Electronic Journal of Probability

Free energy of directed polymers in random environment in $1+1$-dimension at high temperature

Makoto Nakashima

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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 \[ \mathcal{Z} _\beta (t)=\int _\mathbb{R} \mathcal{Z} _\beta (t,x)dx. \]

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 50, 43 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1558404408

Digital Object Identifier
doi:10.1214/19-EJP292

Mathematical Reviews number (MathSciNet)
MR3954790

Zentralblatt MATH identifier
07068781

Subjects
Primary: 82D60: Polymers
Secondary: 82C44: Dynamics of disordered systems (random Ising systems, etc.)

Keywords
directed polymers free energy continuum directed polymer

Rights
Creative Commons Attribution 4.0 International License.

Citation

Nakashima, Makoto. Free energy of directed polymers in random environment in $1+1$-dimension at high temperature. Electron. J. Probab. 24 (2019), paper no. 50, 43 pp. doi:10.1214/19-EJP292. https://projecteuclid.org/euclid.ejp/1558404408


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