iss. 1, pp. 123-150, 1982.
@ARTICLE{BryFroSpe82,
author = {Brydges, David and Fröhlich, Jürg and Spencer, Thomas},
title = {The random walk representation of classical spin systems and correlation inequalities},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {83},
year = {1982},
number = {1},
pages = {123--150},
issn = {0010-3616},
mrclass = {82A67 (81E10)},
mrnumber = {0648362},
mrreviewer = {Gerhard C. Hegerfeldt},
doi = {10.1007/BF01947075},
url = {https://doi.org/10.1007/BF01947075},
zblnumber = {},
} -
[Bis09]
M. Biskup, "Reflection positivity and phase transitions in lattice spin models," in Methods of Contemporary Mathematical Statistical Physics, Springer-Verlag, Berlin, 2009, vol. 1970, pp. 1-86.
@INCOLLECTION{Bis09,
author = {Biskup, Marek},
title = {Reflection positivity and phase transitions in lattice spin models},
booktitle = {Methods of Contemporary Mathematical Statistical Physics},
series = {Lecture Notes in Math.},
volume = {1970},
pages = {1--86},
publisher = {Springer-Verlag, Berlin},
year = {2009},
mrclass = {82B20 (82-02 82B26)},
mrnumber = {2581604},
mrreviewer = {Aernout C. D. van Enter},
doi = {10.1007/978-3-540-92796-9\_1},
url = {https://doi.org/10.1007/978-3-540-92796-9_1},
zblnumber = {1180.82041},
} -
[Dum16]
H. Duminil-Copin, "Random currents expansion of the Ising model," in European Congress of Mathematics, Eur. Math. Soc., Zürich, 2018, pp. 869-889.
@INCOLLECTION{Dum16,
author = {Duminil-Copin, Hugo},
title = {Random currents expansion of the {I}sing model},
booktitle = {European {C}ongress of {M}athematics},
pages = {869--889},
publisher = {Eur. Math. Soc., Zürich},
year = {2018},
mrclass = {82B20 (60K35)},
mrnumber = {3890455},
doi = {10.4171/176-1/39},
url = {https://doi.org/10.4171/176-1/39},
zblnumber = {1403.82005},
} -
[Dum17]
H. Duminil-Copin, "Lectures on the Ising and Potts models on the hypercubic lattice," in Random Graphs, Phase Transitions, and the Gaussian Free Field, Springer, Cham, 2020, vol. 304, pp. 35-161.
@INCOLLECTION{Dum17,
author = {Duminil-Copin, Hugo},
title = {Lectures on the {I}sing and {P}otts models on the hypercubic lattice},
booktitle = {Random Graphs, Phase Transitions, and the {G}aussian Free Field},
series = {Springer Proc. Math. Stat.},
volume = {304},
pages = {35--161},
publisher = {Springer, Cham},
year = {2020},
mrclass = {82B20 (60K35)},
mrnumber = {4043224},
mrreviewer = {Jianping Jiang},
doi = {10.1007/978-3-030-32011-9\_2},
url = {https://doi.org/10.1007/978-3-030-32011-9_2},
zblnumber = {1447.82007},
} -
[DumGosRao18]
H. Duminil-Copin, S. Goswami, and A. Raoufi, "Exponential decay of truncated correlations for the Ising model in any dimension for all but the critical temperature," Comm. Math. Phys., vol. 374, iss. 2, pp. 891-921, 2020.
@ARTICLE{DumGosRao18,
author = {Duminil-Copin, Hugo and Goswami, Subhajit and Raoufi, Aran},
title = {Exponential decay of truncated correlations for the {I}sing model in any dimension for all but the critical temperature},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {374},
year = {2020},
number = {2},
pages = {891--921},
issn = {0010-3616},
mrclass = {82B20 (60K35)},
mrnumber = {4072233},
doi = {10.1007/s00220-019-03633-y},
url = {https://doi.org/10.1007/s00220-019-03633-y},
zblnumber = {07178350},
} -
[DumTas15]
H. Duminil-Copin and V. Tassion, "A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model," Comm. Math. Phys., vol. 343, iss. 2, pp. 725-745, 2016.
