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Directed Polymer in Random Potentials on $d=3+1$
New Phys.: Sae Mulli 2019; 69: 1189~1193
Published online November 29, 2019;
© 2019 New Physics: Sae Mulli.

Jin Min KIM*

Department of Physics, Soongsil University, Seoul 06978, Korea
Correspondence to:
Received July 31, 2019; Revised August 27, 2019; Accepted September 4, 2019.
cc This is an open-access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
We have studied a directed polymer in random media on $d=3+1$. The standard deviation $\Delta E(t)$ of the minimum energy $E(t)$ of the polymer increases as $t^\beta$ for the polymer length $t$ and becomes saturated at $\Delta E(L)$, where $L$ is the system size. Using the relations $\Delta E(t) \sim t^\beta$ and $\Delta E(L) \sim L^\alpha$, $\beta=0.191 \pm 0.006$ and $\alpha=0.324 \pm0.008$ are obtained. The end-to-end distance $\Delta X(t)$ of the polymer is monitored as a function of $t$ and $z=1.696 \pm0.009$ is obtained independently through the relation $\Delta X(t) \sim t^{1/z}$. It is consistent with the value of $z$ estimated from the relation $z=\alpha/\beta$. The relation $\alpha+z=2$ is satisfied very well. We also measured the skewness and the kurtosis of the energy distribution. They are in good agreement with the results of the restricted solid-on-solid model. Thus, the normalized energy distribution should be universal even in $d=3+1$.
PACS numbers: 05.40-a, 64.60.Ht, 68.35.Ct
Keywords: Random media, Directed polymer, End-to-end distance, Restricted solid-on-solid model

November 2019, 69 (11)
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