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2019 Free energy of directed polymers in random environment in $1+1$-dimension at high temperature
Makoto Nakashima
Electron. J. Probab. 24: 1-43 (2019). DOI: 10.1214/19-EJP292

Abstract

We consider the free energy $F(\beta )$ of the directed polymers in random environment in $1+1$-dimension. It is known that $F(\beta )$ is of order $-\beta ^4$ as $\beta \to 0$ [3, 28, 42]. In this paper, we will prove that under a certain dimension free concentration condition on the potential, \[ \lim _{\beta \to 0}\frac{F(\beta )} {\beta ^4}=\lim _{T\to \infty }\frac{1} {T}P_\mathcal{Z} \left [\log \mathcal{Z} _{\sqrt{2} }(T)\right ] =-\frac{1} {6}, \] where $\{\mathcal{Z} _\beta (t,x):t\geq 0,x\in \mathbb{R} \}$ is the unique mild solution to the stochastic heat equation \[ \frac{\partial } {\partial t}\mathcal{Z} =\frac{1} {2}\Delta \mathcal{Z} +\beta \mathcal{Z} {\dot{\mathcal W} },\ \ \lim _{t\to 0}\mathcal{Z} (t,x)dx=\delta _{0}(dx), \] where $\mathcal{W} $ is a time-space white noise and \[ \mathcal{Z} _\beta (t)=\int _\mathbb{R} \mathcal{Z} _\beta (t,x)dx. \]

Citation

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Makoto Nakashima. "Free energy of directed polymers in random environment in $1+1$-dimension at high temperature." Electron. J. Probab. 24 1 - 43, 2019. https://doi.org/10.1214/19-EJP292

Information

Received: 2 April 2018; Accepted: 17 March 2019; Published: 2019
First available in Project Euclid: 21 May 2019

zbMATH: 1419.82078
MathSciNet: MR3954790
Digital Object Identifier: 10.1214/19-EJP292

Subjects:
Primary: 82D60
Secondary: 82C44

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Vol.24 • 2019
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