### Abstract

For a large class of independent (site or bond, short- or long-range) percolation models, we show the following: (1) If the percolation density P_{∞}(p) is discontinuous at p_{c}, then the critical exponent γ (defined by the divergence of expected cluster size, ∑nP_{n}(p) ∼ (P_{c}-P)^{-γ} as p ↑p_{c}) must satisfy γ ≥ 2. (2)γ or γ′ (defined analogously to γ, but as p ↓p_{c}) and δ [P_{n}(p_{c}) ∼ (n^{-1-1/δ}) as n → ∞ ] must satisfy γ, γ′ ≥ 2(1 - 1/δ). These inequalities for γ improve the previously known bound γ ≥ 1(Aizenman and Newman), since δ ≥ 2 (Aizenman and Barsky). Additionally, result 1 may be useful, in standard d-dimensional percolation, for proving rigorously (in d>2) that, as expected, P_{x} has no discontinuity at p_{c}.

Original language | English (US) |
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Pages (from-to) | 359-368 |

Number of pages | 10 |

Journal | Journal of Statistical Physics |

Volume | 45 |

Issue number | 3-4 |

DOIs | |

State | Published - Nov 1986 |

### Fingerprint

### Keywords

- critical exponent inequalities
- Percolation
- rigorous results

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

**Some critical exponent inequalities for percolation.** / Newman, C. M.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 45, no. 3-4, pp. 359-368. https://doi.org/10.1007/BF01021076

}

TY - JOUR

T1 - Some critical exponent inequalities for percolation

AU - Newman, C. M.

PY - 1986/11

Y1 - 1986/11

N2 - For a large class of independent (site or bond, short- or long-range) percolation models, we show the following: (1) If the percolation density P∞(p) is discontinuous at pc, then the critical exponent γ (defined by the divergence of expected cluster size, ∑nPn(p) ∼ (Pc-P)-γ as p ↑pc) must satisfy γ ≥ 2. (2)γ or γ′ (defined analogously to γ, but as p ↓pc) and δ [Pn(pc) ∼ (n-1-1/δ) as n → ∞ ] must satisfy γ, γ′ ≥ 2(1 - 1/δ). These inequalities for γ improve the previously known bound γ ≥ 1(Aizenman and Newman), since δ ≥ 2 (Aizenman and Barsky). Additionally, result 1 may be useful, in standard d-dimensional percolation, for proving rigorously (in d>2) that, as expected, Px has no discontinuity at pc.

AB - For a large class of independent (site or bond, short- or long-range) percolation models, we show the following: (1) If the percolation density P∞(p) is discontinuous at pc, then the critical exponent γ (defined by the divergence of expected cluster size, ∑nPn(p) ∼ (Pc-P)-γ as p ↑pc) must satisfy γ ≥ 2. (2)γ or γ′ (defined analogously to γ, but as p ↓pc) and δ [Pn(pc) ∼ (n-1-1/δ) as n → ∞ ] must satisfy γ, γ′ ≥ 2(1 - 1/δ). These inequalities for γ improve the previously known bound γ ≥ 1(Aizenman and Newman), since δ ≥ 2 (Aizenman and Barsky). Additionally, result 1 may be useful, in standard d-dimensional percolation, for proving rigorously (in d>2) that, as expected, Px has no discontinuity at pc.

KW - critical exponent inequalities

KW - Percolation

KW - rigorous results

UR - http://www.scopus.com/inward/record.url?scp=0002333363&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002333363&partnerID=8YFLogxK

U2 - 10.1007/BF01021076

DO - 10.1007/BF01021076

M3 - Article

VL - 45

SP - 359

EP - 368

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -