Open Access
2013 The near-critical scaling window for directed polymers on disordered trees
Tom Alberts, Marcel Ortgiese
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Electron. J. Probab. 18: 1-24 (2013). DOI: 10.1214/EJP.v18-2036


We study a directed polymer model in a random environment on infinite binary trees. The model is characterized by a phase transition depending on the inverse temperature. We concentrate on the asymptotics of the partition function in the near-critical regime, where the inverse temperature is a small perturbation away from the critical one with the perturbation converging to zero as the system size grows large. Depending on the speed of convergence we observe very different asymptotic behavior. If the perturbation is small then we are inside the critical window and observe the same decay of the partition function as at the critical temperature. If the perturbation is slightly larger the near critical scaling leads to a new range of asymptotic behaviors, which at the extremes match up with the already known rates for the sub- and super-critical regimes. We use our results to identify the size of the fluctuations of the typical energies under the critical Gibbs measure.


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Tom Alberts. Marcel Ortgiese. "The near-critical scaling window for directed polymers on disordered trees." Electron. J. Probab. 18 1 - 24, 2013.


Accepted: 30 January 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1279.82008
MathSciNet: MR3035747
Digital Object Identifier: 10.1214/EJP.v18-2036

Primary: 82B27
Secondary: 60G42 , 82B44

Keywords: Branching random walk , Critical temperature , directed polymers in random environment , multiplicative cascades , near critical scaling

Vol.18 • 2013
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