### Abstract

In this paper we show the that the ground state energy of the one-dimensional discrete random Schrödinger operator with Bernoulli potential is controlled asymptotically as the system size N goes to infinity by the random variable ℓ _{N}, the length the longest consecutive sequence of sites on the lattice with potential equal to zero. Specifically, we will show that for almost every realization of the potential the ground state energy behaves asymptotically as π ^{2}/(ℓ _{N}+1) ^{2} in the sense that the ratio of the quantities goes to one.

Original language | English (US) |
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Pages (from-to) | 529-541 |

Number of pages | 13 |

Journal | Journal of Statistical Physics |

Volume | 147 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2012 |

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### Keywords

- Bernoulli
- Discrete
- Ground state energy
- Longest run
- Schrödinger operator

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics