Electronic Journal of Probability

Subgaussian concentration and rates of convergence in directed polymers

Kenneth Alexander and Nikolaos Zygouras

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We consider directed random polymers in $(d+1)$ dimensions with nearly gamma i.i.d. disorder.  We study the partition function $Z_{N,\omega}$ and establish exponential concentration of $\log Z_{N,\omega}$ about its mean on the subgaussian scale $\sqrt{N/\log N}$ . This is used to show that $\mathbb{E}[ \log Z_{N,\omega}]$ differs from $N$ times the free energy by an amount which is also subgaussian (i.e. $o(\sqrt{N})$), specifically $O( \sqrt{\frac{N}{\log N}}\log \log N)$

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 5, 28 pp.

Accepted: 11 January 2013
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Secondary: 82D60: Polymers 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

directed polymers concentration modified Poincar\'e inequalities coarse graining

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Alexander, Kenneth; Zygouras, Nikolaos. Subgaussian concentration and rates of convergence in directed polymers. Electron. J. Probab. 18 (2013), paper no. 5, 28 pp. doi:10.1214/EJP.v18-2005. https://projecteuclid.org/euclid.ejp/1465064230

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