@ARTICLE{DumTas15,
author = {Duminil-Copin, Hugo and Tassion, Vincent},
title = {A new proof of the sharpness of the phase transition for {B}ernoulli percolation and the {I}sing model},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {343},
year = {2016},
number = {2},
pages = {725--745},
issn = {0010-3616},
mrclass = {60K35 (82B20 82B26 82B43)},
mrnumber = {3477351},
mrreviewer = {Hans-Otto Georgii},
doi = {10.1007/s00220-015-2480-z},
url = {https://doi.org/10.1007/s00220-015-2480-z},
zblnumber = {1342.82026},
} -
[FMRS87]
J. Feldman, J. Magnen, V. Rivasseau, and R. Sénéor, "Construction and Borel summability of infrared $\Phi^4_4$ by a phase space expansion," Comm. Math. Phys., vol. 109, iss. 3, pp. 437-480, 1987.
@ARTICLE{FMRS87,
author = {Feldman, J. and Magnen, J. and Rivasseau, V. and Sénéor, R.},
title = {Construction and {B}orel summability of infrared {$\Phi^4_4$} by a phase space expansion},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {109},
year = {1987},
number = {3},
pages = {437--480},
issn = {0010-3616},
mrclass = {81E08 (81E15)},
mrnumber = {0882810},
mrreviewer = {Claus Montonen},
doi = {10.1007/BF01206146},
url = {https://doi.org/10.1007/BF01206146},
zblnumber = {},
} -
[Fro82]
J. Fröhlich, "On the triviality of $\lambda \varphi ^{4}_{d}$ theories and the approach to the critical point in $d_{\substack{>\\(-)} }4$ dimensions," Nuclear Phys. B, vol. 200, iss. 2, pp. 281-296, 1982.
@ARTICLE{Fro82,
author = {Fröhlich, Jürg},
title = {On the triviality of {$\lambda \varphi \sp{4}\sb{d}$} theories and the approach to the critical point in {$d_{\substack{>\\(-)} }4$} dimensions},
journal = {Nuclear Phys. B},
fjournal = {Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems},
volume = {200},
year = {1982},
number = {2},
pages = {281--296},
issn = {0550-3213},
mrclass = {81E10 (58E99 81E25 82A25)},
mrnumber = {0643591},
mrreviewer = {J. Fernando Perez},
doi = {10.1016/0550-3213(82)90088-8},
url = {https://doi.org/10.1016/0550-3213(82)90088-8},
zblnumber = {},
} -
[FILS78]
J. Fröhlich, R. Israel, E. H. Lieb, and B. Simon, "Phase transitions and reflection positivity. I. General theory and long range lattice models," Comm. Math. Phys., vol. 62, iss. 1, pp. 1-34, 1978.
@ARTICLE{FILS78,
author = {Fröhlich, Jürg and Israel, Robert and Lieb, Elliott H. and Simon, Barry},
title = {Phase transitions and reflection positivity. {I}. {G}eneral theory and long range lattice models},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {62},
year = {1978},
number = {1},
pages = {1--34},
issn = {0010-3616},
mrclass = {82A67},
mrnumber = {0506363},
mrreviewer = {Gerhard C. Hegerfeldt},
doi = {10.1007/BF01940327},
url = {https://doi.org/BF01940327},
zblnumber = {},
} -
[FroSimSpe76]
J. Fröhlich, B. Simon, and T. Spencer, "Infrared bounds, phase transitions and continuous symmetry breaking," Comm. Math. Phys., vol. 50, iss. 1, pp. 79-95, 1976.
@ARTICLE{FroSimSpe76,
author = {Fröhlich, Jürg and Simon, B. and Spencer, Thomas},
title = {Infrared bounds, phase transitions and continuous symmetry breaking},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {50},
year = {1976},
number = {1},
pages = {79--95},
issn = {0010-3616},
mrclass = {82.47},
mrnumber = {0421531},
mrreviewer = {O. Penrose},
doi = {10.1007/BF01608557},
url = {https://doi.org/10.1007/BF01608557},
zblnumber = {},
} -
[GawKup85]
K. Gawedzki and A. Kupiainen, "Massless lattice $\varphi^4_4$ theory: rigorous control of a renormalizable asymptotically free model," Comm. Math. Phys., vol. 99, iss. 2, pp. 197-252, 1985.
