Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Universality and Borel summability of arbitrary quartic tensor models

Thibault Delepouve, Razvan Gurau, and Vincent Rivasseau

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Abstract

We extend the study of melonic quartic tensor models to models with arbitrary quartic interactions. This extension requires a new version of the loop vertex expansion using several species of intermediate fields and iterated Cauchy–Schwarz inequalities. Borel summability is proven, uniformly as the tensor size $N$ becomes large. Every cumulant is written as a sum of explicitly calculated terms plus a remainder, suppressed in $1/N$. Together with the existence of the large $N$ limit of the second cumulant, this proves that the corresponding sequence of probability measures is uniformly bounded and obeys the tensorial universality theorem.

Résumé

Nous étendons l’étude de modèles de tenseurs quartiques meloniques aux modèles avec des interactions quartiques arbitraires. Cette extension nécessite une nouvelle version du développement en vertex à boucles à l’aide de plusieurs nouveaux champs intermédiaires ainsi que l’utilisation répétée d’inégalités de Cauchy–Schwarz. La sommabilité de Borel est prouvée uniformément dans la taille $N$ du tenseur. Chaque cumulant est écrit comme une somme de termes explicitement calculés plus un reste supprimé à grand $N$. L’existence d’une limite finie à grand $N$ du second cumulant est établie et l’on démontre que la suite correspondante de mesures de probabilité est uniformément bornée en $N$ et obéit bien au théorème tensoriel d’universalité comme dans le cas melonique.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 52, Number 2 (2016), 821-848.

Dates
Received: 10 April 2014
Revised: 26 September 2014
Accepted: 8 October 2014
First available in Project Euclid: 4 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1462367895

Digital Object Identifier
doi:10.1214/14-AIHP655

Mathematical Reviews number (MathSciNet)
MR3498011

Zentralblatt MATH identifier
1341.81045

Subjects
Primary: 81T08: Constructive quantum field theory

Keywords
Random tensors Borel summability

Citation

Delepouve, Thibault; Gurau, Razvan; Rivasseau, Vincent. Universality and Borel summability of arbitrary quartic tensor models. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 2, 821--848. doi:10.1214/14-AIHP655. https://projecteuclid.org/euclid.aihp/1462367895


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