## Electronic Journal of Probability

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

Makoto Nakashima

#### 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
Accepted: 17 March 2019
First available in Project Euclid: 21 May 2019

https://projecteuclid.org/euclid.ejp/1558404408

Digital Object Identifier
doi:10.1214/19-EJP292

Mathematical Reviews number (MathSciNet)
MR3954790

Zentralblatt MATH identifier
1419.82078

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

#### 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

#### References

• [1] Tom Alberts, Konstantin Khanin, and Jeremy Quastel. The continuum directed random polymer. J. Stat. Phys., Vol. 154, No. 1-2, pp. 305–326, 2014.
• [2] Tom Alberts, Konstantin Khanin, and Jeremy Quastel. The intermediate disorder regime for directed polymers in dimension $1+1$. Ann. Probab., Vol. 42, No. 3, pp. 1212–1256, 2014.
• [3] Kenneth S. Alexander and Gökhan Yildirim. Directed polymers in a random environment with a defect line. Electron. J. Probab., Vol. 20, pp. no. 6, 1–20, 2015.
• [4] Gideon Amir, Ivan Corwin, and Jeremy Quastel. Probability distribution of the free energy of the continuum directed random polymer in $1+1$ dimensions. Comm. Pure Appl. Math., Vol. 64, No. 4, pp. 466–537, 2011.
• [5] Emil Artin. The gamma function. Translated by Michael Butler. Athena Series: Selected Topics in Mathematics. Holt, Rinehart and Winston, New York-Toronto-London, 1964.
• [6] Quentin Berger and Hubert Lacoin. The high-temperature behavior for the directed polymer in dimension 1+2. http://arxiv.org/abs/1506.09055, To appear in Annales de l’Institut Henri Poincar’e., 2015.
• [7] Quentin Berger and Fabio Lucio Toninelli. On the critical point of the random walk pinning model in dimension $d=3$. Electron. J. Probab., Vol. 15, No. 21, pp. 654–683, 2010.
• [8] Lorenzo Bertini and Giambattista Giacomin. Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys., Vol. 183, No. 3, pp. 571–607, 1997.
• [9] Matthias Birkner. A condition for weak disorder for directed polymers in random environment. Electron. Comm. Probab., Vol. 9, pp. 22–25 (electronic), 2004.
• [10] Matthias Birkner, Andreas Greven, and Frank den Hollander. Collision local time of transient random walks and intermediate phases in interacting stochastic systems. Electron. J. Probab., Vol. 16, No. 20, pp. 552–586, 2011.
• [11] Matthias Birkner and Rongfeng Sun. Annealed vs quenched critical points for a random walk pinning model. Ann. Inst. Henri Poincaré Probab. Stat., Vol. 46, No. 2, pp. 414–441, 2010.
• [12] Matthias Birkner and Rongfeng Sun. Disorder relevance for the random walk pinning model in dimension 3. Ann. Inst. Henri Poincaré Probab. Stat., Vol. 47, No. 1, pp. 259–293, 2011.
• [13] Erwin Bolthausen. A note on the diffusion of directed polymers in a random environment. Comm. Math. Phys., Vol. 123, No. 4, pp. 529–534, 1989.
• [14] Francesco Caravenna, Fabio Lucio Toninelli, and Niccolo Torri. Universality for the pinning model in the weak coupling regime. arXiv preprint arXiv:1505.04927, 2015.
• [15] Philippe Carmona and Yueyun Hu. On the partition function of a directed polymer in a Gaussian random environment. Probab. Theory Related Fields, Vol. 124, No. 3, pp. 431–457, 2002.
• [16] Philippe Carmona and Yueyun Hu. Strong disorder implies strong localization for directed polymers in a random environment. ALEA Lat. Am. J. Probab. Math. Stat., Vol. 2, pp. 217–229, 2006.
• [17] Francis Comets. Directed polymers in random environment. Saint Flour lecture notes, 2016.
• [18] Francis Comets and Vu-Lan Nguyen. Localization in log-gamma polymers with boundaries. Probab. Theory Related Fields, Vol. 166, No. 1-2, pp. 429–461, 2016.
• [19] Francis Comets, Tokuzo Shiga, and Nobuo Yoshida. Directed polymers in a random environment: path localization and strong disorder. Bernoulli, Vol. 9, No. 4, pp. 705–723, 2003.
• [20] Francis Comets, Tokuzo Shiga, and Nobuo Yoshida. Probabilistic analysis of directed polymers in a random environment: a review. In Stochastic analysis on large scale interacting systems, Vol. 39 of Adv. Stud. Pure Math., pp. 115–142. Math. Soc. Japan, Tokyo, 2004.
• [21] Francis Comets and Vincent Vargas. Majorizing multiplicative cascades for directed polymers in random media. ALEA Lat. Am. J. Probab. Math. Stat., Vol. 2, pp. 267–277, 2006.
• [22] Francis Comets and Nobuo Yoshida. Directed polymers in random environment are diffusive at weak disorder. Ann. Probab., Vol. 34, No. 5, pp. 1746–1770, 2006.
• [23] Nicos Georgiou and Timo Seppäläinen. Large deviation rate functions for the partition function in a log-gamma distributed random potential. Ann. Probab., Vol. 41, No. 6, pp. 4248–4286, 2013.
• [24] Giambattista Giacomin, Hubert Lacoin, and Fabio Lucio Toninelli. Disorder relevance at marginality and critical point shift. Vol. 47, No. 1, pp. 148–175, 2011.
• [25] David A Huse and Christopher L Henley. Pinning and roughening of domain walls in ising systems due to random impurities. Physical review letters, Vol. 54, No. 25, pp. 2708–2711, 1985.
• [26] Svante Janson. Gaussian Hilbert spaces, Vol. 129 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1997.
• [27] Ioannis Karatzas and Steven E. Shreve. Brownian motion and stochastic calculus, Vol. 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991.
• [28] Hubert Lacoin. New bounds for the free energy of directed polymers in dimension $1+1$ and $1+2$. Comm. Math. Phys., Vol. 294, No. 2, pp. 471–503, 2010.
• [29] Gregory F. Lawler and Vlada Limic. Random walk: a modern introduction, Vol. 123 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010.
• [30] Michel Ledoux. The concentration of measure phenomenon, Vol. 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001.
• [31] Quansheng Liu and Frédérique Watbled. Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment. Stochastic Process. Appl., Vol. 119, No. 10, pp. 3101–3132, 2009.
• [32] Gregorio R. Moreno Flores. On the (strict) positivity of solutions of the stochastic heat equation. Ann. Probab., Vol. 42, No. 4, pp. 1635–1643, 2014.
• [33] Elchanan Mossel, Ryan O’Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions with low influences: invariance and optimality. Ann. of Math. (2), Vol. 171, No. 1, pp. 295–341, 2010.
• [34] Makoto Nakashima. A remark on the bound for the free energy of directed polymers in random environment in $1+2$ dimension. J. Math. Phys., Vol. 55, No. 9, pp. 093304, 14, 2014.
• [35] Makoto Nakashima. The free energy of the random walk pinning model. Stochastic Process. Appl., Vol. 128, No. 2, pp. 373–403, 2018.
• [36] David Nualart. The Malliavin calculus and related topics. Probability and its Applications (New York). Springer-Verlag, Berlin, second edition, 2006.
• [37] Jeremy Quastel. Introduction to KPZ. In Current developments in mathematics, 2011, pp. 125–194. Int. Press, Somerville, MA, 2012.
• [38] Tomohiro Sasamoto and Herbert Spohn. Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Phys. B, Vol. 834, No. 3, pp. 523–542, 2010.
• [39] Tomohiro Sasamoto and Herbert Spohn. One-dimensional kardar-parisi-zhang equation: an exact solution and its universality. Physical review letters, Vol. 104, No. 23, p. 230602, 2010.
• [40] Timo Seppäläinen. Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab., Vol. 40, No. 1, pp. 19–73, 2012.
• [41] Fabio Lucio Toninelli. Coarse graining, fractional moments and the critical slope of random copolymers. Electron. J. Probab., Vol. 14, No. 20, pp. 531–547, 2009.
• [42] Frédérique Watbled. Sharp asymptotics for the free energy of $1+1$ dimensional directed polymers in an infinitely divisible environment. Electron. Commun. Probab., Vol. 17, No. 53, pp. 9, 2012.