Limit Theorems for Loop Soup Random Variables

Federico Camia, Yves Le Jan, Tulasi Ram Reddy

Research output: Chapter in Book/Report/Conference proceedingChapter

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 languageEnglish (US)
Title of host publicationProgress in Probability
PublisherBirkhauser
Pages219-237
Number of pages19
DOIs
StatePublished - 2021

Publication series

NameProgress in Probability
Volume77
ISSN (Print)1050-6977
ISSN (Electronic)2297-0428

Keywords

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

ASJC Scopus subject areas

  • Statistics and Probability
  • Applied Mathematics
  • Mathematical Physics
  • Mathematics (miscellaneous)

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