@ARTICLE{GawKup85,
author = {Gawedzki, K. and Kupiainen, A.},
title = {Massless lattice {$\varphi^4_4$} theory: rigorous control of a renormalizable asymptotically free model},
journal = {Comm. Math. Phys.},
fjournal = {Comm. Math. Phys.},
volume = {99},
year = {1985},
number = {2},
pages = {197--252},
issn = {0010-3616},
mrclass = {81E25 (81E15)},
mrnumber = {0790736},
mrreviewer = {Giovanni Felder},
doi = {10.1007/BF01212281},
url = {https://doi.org/10.1007/BF01212281},
zblnumber = {},
} -
[GliJaf73]
J. Glimm and A. Jaffe, "Positivity of the $\phi ^{4}_{3}$ Hamiltonian," Fortschr. Physik, vol. 21, pp. 327-376, 1973.
@ARTICLE{GliJaf73,
author = {Glimm, James and Jaffe, Arthur},
title = {Positivity of the {$\phi \sp{4}\sb{3}$} {H}amiltonian},
journal = {Fortschr. Physik},
fjournal = {Fortschritte der Physik. Progress of Physics},
volume = {21},
year = {1973},
pages = {327--376},
issn = {0015-8208},
mrclass = {81.47},
mrnumber = {0408581},
mrreviewer = {R. Hoegh-Krohn},
doi = {10.1002/prop.19730210702},
url = {https://doi.org/10.1002/prop.19730210702},
zblnumber = {},
} -
[Gri67]
R. B. Griffiths, "Correlations in Ising ferromagnets I," J. Math. Phys., vol. 8, iss. 3, pp. 478-483, 1967.
@ARTICLE{Gri67,
author = {Griffiths, Robert B.},
title = {Correlations in {I}sing ferromagnets {I}},
journal = {J. Math. Phys.},
volume = {8},
number = {3},
pages = {478--483},
year = {1967},
doi = {10.1063/1.1705219},
url = {https://doi.org/10.1063/1.1705219},
zblnumber = {},
} -
[Gri69]
R. B. Griffiths, "Rigorous results for Ising ferromagnets of arbitrary spin," J. Mathematical Phys., vol. 10, pp. 1559-1565, 1969.
@ARTICLE{Gri69,
author = {Griffiths, Robert B.},
title = {Rigorous results for {I}sing ferromagnets of arbitrary spin},
journal = {J. Mathematical Phys.},
fjournal = {Journal of Mathematical Physics},
volume = {10},
year = {1969},
pages = {1559--1565},
issn = {0022-2488},
mrclass = {82.00},
mrnumber = {0256682},
mrreviewer = {T. Arai},
doi = {10.1063/1.1665005},
url = {https://doi.org/10.1063/1.1665005},
zblnumber = {},
} -
[GHS70]
R. B. Griffiths, C. A. Hurst, and S. Sherman, "Concavity of magnetization of an Ising ferromagnet in a positive external field," J. Mathematical Phys., vol. 11, pp. 790-795, 1970.
@ARTICLE{GHS70,
author = {Griffiths, Robert B. and Hurst, C. A. and Sherman, S.},
title = {Concavity of magnetization of an {I}sing ferromagnet in a positive external field},
journal = {J. Mathematical Phys.},
fjournal = {Journal of Mathematical Physics},
volume = {11},
year = {1970},
pages = {790--795},
issn = {0022-2488},
mrclass = {81.05},
mrnumber = {0266507},
doi = {10.1063/1.1665211},
url = {https://doi.org/10.1063/1.1665211},
zblnumber = {},
} -
[GriSim73]
B. Simon and R. B. Griffiths, "The $(\phi ^{4})_{2}$ field theory as a classical Ising model," Comm. Math. Phys., vol. 33, pp. 145-164, 1973.
@ARTICLE{GriSim73,
author = {Simon, Barry and Griffiths, Robert B.},
title = {The {$(\phi \sp{4})\sb{2}$} field theory as a classical {I}sing model},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {33},
year = {1973},
pages = {145--164},
issn = {0010-3616},
mrclass = {82.60},
mrnumber = {0428998},
mrreviewer = {Abel Klein},
doi = {10.1007/BF01645626},
url = {https://doi.org/10.1007/BF01645626},
zblnumber = {},
} -
@BOOK{Gri06,
author = {Grimmett, Geoffrey},
title = {The Random-Cluster Model},
series = {Grundlehren Math. Wissen.},
volume = {333},
publisher = {Springer-Verlag, Berlin},
year = {2006},
pages = {xiv+377},
isbn = {978-3-540-32890-2; 3-540-32890-4},
mrclass = {60K35 (60-02 82-02 82B20 82B43)},
mrnumber = {2243761},
mrreviewer = {Olivier Garet},
doi = {10.1007/978-3-540-32891-9},
url = {https://doi.org/10.1007/978-3-540-32891-9},
zblnumber = {1122.60087},
} -
[GRS75]
F. Guerra, L. Rosen, and B. Simon, "The ${\bf P}(\phi)_{2}$ Euclidean quantum field theory as classical statistical mechanics. I," Ann. of Math. (2), vol. 101, iss. 1, pp. 111-189, 1975.
@ARTICLE{GRS75,
author = {Guerra, F. and Rosen, L. and Simon, B.},
title = {The {${\bf P}(\phi)\sb{2}$} {E}uclidean quantum field theory as classical statistical mechanics. {I}},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {101},
year = {1975},
number = {1},
pages = {111--189},
issn = {0003-486X},
mrclass = {82.46 (81.46)},
mrnumber = {0378670},
mrreviewer = {R. A. Minlos},
doi = {10.2307/1970988},
url = {https://doi.org/10.2307/1970988},
zblnumber = {},
} -
[Hai14]
M. Hairer, "A theory of regularity structures," Invent. Math., vol. 198, iss. 2, pp. 269-504, 2014.
@ARTICLE{Hai14,
author = {Hairer, M.},
title = {A theory of regularity structures},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {198},
year = {2014},
number = {2},
pages = {269--504},
issn = {0020-9910},
mrclass = {60H15 (35R60 60H40 81S20 82C28)},
mrnumber = {3274562},
mrreviewer = {Dora Seleši},
doi = {10.1007/s00222-014-0505-4},
url = {https://doi.org/10.1007/s00222-014-0505-4},
zblnumber = {1332.60093},
} -
[Ham57]
J. M. Hammersley, "Percolation processes: Lower bounds for the critical probability," Ann. Math. Statist., vol. 28, pp. 790-795, 1957.
@ARTICLE{Ham57,
author = {Hammersley, J. M.},
title = {Percolation processes: {L}ower bounds for the critical probability},
journal = {Ann. Math. Statist.},
fjournal = {Annals of Mathematical Statistics},
volume = {28},
year = {1957},
pages = {790--795},
issn = {0003-4851},
mrclass = {60.00 (82.00)},
mrnumber = {0101564},
mrreviewer = {G. Newell},
doi = {10.1214/aoms/1177706894},
url = {https://doi.org/10.1214/aoms/1177706894},
zblnumber = {0091.13903},
} -
[HarTas87]
T. Hara and H. Tasaki, "A rigorous control of logarithmic corrections in four-dimensional $\phi^4$ spin systems. II. Critical behavior of susceptibility and correlation length," J. Statist. Phys., vol. 47, iss. 1-2, pp. 99-121, 1987.
@ARTICLE{HarTas87,
author = {Hara, Takashi and Tasaki, Hal},
title = {A rigorous control of logarithmic corrections in four-dimensional {$\phi^4$} spin systems. {II}. {C}ritical behavior of susceptibility and correlation length},
journal = {J. Statist. Phys.},
fjournal = {Journal of Statistical Physics},
volume = {47},
year = {1987},
number = {1-2},
pages = {99--121},
issn = {0022-4715},
mrclass = {82A25 (81E25 82A05)},
mrnumber = {0892925},
doi = {10.1007/BF01009036},
url = {https://doi.org/10.1007/BF01009036},
zblnumber = {},
} -
[Heg77]
G. C. Hegerfeldt, "Correlation inequalities for Ising ferromagnets with symmetries," Comm. Math. Phys., vol. 57, iss. 3, pp. 259-266, 1977.
@ARTICLE{Heg77,
author = {Hegerfeldt, Gerhard C.},
title = {Correlation inequalities for {I}sing ferromagnets with symmetries},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {57},
year = {1977},
number = {3},
pages = {259--266},
issn = {0010-3616},
mrclass = {82.60},
mrnumber = {0503339},
doi = {10.1007/BF01614166},
url = {https://doi.org/10.1007/BF01614166},
zblnumber = {},
} -
[JW] A. Jaffe and E. Witten, "Quantum Yang-Mills theory," in The Millennium Prize Problems, Clay Math. Inst., Cambridge, MA, 2006, pp. 129-152.
