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.
|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|
|Number of pages||19|
|ISBN (Print)||9783030607531, 9783030607562|
|Publication status||Published - 2021|
|Name||Progress in Probability|
Bibliographical notePublisher 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