Revisiting Directed Polymers with heavy-tailed disorder

Abstract

In this mostly numerical study, we revisit the statistical properties of the ground state of a directed polymer in a $d=1+1$ "hilly" disorder landscape, i.e. when the quenched disorder has power-law tails. When disorder is Gaussian, the polymer minimizes its total energy through a collective optimization, where the energy of each visited site only weakly contributes to the total. Conversely, a hilly landscape forces the polymer to distort and explore a larger portion of space to reach some particularly deep energy sites. As soon as the fifth moment of the disorder diverges, this mechanism radically changes the standard "KPZ" scaling behaviour of the directed polymer, and new exponents prevail. After confirming again that the Flory argument accurately predicts these exponent in the tail-dominated phase, we investigate several other statistical features of the ground state that shed light on this unusual transition and on the accuracy of the Flory argument. We underline the theoretical challenge posed by this situation, which paradoxically becomes even more acute above the upper critical dimension.

10 pages, 13 figures

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• Analytical Methods and Field Theory for Disordered Systems
• High temperature limits for $(1+1)$-dimensional directed polymer with heavy-tailed disorder
• Non-directed polymers in heavy-tail random environment in dimension $d\ge 2$
• Optimal paths on the road network as directed polymers
• Localization of a one-dimensional simple random walk among power-law renewal obstacles