@INCOLLECTION{JW,
author = {Jaffe, Arthur and Witten, Edward},
title = {Quantum {Y}ang-{M}ills theory},
booktitle = {The Millennium Prize Problems},
pages = {129--152},
publisher = {Clay Math. Inst., Cambridge, MA},
year = {2006},
mrclass = {81T13},
mrnumber = {2238278},
zblnumber = {1194.81002},
} -
[Kes86]
H. Kesten, "The incipient infinite cluster in two-dimensional percolation," Probab. Theory Related Fields, vol. 73, iss. 3, pp. 369-394, 1986.
@ARTICLE{Kes86,
author = {Kesten, Harry},
title = {The incipient infinite cluster in two-dimensional percolation},
journal = {Probab. Theory Related Fields},
fjournal = {Probability Theory and Related Fields},
volume = {73},
year = {1986},
number = {3},
pages = {369--394},
issn = {0178-8051},
mrclass = {60K35},
mrnumber = {0859839},
mrreviewer = {R. T. Smythe},
doi = {10.1007/BF00776239},
url = {https://doi.org/10.1007/BF00776239},
zblnumber = {0584.60098},
} -
[KrRu] M. G. Kreuin and M. A. Rutman, "Linear operators leaving invariant a cone in a Banach space," Amer. Math. Soc. Translation, vol. 1950, iss. 26, p. 128, 1950.
@ARTICLE{KrRu,
author = {Kreĭn, M. G. and Rutman, M. A.},
title = {Linear operators leaving invariant a cone in a {B}anach space},
journal = {Amer. Math. Soc. Translation},
fjournal = {American Mathematical Society Translations},
volume = {1950},
year = {1950},
number = {26},
pages = {128},
issn = {0065-9290},
mrclass = {46.3X},
mrnumber = {0038008},
zblnumber = {},
} -
@BOOK{Law13,
author = {Lawler, Gregory F.},
title = {Intersections of Random Walks},
series = {Modern Birkhäuser Classics},
note = {reprint of the 1996 edition},
publisher = {Birkhäuser/Springer, New York},
year = {2013},
pages = {iv+223},
isbn = {978-1-4614-5971-2; 978-1-4614-5972-9},
mrclass = {60G50 (60G17 60K99)},
mrnumber = {2985195},
doi = {10.1007/978-1-4614-5972-9},
url = {https://doi.org/10.1007/978-1-4614-5972-9},
zblnumber = {1253.60003},
} -
[Lie80]
E. H. Lieb, "A refinement of Simon’s correlation inequality," Comm. Math. Phys., vol. 77, iss. 2, pp. 127-135, 1980.
@ARTICLE{Lie80,
author = {Lieb, Elliott H.},
title = {A refinement of {S}imon's correlation inequality},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {77},
year = {1980},
number = {2},
pages = {127--135},
issn = {0010-3616},
mrclass = {82A05},
mrnumber = {0589427},
mrreviewer = {Francesco Guerra},
doi = {10.1007/BF01982712},
url = {https://doi.org/10.1007/BF01982712},
zblnumber = {},
} -
[MesMir77]
A. Messager and S. Miracle-Sole, "Correlation functions and boundary conditions in the Ising ferromagnet," J. Statist. Phys., vol. 17, iss. 4, pp. 245-262, 1977.
@ARTICLE{MesMir77,
author = {Messager, A. and Miracle-Sole, S.},
title = {Correlation functions and boundary conditions in the {I}sing ferromagnet},
journal = {J. Statist. Phys.},
fjournal = {Journal of Statistical Physics},
volume = {17},
year = {1977},
number = {4},
pages = {245--262},
issn = {0022-4715},
mrclass = {82.47},
mrnumber = {0676312},
doi = {10.1007/BF01040105},
url = {https://doi.org/10.1007/BF01040105},
zblnumber = {},
} -
[New75]
C. M. Newman, "Inequalities for Ising models and field theories which obey the Lee-Yang theorem," Comm. Math. Phys., vol. 41, iss. 1, pp. 1-9, 1975.
@ARTICLE{New75,
author = {Newman, Charles M.},
title = {Inequalities for {I}sing models and field theories which obey the {L}ee-{Y}ang theorem},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {41},
year = {1975},
number = {1},
pages = {1--9},
issn = {0010-3616},
mrclass = {82.60},
mrnumber = {0376061},
mrreviewer = {C. A. Hurst},
doi = {10.1007/BF01608542},
url = {https://doi.org/10.1007/BF01608542},
zblnumber = {},
} -
[OS73]
K. Osterwalder and R. Schrader, "Axioms for Euclidean Green’s functions. I," Comm. Math. Phys., vol. 31, iss. 2, pp. 83-112, 1973.
