Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The quenched limiting distributions of a charged-polymer model

Nadine Guillotin-Plantard and Renato Soares dos Santos

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The limit distributions of the charged-polymer Hamiltonian of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths and Higgs [Gaussian case] are considered. Two sources of randomness enter in the definition: a random field $q=(q_{i})_{i\geq1}$ of i.i.d. random variables, which is called the random charges, and a random walk $S=(S_{n})_{n\in\mathbb{N}}$ evolving in $\mathbb{Z}^{d}$, independent of the charges. The energy or Hamiltonian $K=(K_{n})_{n\geq2}$ is then defined as

\[K_{n}:=\sum_{1\leq i<j\leq n}q_{i}q_{j}\mathbf{1}_{\{S_{i}=S_{j}\}}.\] The law of $K$ under the joint law of $q$ and $S$ is called “annealed,” and the conditional law given $q$ is called “quenched.” Recently, strong approximations under the annealed law were proved for $K$. In this paper we consider the limit distributions of $K$ under the quenched law.


Les lois limites de l’hamiltonien dans le modèle de polymère chargé introduit par Kantor et Kardar dans le cas Bernoulli et par Derrida, Griffiths et Higgs dans le cas gaussien sont considérées. Deux aléas interviennent dans la définition : un champ aléatoire $q=(q_{i})_{i\geq1}$ de variables aléatoires i.i.d., appelées charges et une marche aléatoire $S=(S_{n})_{n\in\mathbb{N}}$ dans $\mathbb{Z}^{d}$, indépendante des charges. L’énergie ou hamiltonien $K=(K_{n})_{n\geq2}$ est définie par

\[K_{n}:=\sum_{1\leq i<j\leq n}q_{i}q_{j}\mathbf{1}_{\{S_{i}=S_{j}\}}.\] La loi de $K$ sous la loi conjointe de $q$ et $S$ est appelée « annealed » et la loi conditionnelle sachant $q$ est appelée « quenched ». Récemment, des approximations fortes sous la loi annealed ont été prouvées pour $K$. Dans ce papier, nous considérons les lois limites de $K$ sous la loi quenched.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 2 (2016), 703-725.

Received: 22 January 2014
Revised: 24 September 2014
Accepted: 7 October 2014
First available in Project Euclid: 4 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F05: Central limit and other weak theorems

Random walk Polymer model Self-intersection local time Limit theorems Law of the iterated logarithm Martingale


Guillotin-Plantard, Nadine; dos Santos, Renato Soares. The quenched limiting distributions of a charged-polymer model. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 2, 703--725. doi:10.1214/14-AIHP654.

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