## Abstract

We prove that the one-site distribution of Gibbs states (for any finite spin set S) on the Bethe lattice is given by the points satisfying the equation π=T^{2}π, where T=h·A·φ{symbol}, with φ{symbol}(x)=x^{(q-1}/q, h(x)=(x∥x∥_{q})^{q}, A=(a(r, s):r, s∈S), and {Mathematical expression} We also show that for A a symmetric, irreducible operator the nonlinear evolution on probability vectors x(n+1)=Ax(n)^{p}∥Ax(n)^{p}∥_{1} with p>0 has limit points ξ of period≤2. We show that A positive definite implies limit points are fixed points that satisfy the equation Aξ^{p}=λξ. The main tool is the construction of a Liapunov functional by means of convex analysis techniques.

Original language | English |
---|---|

Pages (from-to) | 267-285 |

Number of pages | 19 |

Journal | Journal of Statistical Physics |

Volume | 52 |

Issue number | 1-2 |

DOIs | |

State | Published - Jul 1988 |

Externally published | Yes |

## Keywords

- Bethe lattice
- Gibbs states
- Liapunov functional
- automata networks
- convex function
- cyclically monotone function
- dynamical systems
- spin vector
- subdifferential