### Abstract

We study first-passage percolation models and their higher dimensional analogs - models of surfaces with random weights. We prove that under very general conditions the number of lines or, in the second case, hypersurfaces which locally minimize the sum of the random weights is with probability one equal to 0 or with probability one equal to + ∞. As corollaries we show that in any dimension d ≥ 2 the number of ground states of an Ising ferromagnet with random coupling constants equals (with probability one) 2 or + ∞. Proofs employ simple large-deviation estimates and ergodic arguments.

Original language | English (US) |
---|---|

Pages (from-to) | 439-447 |

Number of pages | 9 |

Journal | Journal of Statistical Physics |

Volume | 87 |

Issue number | 1-2 |

State | Published - Apr 1997 |

### Fingerprint

### Keywords

- Disordered systems
- First-passage percolation
- Geodesics
- Ground states
- Large deviations
- Minimal hypersurfaces

### ASJC Scopus subject areas

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

### Cite this

*Journal of Statistical Physics*,

*87*(1-2), 439-447.

**On the number of infinite geodesics and ground states in disordered systems.** / Wehr, Jan.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 87, no. 1-2, pp. 439-447.

}

TY - JOUR

T1 - On the number of infinite geodesics and ground states in disordered systems

AU - Wehr, Jan

PY - 1997/4

Y1 - 1997/4

N2 - We study first-passage percolation models and their higher dimensional analogs - models of surfaces with random weights. We prove that under very general conditions the number of lines or, in the second case, hypersurfaces which locally minimize the sum of the random weights is with probability one equal to 0 or with probability one equal to + ∞. As corollaries we show that in any dimension d ≥ 2 the number of ground states of an Ising ferromagnet with random coupling constants equals (with probability one) 2 or + ∞. Proofs employ simple large-deviation estimates and ergodic arguments.

AB - We study first-passage percolation models and their higher dimensional analogs - models of surfaces with random weights. We prove that under very general conditions the number of lines or, in the second case, hypersurfaces which locally minimize the sum of the random weights is with probability one equal to 0 or with probability one equal to + ∞. As corollaries we show that in any dimension d ≥ 2 the number of ground states of an Ising ferromagnet with random coupling constants equals (with probability one) 2 or + ∞. Proofs employ simple large-deviation estimates and ergodic arguments.

KW - Disordered systems

KW - First-passage percolation

KW - Geodesics

KW - Ground states

KW - Large deviations

KW - Minimal hypersurfaces

UR - http://www.scopus.com/inward/record.url?scp=0031115690&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031115690&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031115690

VL - 87

SP - 439

EP - 447

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-2

ER -