DOI: 10.1007/bf01212322. Sharpness of the phase transition in percolation models

Sharpness of the phase transition in percolation models

10.1007/bf01212322

Crossref journal-article

Springer Science and Business Media LLC

Communications in Mathematical Physics (297)

Bibliography

Aizenman, M., & Barsky, D. J. (1987). Sharpness of the phase transition in percolation models. Communications in Mathematical Physics, 108(3), 489â526.

Authors 2

Michael Aizenman (first)
David J. Barsky (additional)

References 23 Referenced 235

Kesten, H.: The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys.74, 41?59 (1980) (10.1007/BF01197577) / Commun. Math. Phys. by H. Kesten (1980)
Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 229?237 (1981) (10.1007/BF00535742) / Z. Wahrscheinlichkeitstheor. Verw. Geb. by L. Russo (1981)
Aizenman, M., Newman, C.M.: Discontinuity of the percolation density in one-dimensional 1/|x?y|2 percolation models. Commun. Math. Phys.107, 611?647 (1986) (10.1007/BF01205489) / Commun. Math. Phys. by M. Aizenman (1986)
Aizenman, M., Chayes, J.T., Chayes, L., Imbrie, J., Newman, C.M.: An intermediate phase with slow decay of correlations in one-dimensional 1/|x?y|2 Ising and Potts models (in preparation)
Aizenman, M.: Contribution in: Statistical physics and dynamical systems (Proceedings Kösheg 1984). Fritz, J., Jaffe, A., Szasz, D. (eds.). Boston: Birkhäuser 1985 / Statistical physics and dynamical systems (Proceedings Kösheg 1984) by M. Aizenman (1985)
Chayes, J.T., Chayes, L.: Critical points and intermediate phases on wedges of ?d. J. Phys. A (to appear)
Hammersley, J.M.: Percolation processes. Lower bounds for the critical probability. Ann. Math. Statist.28, 790?795 (1957) (10.1214/aoms/1177706894) / Ann. Math. Statist. by J.M. Hammersley (1957)
Aizenman, M., Newman, C.M.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys.36, 107?143 (1984) (10.1007/BF01015729) / J. Stat. Phys. by M. Aizenman (1984)
Newman, C.M., Schulman, L.S.: One-dimensional 1/|j ? i| s percolation models: The existence of a transition fors?2. Commun. Math. Phys.104, 547?571 (1986) (10.1007/BF01211064) / Commun. Math. Phys. by C.M. Newman (1986)
Griffiths, R.B., Hurst, C.A., Sherman, S.: Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys.11, 790?795 (1970) (10.1063/1.1665211) / J. Math. Phys. by R.B. Griffiths (1970)
Newman, C.M.: Shock waves and mean field bounds. Concavity and analyticity of the magnetization at low temperature. Appendix to contribution in Proceedings of the SIAM workshop on multiphase flow, G. Papanicolau (ed.) (to appear)
Aizenman, M., Fernández, R.: On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys.44, 393?454 (1986) (10.1007/BF01011304) / J. Stat. Phys. by M. Aizenman (1986)
Harris, A.B., Lubensky, T.C., Holcomb, W.K., Dasgupta, C.: Renormalization group approach to percolation problems. Phys. Rev. Lett.35, 327?330 (1975) (10.1103/PhysRevLett.35.327) / Phys. Rev. Lett. by A.B. Harris (1975)
Chayes, J.T., Chayes, L.: An inequality for the infinite cluster density in Bernoulli percolation. Phys. Rev. Lett.56, 1619?1622 (1986) (10.1103/PhysRevLett.56.1619) / Phys. Rev. Lett. by J.T. Chayes (1986)
Fröhlich, J., Sokal, A.D.: The random walk representation of classical spin systems and correlation inequalities. III. Nonzero magnetic field (in preparation)
Fernández, R., Fröhlich, J., Sokal, A.D.: Random-walk models and random-walk representations of classical lattice spin systems (in preparation)
Griffiths, R.B.: Correlations in Ising ferromagnets. II. External magnetic fields. J. Math. Phys.8, 484?489 (1967) (10.1063/1.1705220) / J. Math. Phys. by R.B. Griffiths (1967)
Aizenman, M., Kesten, H., Newman, C.M.: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Submitted to Commun. Math. Phys. (10.1007/BF01219071)
van den Berg, J., Kesten, H.: Inequalities with applications to percolation and reliability. J. Appl. Probab.22, 556?569 (1985) (10.2307/3213860) / J. Appl. Probab. by J. Berg van den (1985)
Kesten, H.: Percolation theory for mathematicians. Boston: Birkhäuser 1982 (10.1007/978-1-4899-2730-9) / Percolation theory for mathematicians by H. Kesten (1982)
Griffeath, D.: The basic contact process. Stochastic Processes Appl.11, 151?185 (1981) (10.1016/0304-4149(81)90002-8) / Stochastic Processes Appl. by D. Griffeath (1981)
Aizenman, M., Barsky, D.J., Fernández, R.: The phase transition in a general class of Ising-type models in sharp. Submitted to J. Stat. Phys. (10.1007/BF01007515)
Chayes, J.T., Chayes, L., Newman, C.M.: Bernoulli percolation above threshold: an invasion percolation analysis. Ann. Probab. (to appear) (10.1214/aop/1176991976)