@ARTICLE{OS73,
author = {Osterwalder, Konrad and Schrader, Robert},
title = {Axioms for {E}uclidean {G}reen's functions. {I}},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {31},
year = {1973},
number = {2},
pages = {83--112},
issn = {0010-3616},
mrclass = {81.46},
mrnumber = {0329492},
mrreviewer = {W. Güttinger},
doi = {10.1007/BF01645738},
url = {https://doi.org/10.1007/BF01645738},
zblnumber = {0274.46047},
} -
[OS75]
K. Osterwalder and R. Schrader, "Axioms for Euclidean Green’s functions. II," Comm. Math. Phys., vol. 42, pp. 281-305, 1975.
@ARTICLE{OS75,
author = {Osterwalder, Konrad and Schrader, Robert},
title = {Axioms for {E}uclidean {G}reen's functions. {II}},
note = {With an appendix by Stephen Summers},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {42},
year = {1975},
pages = {281--305},
issn = {0010-3616},
mrclass = {81.46},
mrnumber = {0376002},
mrreviewer = {W. Güttinger},
doi = {10.1007/BF01608978},
url = {https://doi.org/10.1007/BF01608978},
zblnumber = {0303.46034},
} -
[Sch77]
R. Schrader, "New correlation inequalities for the Ising model and $P(\phi )$ theories," Phys. Rev. B (3), vol. 15, iss. 5, pp. 2798-2803, 1977.
@ARTICLE{Sch77,
author = {Schrader, Robert},
title = {New correlation inequalities for the {I}sing model and {$P(\phi )$} theories},
journal = {Phys. Rev. B (3)},
fjournal = {Physical Review. B. Condensed Matter. Third Series},
volume = {15},
year = {1977},
number = {5},
pages = {2798--2803},
issn = {0163-1829},
mrclass = {82.60},
mrnumber = {0449416},
mrreviewer = {J. E. Lowther},
doi = {10.1103/PhysRevB.15.2798},
url = {https://link.aps.org/doi/10.1103/PhysRevB.15.2798},
zblnumber = {},
} -
[SchMatLie64]
T. D. Schultz, D. C. Mattis, and E. H. Lieb, "Two-dimensional Ising model as a soluble problem of many fermions," Rev. Modern Phys., vol. 36, pp. 856-871, 1964.
@ARTICLE{SchMatLie64,
author = {Schultz, T. D. and Mattis, D. C. and Lieb, E. H.},
title = {Two-dimensional {I}sing model as a soluble problem of many fermions},
journal = {Rev. Modern Phys.},
fjournal = {Reviews of Modern Physics},
volume = {36},
year = {1964},
pages = {856--871},
issn = {0034-6861},
mrclass = {82.17},
mrnumber = {0180274},
mrreviewer = {R. B. Potts},
doi = {10.1103/RevModPhys.36.856},
url = {https://doi.org/10.1103/RevModPhys.36.856},
zblnumber = {},
} -
[Sim80]
B. Simon, "Correlation inequalities and the decay of correlations in ferromagnets," Comm. Math. Phys., vol. 77, iss. 2, pp. 111-126, 1980.
@ARTICLE{Sim80,
author = {Simon, Barry},
title = {Correlation inequalities and the decay of correlations in ferromagnets},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {77},
year = {1980},
number = {2},
pages = {111--126},
issn = {0010-3616},
mrclass = {82A05 (82A25)},
mrnumber = {0589426},
mrreviewer = {Francesco Guerra},
doi = {10.1007/BF01982711},
url = {https://doi.org/10.1007/BF01982711},
zblnumber = {},
} -
[Sok82] A. D. Sokal, "An alternate constructive approach to the $\varphi ^{4}_{3}$ quantum field theory, and a possible destructive approach to $\varphi ^{4}_{4}$," Ann. Inst. H. Poincaré Sect. A (N.S.), vol. 37, iss. 4, pp. 317-398, 1982.
