## 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.

## Keywords

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