### Abstract

Real-space renormalization group maps, e.g., the majority rule transformation, map Ising-type models to Ising-type models on a coarser lattice. We show that each coefficient in the renormalized Hamiltonian in the lattice-gas variables depends on only a finite number of values of the renormalized Hamiltonian. We introduce a method which computes the values of the renormalized Hamiltonian with high accuracy and so computes the coefficients in the lattice-gas variables with high accuracy. For the critical nearest neighbor Ising model on the square lattice with the majority rule transformation, we compute over 1,000 different coefficients in the lattice-gas variable representation of the renormalized Hamiltonian and study the decay of these coefficients. We find that they decay exponentially in some sense but with a slow decay rate. We also show that the coefficients in the spin variables are sensitive to the truncation method used to compute them.

Original language | English (US) |
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Pages (from-to) | 409-426 |

Number of pages | 18 |

Journal | Journal of Statistical Physics |

Volume | 140 |

Issue number | 3 |

DOIs | |

State | Published - Jun 9 2010 |

### Keywords

- Ising model
- Lattice gas variables
- Majority rule
- Renormalization group

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics