Universality of the directed polymer model

Abstract

The universality of the directed polymer model and the analogous KPZ equation is supported by numerical simulations using non-Gaussian random probability distributions in two, three and four dimensions. It is shown that although in the non-Gaussian cases the finite size estimates of the energy exponents are below the persumed universal values, these estimates increase with the system size, and the further they are below the universal values, the higher is their rate of increase. The results are explained in terms of the efficiency of variance reduction during the optimization process. 05.40-aThe directed polymer model [1] and the analogous KPZ equation [2] have drawn significant attention recently [3]. The original introduction of the model [1] considered directed polymers in random media, and more recently it has been applied to model other processes such as tearing and cracks formation [4]. The study of the model is focused on the characteristics of the optimal path which connects two distant points in a random environment. Passing through any point of that environment is associated with a random valued cost, usually expressed in units of energy or time. Unlike the simple case of constant cost, the path of minimal cost (the optimal path) is usually not the straight line which connects the two points, and its mean transversal distance from the straight line which connects its endpoints grows as a power law with the distance between the endpoints. Another power law behaviour is associated with the relation between the variance of the energy cost (or time cost), and the distance between the endpoints. These two power laws are expressed in terms of their exponents, which are connected by a scaling relation. The random environment is usually simulated by a lattice whose bonds (or sites) are assigned with random values, and it is generally believed that the characteristics of the optimal path are universal, i.e. apart from very special cases, the exponents depend only on the dimension of the lattice and not on the details of the lattice structure, or the type of randomness associated with it.The universality hypothesis was recently challenged by numerical results [5] which showed that for dimensions higher than 2, the values of the estimated exponents do depend on the type of the probability distribution which is associated with the random lattice. In particular, nonGaussian distributions yield lower values for the exponents compared to Gaussian distribution values, which are the persumed universal values. This publication has raised strong opposition in the form of three recent articles [6,7,8], which argued that the estimates presented in [5] are affected by finite size effects, and thus they are not the asymptotic values of the exponents in these cases. Two of these articles [6,7] independently suggested that the estimates presented in [5] are disturbed by directed percolation effects. However, none of these opposing articles presented any alternative numerical estimates to those presented in [5].This article pre...

Universal properties of shortest paths in isotropically correlated random potentials

“…For uncorrelated randomness we obtain ν = 0.62 ± 0.02 and ω = 0.22 ± 0.01 in agreement with [1,2,17]. Both ν and ω increase monotonuously with ρ, i.e.…”

“…The former is still discussed [1,13,14,15,16]. As far as the latter is concerned, it seems to be clear that the universal properties are not changed if the randomness is uncorrelated.…”

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Statistical theory for the Kardar-Parisi-Zhang equation in(1+1)dimensions

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