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Register a new userStatistical properties of one-dimensional directed polymers in a random potential
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Authors
Victor Dotsenko
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This review is devoted to the detailed consideration of the universal statistical properties of one-dimensional directed polymers in a random potential. In terms of the replica Bethe ansatz technique we derive several exact results for different types of the free energy probability distribution functions. In the second part of the review we discuss the problems which are still waiting for their solutions. Several mathematical appendices in the ending part of the review contain various technical details of the performed calculations.
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Zero temperature limit in (1+1) directed polymers with finite range correlated random potential is studied. In terms of the standard replica technique it is demonstrated that in this limit the considered system reveals the one-step replica symmetry breaking structure similar to the one which takes place in the Random Energy Model. In particular, it is shown that at the temperature $T_{*} sim (u R)^{1/3}$ (where $u$ and $R$ are the strength and the correlation length of the random potential) there is a crossover from the high- to the low-temperature regime. Namely, in the high-temperature regime at $T >> T_{*}$ the model is equivalent to the one with the $delta$-correlated potential where the non-universal prefactor of the free energy is proportional to $T^{-2/3}$, while at $T << T_{*}$ this non-universal prefactor saturates at a finite (temperature independent) value.
The asymptotic analytic expression for the two-time free energy distribution function in (1+1) random directed polymers is derived in the limit when the two times are close to each other
In this paper in terms of the replica method we consider the high temperature limit of (2+1) directed polymers in a random potential and propose an approach which allows to compute the scaling exponent $theta$ of the free energy fluctuations as well as the left tail of its probability distribution function. It is argued that $theta = 1/2$ which is different from the zero-temperature numerical value which is close to 0.241. This result implies that unlike the $(1+1)$ system in the two-dimensional case the free energy scaling exponent is non-universal being temperature dependent.
Using the replica method, we develop an analytical approach to compute the characteristic function for the probability $mathcal{P}_N(K,lambda)$ that a large $N times N$ adjacency matrix of sparse random graphs has $K$ eigenvalues below a threshold $lambda$. The method allows to determine, in principle, all moments of $mathcal{P}_N(K,lambda)$, from which the typical sample to sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with $N gg 1$ for $|lambda| > 0$, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdos-Renyi and regular random graphs, both exhibiting a prefactor with a non-monotonic behavior as a function of $lambda$. These results contrast with rotationally invariant random matrices, where the index variance scales only as $ln N$, with an universal prefactor that is independent of $lambda$. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.
The joint statistical properties of two free energies computed at two different temperatures in {it the same sample} of $(1+1)$ directed polymers is studied in terms of the replica technique. The scaling dependence of the reduced free energies difference ${cal F} = F(T_{1})/T_{1} - F(T_{2})/T_{2}$ on the two temperatures $T_{1}$ and $T_{2}$ is derived. In particular, it is shown that if the two temperatures $T_{1} , < , T_{2}$ are close to each other the typical value of the fluctuating part of the reduced free energies difference ${cal F}$ is proportional to $(1 - T_{1}/T_{2})^{1/3}$. It is also shown that the left tail asymptotics of this free energy difference probability distribution function coincides with the corresponding tail of the TW distribution.
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