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Subgaussian concentration and rates of convergence in directed polymers. (English) Zbl 1281.82036

Summary: 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 sub-Gaussian 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 sub-Gaussian (i.e. \(O(\sqrt{N}))\), specifically \(O\left( \sqrt{\frac{N}{\log N}}\log \log N\right)\).

MSC:

82D60 Statistical mechanics of polymers
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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