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 \(1/\log \delta \), 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.
This article is dedicated to the memory of Vladas Sidoravicius, colleague and friend.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Belisle, C.: Windings of random walks. Ann. Probab. 17, 1377–1402 (1989)
van de Brug, T., Camia, F., Lis, M.: Spin systems from loop soups. Electron. J. Probab. 23, 17 pp. (2018)
Budd, T.: The peeling process in random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen) (2018). Preprint arXiv:1809.02012
Budd, T.: Winding of simple walks on the square lattice. J. Comb. Theory Ser. A. 172, 105191 (2019)
Camia, F.: Scaling limits, Brownian loops, and conformal fields. In: Contucci, P., Giardinà, C. (eds.) Advances in Disordered Systems, Random Processes and Some Applications, pp. 205–269. Cambridge University Press, Cambridge (2017)
Camia, F., Gandolfi, A., Kleban, M.: Conformal correlation functions in the Brownian loop soup. Nuclear Phys. B 902, 483–507 (2016)
Lawler, G.F.: Topics in loop measures and the loop-erased walk. Probab. Surveys 15, 28–101 (2018)
Lawler, G.F., Limic V.: Random Walk: A Modern Introduction. Cambridge University Press, Cambridge (2010)
Lawler, G.F., Trujillo Ferreras, J.A.: Random walk loop soup. Trans. Amer. Math. Soc. 359, 767–787 (2007)
Lawler, G.F., Werner, W.: The Brownian loop soup. Probab. Theory Relat. Fields 128, 565–588 (2004)
Le Jan, Y.: Markov Paths, Loops and Fields. Lecture Notes in Mathematics, vol. 206. Springer, Heidelberg (2011)
Le Jan, Y.: Brownian winding fields. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds.) Séminaire de Probabilités L. Lecture Notes in Mathematics, pp. 487–492, vol. 2252. Springer, Cham (2019)
Le Jan, Y.: Brownian loops topology. Potential Analy. 53, 223–229 (2019)
Lupu, T.: Topological expansion in isomorphisms with random walks for matrix valued fields (2019). Preprint arXiv:1908.06732
Schapira, B., Young, R.: Windings of planar random walks and averaged Dehn function. Ann. Inst. H. Poincaré Probab. Statist. 47, 130–147 (2011)
Spitzer, F.: Some theorems concerning 2-dimensional Brownian motion. Trans. Amer. Math. Soc. 87, 187–197 (1958)
Symanzik, K.: Euclidean quantum field theory. Rend. Scu. Int. Fis. Enrico. Fermi. 45, 152–226 (1969)
Sznitman, A.-S.: Topics in Occupation Times and Gaussian Free Fields. European Mathematical Society, Zürich (2012)
Yor, M.: Loi de l’indice du lacet Brownien, et distribution de Hartman-Watson. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 53, 71–95 (1980)
Yor, M.: Some Aspects of Brownian Motion. Part I: Some Special Functionals. Birkhäuser, Basel (1992)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Camia, F., Jan, Y.L., Reddy, T.R. (2021). Limit Theorems for Loop Soup Random Variables. In: Vares, M.E., Fernández, R., Fontes, L.R., Newman, C.M. (eds) In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius. Progress in Probability, vol 77. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-60754-8_11
Download citation
DOI: https://doi.org/10.1007/978-3-030-60754-8_11
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-60753-1
Online ISBN: 978-3-030-60754-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)