Limit Theorems for Loop Soup Random Variables

Federico Camia*, Yves Le Jan, Tulasi Ram Reddy

*Corresponding author for this work

Research output: Chapter in Book / Report / Conference proceedingChapterAcademicpeer-review


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 languageEnglish
Title of host publicationIn and Out of Equilibrium 3: Celebrating Vladas Sidoravicius
EditorsMaria Eulália Vares, Roberto Fernández, Luiz Renato Fontes, Charles M. Newman
Number of pages19
ISBN (Electronic)9783030607548
ISBN (Print)9783030607531, 9783030607562
Publication statusPublished - 2021

Publication series

NameProgress in Probability
ISSN (Print)1050-6977
ISSN (Electronic)2297-0428

Bibliographical note

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


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


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