### Abstract

We review the derivation of Wilson's differential equation in (infinitely) many variables, which describes the infinitesimal change in an effective potential of a statistical mechanical model or quantum field theory when an infinitesimal "integration out" is performed. We show that this equation can be solved for short times by a very elementary method when the initial data are bounded and analytic. The resulting series solutions are generalizations of the Mayer expansion in statistical mechanics. The differential equation approach gives a remarkable identity for "connected parts" and precise estimates which include criteria for convergence of iterated Mayer expansions. Applications include the Yukawa gas in two dimensions past the Β=4 π threshold and another derivation of some earlier results of Göpfert and Mack.

Original language | English (US) |
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Pages (from-to) | 19-49 |

Number of pages | 31 |

Journal | Journal of Statistical Physics |

Volume | 48 |

Issue number | 1-2 |

DOIs | |

State | Published - Jul 1 1987 |

Externally published | Yes |

### Keywords

- Multiscale Mayer expansions
- renormalization group
- tree graph identities

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Statistical Physics*,

*48*(1-2), 19-49. https://doi.org/10.1007/BF01010398