Electronic Journal of Probability

The near-critical scaling window for directed polymers on disordered trees

Tom Alberts and Marcel Ortgiese

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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.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 19, 24 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B27: Critical phenomena
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 60G42: Martingales with discrete parameter

Directed polymers in random environment branching random walk multiplicative cascades critical temperature near critical scaling

This work is licensed under a Creative Commons Attribution 3.0 License.


Alberts, Tom; Ortgiese, Marcel. The near-critical scaling window for directed polymers on disordered trees. Electron. J. Probab. 18 (2013), paper no. 19, 24 pp. doi:10.1214/EJP.v18-2036. https://projecteuclid.org/euclid.ejp/1465064244

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