### Abstract

The problem of diagonalizing a class of complicated matrices, to be called ultrametric matrices, is investigated. These matrices appear at various stages in the description of disordered systems with many equilibrium phases by the technique of replica-symmetry breaking. The residual symmetry, remaining after the breaking of permutation symmetry between replicas, allows us to bring all ultrametric matrices to a block diagonal form by a common similarity transformation. A large number of these blocks are, in fact, of size 1*1, i.e, in a vast sector the transformation actually diagonalizes the matrix. In the other sectors we end up with blocks of size (R+1)*(R+1), where R is the number of replica-symmetry-breaking steps. These blocks cannot be further reduced without giving more information, in addition to ultrametric symmetry, about the matrix. Similar results for the inverse of a generic ultrametric matrix are also derived.

Original language | English |
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Article number | 008 |

Pages (from-to) | 7569-7595 |

Number of pages | 27 |

Journal | Journal of Physics A: General Physics |

Volume | 27 |

Issue number | 23 |

DOIs | |

Publication status | Published - Dec 1 1994 |

### ASJC Scopus subject areas

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

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## Cite this

*Journal of Physics A: General Physics*,

*27*(23), 7569-7595. [008]. https://doi.org/10.1088/0305-4470/27/23/008