## Abstract

This article deals with limit theorems for certain loop variables for loop soups whose intensity approaches infinity. We first consider random walk loop soups on finite graphs and obtain a central limit theorem when the loop variable is the sum over all loops of the integral of each loop against a given one-form on the graph. An extension of this result to the noncommutative case of loop holonomies is also discussed. As an application of the first result, we derive a central limit theorem for windings of loops around the faces of a planar graph. More precisely, we show that the winding field generated by a random walk loop soup, when appropriately normalized, has a Gaussian limit as the loop soup intensity tends to ∞, and we give an explicit formula for the covariance kernel of the limiting field. We also derive a Spitzer-type law for windings of the Brownian loop soup, i.e., we show that the total winding around a point of all loops of diameter larger than δ, when multiplied by 1 ∕ log δ, converges in distribution to a Cauchy random variable as δ → 0. The random variables analyzed in this work have various interpretations, which we highlight throughout the paper.

Original language | English |
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Title of host publication | In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius |

Editors | Maria Eulália Vares, Roberto Fernández, Luiz Renato Fontes, Charles M. Newman |

Publisher | Birkhauser |

Pages | 219-237 |

Number of pages | 19 |

ISBN (Electronic) | 9783030607548 |

ISBN (Print) | 9783030607531, 9783030607562 |

DOIs | |

Publication status | Published - 2021 |

### Publication series

Name | Progress in Probability |
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Volume | 77 |

ISSN (Print) | 1050-6977 |

ISSN (Electronic) | 2297-0428 |

### Bibliographical note

Publisher Copyright:© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

### Funding

Acknowledgments The first author thanks David Brydges for an enlightening discussion during the workshop “Random Structures in High Dimensions” held in June–July 2016 at the Casa Matemática Oaxaca (CMO) in Oaxaca, Mexico. All authors thank an anonymous referee for a careful reading of the manuscript and useful suggestions. The research presented in this paper was carried out while the third author was a postdoctoral associate in the Division of Science of NYU Abu Dhabi. The second author acknowledges the support of NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai.

Funders | Funder number |
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NYU-ECNU Institute of Mathematical Sciences |

## Keywords

- Limit theorems
- Loop holonomies
- Loop soups
- Spitzer’s law
- Winding field
- Winding number