Dates

Type	When
Created	20 years, 6 months ago (Feb. 25, 2005, 11:37 a.m.)
Deposited	6 years, 3 months ago (May 2, 2019, 5:47 a.m.)
Indexed	1 week, 5 days ago (Aug. 19, 2025, 6:36 a.m.)
Issued	38 years ago (Sept. 1, 1987)
Published	38 years ago (Sept. 1, 1987)
Published Print	38 years ago (Sept. 1, 1987)

Funders 0

None

BibTeX

@article{Aizenman_1987, title={Sharpness of the phase transition in percolation models}, volume={108}, ISSN={1432-0916}, url={http://dx.doi.org/10.1007/bf01212322}, DOI={10.1007/bf01212322}, number={3}, journal={Communications in Mathematical Physics}, publisher={Springer Science and Business Media LLC}, author={Aizenman, Michael and Barsky, David J.}, year={1987}, month=sep, pages={489–526} }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        8,
        19
      ]
    ],
    "date-time": "2025-08-19T10:36:24Z",
    "timestamp": 1755599784834
  },
  "reference-count": 23,
  "publisher": "Springer Science and Business Media LLC",
  "issue": "3",
  "license": [
    {
      "start": {
        "date-parts": [
          [
            1987,
            9,
            1
          ]
        ],
        "date-time": "1987-09-01T00:00:00Z",
        "timestamp": 557452800000
      },
      "content-version": "tdm",
      "delay-in-days": 0,
      "URL": "http://www.springer.com/tdm"
    }
  ],
  "content-domain": {
    "domain": [],
    "crossmark-restriction": false
  },
  "published-print": {
    "date-parts": [
      [
        1987,
        9
      ]
    ]
  },
  "DOI": "10.1007/bf01212322",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2005,
        2,
        25
      ]
    ],
    "date-time": "2005-02-25T16:37:44Z",
    "timestamp": 1109349464000
  },
  "page": "489-526",
  "source": "Crossref",
  "is-referenced-by-count": 235,
  "title": "Sharpness of the phase transition in percolation models",
  "prefix": "10.1007",
  "volume": "108",
  "author": [
    {
      "given": "Michael",
      "family": "Aizenman",
      "sequence": "first",
      "affiliation": []
    },
    {
      "given": "David J.",
      "family": "Barsky",
      "sequence": "additional",
      "affiliation": []
    }
  ],
  "member": "297",
  "reference": [
    {
      "key": "CR1",
      "doi-asserted-by": "crossref",
      "first-page": "41",
      "DOI": "10.1007/BF01197577",
      "volume": "74",
      "author": "H. Kesten",
      "year": "1980",
      "unstructured": "Kesten, H.: The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys.74, 41?59 (1980)",
      "journal-title": "Commun. Math. Phys."
    },
    {
      "key": "CR2",
      "doi-asserted-by": "crossref",
      "first-page": "229",
      "DOI": "10.1007/BF00535742",
      "volume": "56",
      "author": "L. Russo",
      "year": "1981",
      "unstructured": "Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 229?237 (1981)",
      "journal-title": "Z. Wahrscheinlichkeitstheor. Verw. Geb."
    },
    {
      "key": "CR3",
      "doi-asserted-by": "crossref",
      "first-page": "611",
      "DOI": "10.1007/BF01205489",
      "volume": "107",
      "author": "M. Aizenman",
      "year": "1986",
      "unstructured": "Aizenman, M., Newman, C.M.: Discontinuity of the percolation density in one-dimensional 1/|x?y|2 percolation models. Commun. Math. Phys.107, 611?647 (1986)",
      "journal-title": "Commun. Math. Phys."
    },
    {
      "key": "CR4",
      "unstructured": "Aizenman, M., Chayes, J.T., Chayes, L., Imbrie, J., Newman, C.M.: An intermediate phase with slow decay of correlations in one-dimensional 1/|x?y|2 Ising and Potts models (in preparation)"
    },
    {
      "key": "CR5",
      "volume-title": "Statistical physics and dynamical systems (Proceedings K\u00f6sheg 1984)",
      "author": "M. Aizenman",
      "year": "1985",
      "unstructured": "Aizenman, M.: Contribution in: Statistical physics and dynamical systems (Proceedings K\u00f6sheg 1984). Fritz, J., Jaffe, A., Szasz, D. (eds.). Boston: Birkh\u00e4user 1985"
    },
    {
      "key": "CR6",
      "unstructured": "Chayes, J.T., Chayes, L.: Critical points and intermediate phases on wedges of ?d. J. Phys. A (to appear)"
    },
    {
      "key": "CR7",
      "doi-asserted-by": "crossref",
      "first-page": "790",
      "DOI": "10.1214/aoms/1177706894",
      "volume": "28",
      "author": "J.M. Hammersley",
      "year": "1957",
      "unstructured": "Hammersley, J.M.: Percolation processes. Lower bounds for the critical probability. Ann. Math. Statist.28, 790?795 (1957)",
      "journal-title": "Ann. Math. Statist."
    },
    {
      "key": "CR8",
      "doi-asserted-by": "crossref",
      "first-page": "107",
      "DOI": "10.1007/BF01015729",
      "volume": "36",
      "author": "M. Aizenman",
      "year": "1984",
      "unstructured": "Aizenman, M., Newman, C.M.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys.36, 107?143 (1984)",
      "journal-title": "J. Stat. Phys."
    },
    {
      "key": "CR9",
      "doi-asserted-by": "crossref",
      "first-page": "547",
      "DOI": "10.1007/BF01211064",
      "volume": "104",
      "author": "C.M. Newman",
      "year": "1986",
      "unstructured": "Newman, C.M., Schulman, L.S.: One-dimensional 1/|j ? i| s percolation models: The existence of a transition fors?2. Commun. Math. Phys.104, 547?571 (1986)",
      "journal-title": "Commun. Math. Phys."
    },
    {
      "key": "CR10",
      "doi-asserted-by": "crossref",
      "first-page": "790",
      "DOI": "10.1063/1.1665211",
      "volume": "11",
      "author": "R.B. Griffiths",
      "year": "1970",
      "unstructured": "Griffiths, R.B., Hurst, C.A., Sherman, S.: Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys.11, 790?795 (1970)",
      "journal-title": "J. Math. Phys."
    },
    {
      "key": "CR11",
      "unstructured": "Newman, C.M.: Shock waves and mean field bounds. Concavity and analyticity of the magnetization at low temperature. Appendix to contribution in Proceedings of the SIAM workshop on multiphase flow, G. Papanicolau (ed.) (to appear)"
    },
    {
      "key": "CR12",
      "doi-asserted-by": "crossref",
      "first-page": "393",
      "DOI": "10.1007/BF01011304",
      "volume": "44",
      "author": "M. Aizenman",
      "year": "1986",
      "unstructured": "Aizenman, M., Fern\u00e1ndez, R.: On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys.44, 393?454 (1986)",
      "journal-title": "J. Stat. Phys."
    },
    {
      "key": "CR13",
      "doi-asserted-by": "crossref",
      "first-page": "327",
      "DOI": "10.1103/PhysRevLett.35.327",
      "volume": "35",
      "author": "A.B. Harris",
      "year": "1975",
      "unstructured": "Harris, A.B., Lubensky, T.C., Holcomb, W.K., Dasgupta, C.: Renormalization group approach to percolation problems. Phys. Rev. Lett.35, 327?330 (1975)",
      "journal-title": "Phys. Rev. Lett."
    },
    {
      "key": "CR14",
      "doi-asserted-by": "crossref",
      "first-page": "1619",
      "DOI": "10.1103/PhysRevLett.56.1619",
      "volume": "56",
      "author": "J.T. Chayes",
      "year": "1986",
      "unstructured": "Chayes, J.T., Chayes, L.: An inequality for the infinite cluster density in Bernoulli percolation. Phys. Rev. Lett.56, 1619?1622 (1986)",
      "journal-title": "Phys. Rev. Lett."
    },
    {
      "key": "CR15",
      "unstructured": "Fr\u00f6hlich, J., Sokal, A.D.: The random walk representation of classical spin systems and correlation inequalities. III. Nonzero magnetic field (in preparation)"
    },
    {
      "key": "CR16",
      "unstructured": "Fern\u00e1ndez, R., Fr\u00f6hlich, J., Sokal, A.D.: Random-walk models and random-walk representations of classical lattice spin systems (in preparation)"
