@inbook{de1899eca22045b0be3b0fc8b3582ab5,

title = "Limit Theorems for Loop Soup Random Variables",

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.",

keywords = "Limit theorems, Loop holonomies, Loop soups, Spitzer{\textquoteright}s law, Winding field, Winding number",

author = "Federico Camia and Jan, {Yves Le} and Reddy, {Tulasi Ram}",

note = "Funding Information: 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 Matem{\'a}tica 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. Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.",

year = "2021",

doi = "10.1007/978-3-030-60754-8_11",

language = "English (US)",

series = "Progress in Probability",

publisher = "Birkhauser",

pages = "219--237",

booktitle = "Progress in Probability",

}