A new "static" renormalization group approach to stochastic models of fluctuating surfaces with spatially quenched noise is proposed in which only timeindependent quantities are involved. As examples, quenched versions of the Kardar-Parisi-Zhang model and its Pavlik's modification, the Hwa-Kardar model of selforganized criticality, and Pastor-Satorras-Rothman model of landscape erosion are studied. It is shown that the upper critical dimension in the quenched models is shifted by two units upwards in comparison to their counterparts with white in-time noise. Possible scaling regimes associated with fixed points of the renormalization group equations are found and the critical exponents are derived to the leading order of the corresponding ε expansions. Some exact values and relations for these exponents are obtained.
The effects of a randomly moving environment on a randomly growing interface are studied by the field theoretic renormalization group analysis. The kinetic growth of an interface (kinetic roughening) is described by the Kardar-Parisi-Zhang stochastic differential equation while the velocity field of the moving medium is modelled by the Navier-Stokes equation with an external random force. It is found that the large-scale, long-time (infrared) asymptotic behavior of the system is divided into four nonequilibrium universality classes related to the four types of the renormalization group equations fixed points. In addition to the previously established regimes of asymptotic behavior (ordinary diffusion, ordinary kinetic growth process, and passively advected scalar field), a new nontrivial regime is found. The fixed point coordinates, their regions of stability and the critical dimensions related to the critical exponents (e.g. roughness exponent) are calculated to the first order of the expansion in ε = 2 − d where d is a space dimension (one-loop approximation) or exactly. The new regime possesses a feature typical to the the Kardar-Parisi-Zhang model: the fixed point corresponding to the regime cannot be reached from a physical starting point. Thus, physical interpretation is elusive.
Abstract. Critical behaviour of the O(n)-symmetric φ4 -model with an antisymmetric tensor order parameter is studied by means of the field-theoretic renormalization group (RG) in the leading order of the ε = 4 − d-expansion (one-loop approximation). For n = 2 and 3 the model is equivalent to the scalar and the O(3)-symmetric vector models, for n ≥ 4 it involves two independent interaction terms and two coupling constants. It is shown that for n > 4 the RG equations have no infrared (IR) attractive fixed points and their solutions (RG flows) leave the stability region of the model. This means that fluctuations of the order parameter change the nature of the phase transition from the second-order type (suggested by the mean-field theory) to the first-order one. For n = 4, the IR attractive fixed point exists and the IR behaviour is non-universal: if the coupling constants belong to the basin of attraction for the IR point, the phase transition is of the second order and the IR critical scaling regime realizes. The corresponding critical exponents ν and η are presented in the order ε and ε 2 , respectively. Otherwise the RG flows pass outside the stability region and the first-order transition takes place.
При помощи ренормгруппового подхода рассматривается критическое поведение O(n)-симметричной модели φ 4 с антисимметричным тензорным параметром порядка. Ранее данная модель изучалась в рамках трёхпетлевого приближения посредством методов ε-разложения с пересуммированием конформ-борелевским методом и псевдо-ε-разложения. Было показано, что результаты данных подходов качественно согласуются друг с другом, однако полученные значения критических индексов могут довольно существенно отличаться. Более того, оставался открытым вопрос о чувствительности даже качественных результатов по отношению к учёту следующих порядков теории возмущений. В настоящей работе ренормгрупповые функции и критические индексы модели вычислены с четырёхпетлевой точностью в рамках обоих подходов. Их анализ показывает, что результаты применения различных подходов качественно согласуются и между собой и с результатами трёхпетлевого анализа. Библиогр. 29 назв. Табл. 3. Ключевые слова: ренормализационная группа, критическое поведение.
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