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May 2021 Gaussian fluctuations for the directed polymer partition function in dimension d3 and in the whole L2-region
Clément Cosco, Shuta Nakajima
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Ann. Inst. H. Poincaré Probab. Statist. 57(2): 872-889 (May 2021). DOI: 10.1214/20-AIHP1100

Abstract

We consider the discrete directed polymer model with i.i.d. environment and we study the fluctuations of the tail n(d2)/4(WWn) of the normalized partition function. It was proven by Comets and Liu (J. Math. Anal. Appl. 455 (2017) 312–335), that for sufficiently high temperature, the fluctuations converge in distribution towards the product of the limiting partition function and an independent Gaussian random variable. We extend the result to the whole L2-region of temperature, which is predicted to be the maximal high-temperature region where the Gaussian fluctuations should occur under the considered scaling. To do so, we manage to avoid the heavy 4th-moment computation and instead rely on the local limit theorem for polymers (Fund. Math. 147 (1995) 173–180; Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 521–534) and homogenization.

Nous considérons le polymère dirigé discret en environnement i.i.d. et nous étudions les fluctuations de la queue n(d2)/4(WWn) de la fonction de partition normalisée. Comets et Liu (J. Math. Anal. Appl. 455 (2017) 312–335) ont montré que pour une température suffisamment haute, les fluctuations convergent en loi vers le produit de la fonction de partition limite et d’une variable aléatoire gaussienne indépendante. Dans cet article, nous étendons le résultat à toute la région de température L2. Il est attendu que ce soit la région optimale où les fluctuations gaussiennes apparaissent sous le changement d’échelle considéré. Pour y parvenir, nous évitons le calcul de moments d’ordre 4 et nous utilisons à la place le théorème local limite pour les polymères (Fund. Math. 147 (1995) 173–180 ; Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 521–534) ainsi qu’un argument d’homogénéisation.

Acknowledgements

The authors would like to thank Francis Comets for giving them the opportunity to meet, as well as for his careful reading and helpful comments. The authors are also grateful to NYU Shanghai, for the hospitality during the stay where the present work was initiated.

Citation

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Clément Cosco. Shuta Nakajima. "Gaussian fluctuations for the directed polymer partition function in dimension d3 and in the whole L2-region." Ann. Inst. H. Poincaré Probab. Statist. 57 (2) 872 - 889, May 2021. https://doi.org/10.1214/20-AIHP1100

Information

Received: 24 October 2019; Revised: 23 July 2020; Accepted: 28 August 2020; Published: May 2021
First available in Project Euclid: 13 May 2021

Digital Object Identifier: 10.1214/20-AIHP1100

Subjects:
Primary: 60K37, 60K37
Secondary: 60F05

Rights: Copyright © 2021 Association des Publications de l’Institut Henri Poincaré

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Vol.57 • No. 2 • May 2021
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