High temperature limits for $(1+1)$-dimensional directed polymer with heavy-tailed disorder

Abstract

The directed polymer model at intermediate disorder regime was introduced by Alberts-Khanin-Quastel~\cite{AKQ12}. It was proved that at inverse temperature $\beta n^{-\gamma }$ with $\gamma =1/4$ the partition function, centered appropriately, converges in distribution and the limit is given in terms of the solution of the stochastic heat equation. This result was obtained under the assumption that the disorder variables posses exponential moments, but its universality was also conjectured under the assumption of six moments. We show that this conjecture is valid and we further extend it by exhibiting classes of different universal limiting behaviors in the case of less than six moments. We also explain the behavior of the scaling exponent for the log-partition function under different moment assumptions and values of $\gamma$.

arXiv comment33 pages, 1 figure

References in corpus (5)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• The Intermediate Disorder Regime for Directed Polymers in Dimension 1+1
• Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
• Revisiting Directed Polymers with heavy-tailed disorder
• Fluctuation exponents for directed polymers in the intermediate disorder regime