A remark on the bound for the free energy of directed polymers in random environment in 1+2 dimension

Abstract

We consider the behavior of the quantity $p(\beta )$; the free energy of directed polymers in random environment in $1+2$ dimension, where $\beta$ is inverse temperature. We know that the free energy is strictly negative when $\beta$ is not zero. In this paper, we will prove that $p(\beta )$ is bounded from above by $-\exp \left(-\frac{c_{\epsilon }}{{\beta }^{2+\epsilon }}\right)$ for small $\beta$, where $c_{\epsilon }>0$ is a constant depending on $\epsilon >0$. Also, we will suggest a strategy to get a sharper asymptotics.

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Cited by in corpus (2)

• Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift
• High temperature behaviors of the directed polymer on a cylinder