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.


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.

FundersFunder number
NYU-ECNU Institute of Mathematical Sciences


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


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