Subgaussian concentration and rates of convergence in directed polymers

Abstract

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 $O(\sqrt{\frac{N}{\log N}}\log \log N)$.

Minor changes. Appears in Electronic Journal of Probability, 18, 2013, no. 5

References in corpus (5)

• Shape Fluctuations and Random Matrices
• Scaling for a one-dimensional directed polymer with boundary conditions
• Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity
• Disorder chaos and multiple valleys in spin glasses
• Subgaussian rates of convergence of means in directed first passage percolation

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• Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation
• The scaling relation chi = 2 xi - 1 for directed polymers in a random environment
• Localization for random walks among random obstacles in a single Euclidean ball
• Sublinear variance in Euclidean first-passage percolation
• Localization and free energy asymptotics in disordered statistical mechanics and random growth models
• Entropy reduction in Euclidean first-passage percolation