    },
    {
      "key": "CR17",
      "doi-asserted-by": "crossref",
      "first-page": "484",
      "DOI": "10.1063/1.1705220",
      "volume": "8",
      "author": "R.B. Griffiths",
      "year": "1967",
      "unstructured": "Griffiths, R.B.: Correlations in Ising ferromagnets. II. External magnetic fields. J. Math. Phys.8, 484?489 (1967)",
      "journal-title": "J. Math. Phys."
    },
    {
      "key": "CR18",
      "doi-asserted-by": "crossref",
      "unstructured": "Aizenman, M., Kesten, H., Newman, C.M.: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Submitted to Commun. Math. Phys.",
      "DOI": "10.1007/BF01219071"
    },
    {
      "key": "CR19",
      "doi-asserted-by": "crossref",
      "first-page": "556",
      "DOI": "10.2307/3213860",
      "volume": "22",
      "author": "J. Berg van den",
      "year": "1985",
      "unstructured": "van den Berg, J., Kesten, H.: Inequalities with applications to percolation and reliability. J. Appl. Probab.22, 556?569 (1985)",
      "journal-title": "J. Appl. Probab."
    },
    {
      "key": "CR20",
      "doi-asserted-by": "crossref",
      "DOI": "10.1007/978-1-4899-2730-9",
      "volume-title": "Percolation theory for mathematicians",
      "author": "H. Kesten",
      "year": "1982",
      "unstructured": "Kesten, H.: Percolation theory for mathematicians. Boston: Birkh\u00e4user 1982"
    },
    {
      "key": "CR21",
      "doi-asserted-by": "crossref",
      "first-page": "151",
      "DOI": "10.1016/0304-4149(81)90002-8",
      "volume": "11",
      "author": "D. Griffeath",
      "year": "1981",
      "unstructured": "Griffeath, D.: The basic contact process. Stochastic Processes Appl.11, 151?185 (1981)",
      "journal-title": "Stochastic Processes Appl."
    },
    {
      "key": "CR22",
      "doi-asserted-by": "crossref",
      "unstructured": "Aizenman, M., Barsky, D.J., Fern\u00e1ndez, R.: The phase transition in a general class of Ising-type models in sharp. Submitted to J. Stat. Phys.",
      "DOI": "10.1007/BF01007515"
    },
    {
      "key": "CR23",
      "doi-asserted-by": "crossref",
      "unstructured": "Chayes, J.T., Chayes, L., Newman, C.M.: Bernoulli percolation above threshold: an invasion percolation analysis. Ann. Probab. (to appear)",
      "DOI": "10.1214/aop/1176991976"
    }
  ],
  "container-title": "Communications in Mathematical Physics",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "http://link.springer.com/content/pdf/10.1007/BF01212322.pdf",
      "content-type": "application/pdf",
      "content-version": "vor",
      "intended-application": "text-mining"
    },
    {
      "URL": "http://link.springer.com/article/10.1007/BF01212322/fulltext.html",
      "content-type": "text/html",
      "content-version": "vor",
      "intended-application": "text-mining"
    },
    {
      "URL": "http://link.springer.com/content/pdf/10.1007/BF01212322",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2019,
        5,
        2
      ]
    ],
    "date-time": "2019-05-02T09:47:02Z",
    "timestamp": 1556790422000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "http://link.springer.com/10.1007/BF01212322"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        1987,
        9
      ]
    ]
  },
  "references-count": 23,
  "journal-issue": {
    "issue": "3",
    "published-print": {
      "date-parts": [
        [
          1987,
          9
        ]
      ]
    }
  },
  "alternative-id": [
    "BF01212322"
  ],
  "URL": "http://dx.doi.org/10.1007/BF01212322",
  "relation": {},
  "ISSN": [
    "0010-3616",
    "1432-0916"
  ],
  "subject": [],
  "container-title-short": "Commun.Math. Phys.",
  "published": {
    "date-parts": [
      [
        1987,
        9
      ]
    ]
  }
}