@ARTICLE{Sok82,
author = {Sokal, Alan D.},
title = {An alternate constructive approach to the {$\varphi \sp{4}\sb{3}$} quantum field theory, and a possible destructive approach to {$\varphi \sp{4}\sb{4}$}},
journal = {Ann. Inst. H. Poincaré Sect. A (N.S.)},
fjournal = {Annales de l'Institut Henri Poincaré. Section A. Physique Théorique. Nouvelle Série},
volume = {37},
year = {1982},
number = {4},
pages = {317--398},
issn = {0246-0211},
mrclass = {81E08 (81E25)},
mrnumber = {0693644},
mrreviewer = {J. Fernando Perez},
zblnumber = {},
} -
[Sok79]
A. D. Sokal, "A rigorous inequality for the specific heat of an Ising or $\varphi ^{4}$ ferromagnet," Phys. Lett. A, vol. 71, iss. 5-6, pp. 451-453, 1979.
@article{Sok79,
author = {Sokal, Alan D.},
TITLE = {A rigorous inequality for the specific heat of an {I}sing or {$\varphi ^{4}$} ferromagnet},
JOURNAL = {Phys. Lett. A},
FJOURNAL = {Physics Letters. A},
VOLUME = {71},
YEAR = {1979},
NUMBER = {5-6},
PAGES = {451--453},
ISSN = {0375-9601},
MRCLASS = {82A05 (81E10 81E25 82A67)},
MRNUMBER = {588470},
DOI = {10.1016/0375-9601(79)90633-9},
URL = {https://doi.org/10.1016/0375-9601(79)90633-9},
} -
[Sym69] K. Symanzik, "Euclidean quantum field theory," in Local Quantum Theory, Jost, R., Ed., Academic Press, New York, 1969, vol. 45, pp. 152-226.
@incollection{Sym69,
author = {Symanzik, K.},
title = {Euclidean quantum field theory},
booktitle = {Local {Q}uantum {T}heory},
editor = {{J}ost, R.},
publisher = {Academic Press, New York},
journal = {Rend. Scu. Int. Fis. Enrico Fermi},
volume = {45},
pages = {152--226},
year = {1969},
zblnumber = {},
} -
[HofJar04]
R. van der Hofstad and A. A. Járai, "The incipient infinite cluster for high-dimensional unoriented percolation," J. Statist. Phys., vol. 114, iss. 3-4, pp. 625-663, 2004.
@ARTICLE{HofJar04,
author = {{van der Hofstad},
Remco and J\'{a}rai, Antal A.},
title = {The incipient infinite cluster for high-dimensional unoriented percolation},
journal = {J. Statist. Phys.},
fjournal = {Journal of Statistical Physics},
volume = {114},
year = {2004},
number = {3-4},
pages = {625--663},
issn = {0022-4715},
mrclass = {82B43},
mrnumber = {2035627},
mrreviewer = {Raphaël Cerf},
doi = {10.1023/B:JOSS.0000012505.39213.6a},
url = {https://doi.org/10.1023/B:JOSS.0000012505.39213.6a},
zblnumber = {1061.82016},
} -
[Wig56]
A. S. Wightman, "Quantum field theory in terms of vacuum expectation values," Phys. Rev. (2), vol. 101, pp. 860-866, 1956.
@ARTICLE{Wig56,
author = {Wightman, A. S.},
title = {Quantum field theory in terms of vacuum expectation values},
journal = {Phys. Rev. (2)},
fjournal = {Physical Review. Series II},
volume = {101},
year = {1956},
pages = {860--866},
issn = {0031-899X},
mrclass = {81.0X},
mrnumber = {0083914},
mrreviewer = {M. Cini},
doi = {10.1103/PhysRev.101.860},
url = {https://doi.org/10.1103/PhysRev.101.860},
zblnumber = {0074.22902},
} -
[Wil71]
K. G. Wilson, "Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture," Phys. Rev. B (3), vol. 4, pp. 3174-3183, 1971.
@ARTICLE{Wil71,
author = {Wilson, K. G.},
title = {Renormalization group and critical phenomena. {I. R}enormalization group and the {K}adanoff scaling picture},
fjournal = {PHysical Review B},
journal = {Phys. Rev. B (3)},
volume = {4},
year = {1971},
pages = {3174--3183},
publisher = {American Physical Society (APS), New York, NY},
doi = {10.1103/PhysRevB.4.3174},
url = {https://doi.org/10.1103/PhysRevB.4.3174},
zblnumber = {1236.82017},
}