TY - JOUR
T1 - The near-critical scaling window for directed polymers on disordered trees
AU - Alberts, Tom
AU - Ortgiese, Marcel
PY - 2013
Y1 - 2013
N2 - 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.
AB - 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.
UR - https://doi.org/10.1214/EJP.v18-2036
UR - https://doi.org/10.1214/EJP.v18-2036
U2 - 10.1214/EJP.v18-2036
DO - 10.1214/EJP.v18-2036
M3 - Article
SN - 1083-6489
VL - 18
JO - Electronic Journal of Probability
JF - Electronic Journal of Probability
M1 - 19
ER -