Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Lace expansion for dummies

Erwin Bolthausen, Remco van der Hofstad, and Gady Kozma

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Abstract

We show Green’s function asymptotic upper bound for the two-point function of weakly self-avoiding walk in $d>4$, revisiting a classic problem. Our proof relies on Banach algebras to analyse the lace-expansion fixed point equation and is simpler than previous approaches in that it avoids Fourier transforms.

Résumé

Nous montrons une domination asymptotique de la fonction à deux points de la marche faiblement auto-évitante en dimension $d>4$ par la fonction de Green, revisitant ainsi un problème classique. Notre preuve s’appuie sur des techniques d’algèbres de Banach pour analyser le point fixe de l’équation de développement en lacets. Elle est plus simple que les approches précédentes car elle ne passe pas par la transformée de Fourier.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 141-153.

Dates
Received: 26 April 2016
Revised: 27 August 2016
Accepted: 22 September 2016
First available in Project Euclid: 19 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1519030822

Digital Object Identifier
doi:10.1214/16-AIHP797

Mathematical Reviews number (MathSciNet)
MR3765883

Zentralblatt MATH identifier
06880048

Subjects
Primary: 84B41 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Self-avoiding walk Lace expansion Banach algebra Deconvolution Edgeworth expansion

Citation

Bolthausen, Erwin; van der Hofstad, Remco; Kozma, Gady. Lace expansion for dummies. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 141--153. doi:10.1214/16-AIHP797. https://projecteuclid.org/euclid.aihp/1519030822


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