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The phase transitions and critical properties of two types of inhomogeneous systems are reviewed. In one case, the local critical behaviour results from the particular shape of the system. Here scale-invariant forms like wedges or cones are considered as well as general parabolic shapes. In the other case the system contains defects, either narrow ones in the form of lines or stars, or extended ones where the couplings deviate from their bulk values according to power laws. In each case the perturbation may be irrelevant, marginal or relevant. In the marginal case one finds local exponents which depend on a parameter. In the relevant case unusual stretched exponential behaviour and/or local first order transitions appear. The discussion combines mean field theory, scaling considerations, conformal transformations and perturbation theory. A number of examples are Ising models for which exact results can be obtained. Some walks and polymer problems are considered, too.

We study first-and second-order phase transitions of ferromagnetic lattice models on scalefree networks, with a degree exponent γ. Using the example of the q-state Potts model we derive a general self-consistency relation within the frame of the Weiss molecular-field approximation, which presumably leads to exact critical singularities. Depending on the value of γ, we have found three different regimes of the phase diagram. As a general trend first-order transitions soften with decreasing γ and the critical singularities at the second-order transitions are γ dependent.Complex networks, which have more complicated connectivity structure than periodic lattices (PLs) have attracted considerable interest recently [1,2]. This research is motivated by empirical data collected and analyzed in different fields. Small-world (SW) networks [3], which can be generated from PLs by replacing a fraction p of bonds by new random links of arbitrary lengths, are suitable to model neural networks [4] and transportation systems [5]. On the other hand, scale-free (SF) networks [6] are realized among others in social systems [7], in protein interaction networks [8], in the internet [9] and in the world-wide web [10]. In a SF network the degree distribution P D (k), where k is the number of links connected to a vertex, has an asymptotic power-law decay P D (k) ∼ k −γ , thus there is no characteristic scale involved. In natural and artificial networks the value of the degree exponent is usually in the range 2 < γ < 3 [11].Cooperative processes such as spread of epidemic disease [12], percolation [13], Ising model [14,15], etc. have also been studied in the SW and the SF networks. For SW networks numerical studies show [16] that any finite fraction of new, long-range bonds, p > 0, brings the transition into the classical, mean-field (MF) universality class. It is understandable since for systems with longrange interactions the MF approximation is exact. In the SF networks, where links between remote sites exists, too, at first thought one could expect also a traditional MF critical behavior. In specific problems, however, it turned out that it is only true for losely connected networks, when the degree exponent γ is large enough. Otherwise the critical singularities of the transition are model independent, but nonuniversal; the critical exponents continuously depend on the value of the degree exponent. In particular, for 2 < γ ≤ 3, when k 2 is divergent the systems are in their ordered phase for any value of the control parameter (temperature, percolation probability, transition rate, etc.), and the critical properties can be investigated in the limit of infinitely strong fluctuations.Till now investigations on cooperative processes in the SF networks are almost exclusively limited to continuous phase transitions. However, in many problems the phase transitions on PLs are first order and it seems natural to ask what happens with these transitions on the SF networks? There is a general tendency that the discontinuities (e.g., the latent ...

Abstract. We consider the behaviour of a critical system in the presence of a gradient perturbation of the couplings. In the direction of the gradient an interface region separates the ordered phase from the disordered one. We develop a scaling theory for the density profiles induced by the gradient perturbation which involves a characteristic length given by the width of the interface region. The scaling predictions are tested in the framework of the mean-field Ginzburg-Landau theory. Then we consider the Ising quantum chain in a linearly varying transverse field which corresponds to the extreme anisotropic limit of a classical two-dimensional Ising model. The quantum Hamiltonian can be diagonalized exactly in the scaling limit where the eigenvalue problem is the same as for the quantum harmonic oscillator. The energy density, the magnetization profile and the two-point correlation function are studied either analytically or by exact numerical calculations. Their scaling behaviour are in agreement with the predictions of the scaling theory.

We consider the Ising model and the directed walk on two-dimensional layered lattices and show that the two problems are inherently related: The zero-field thermodynamical properties of the Ising model are contained in the spectrum of the transfer matrix of the directed walk. The critical properties of the two models are connected to the scaling behavior of the eigenvalue spectrum of the transfer matrix which is studied exactly through renormalization for different self-similar distributions of the couplings. The models show very rich bulk and surface critical behaviors with nonuniversal critical exponents, coupling-dependent anisotropic scaling, first-order surface transition, and stretched exponential critical correlations. It is shown that all the nonuniversal critical exponents obtained for the aperiodic Ising models satisfy scaling relations and can be expressed as functions of varying surface magnetic exponents.

We consider semi-infinite two-dimensional layered Ising models in the extreme anisotropic limit with an aperiodic modulation of the couplings. Using substitution rules to generate the aperiodic sequences, we derive functional equations for the surface magnetization. These equations are solved by iteration and the critical exponent βs can be determined exactly. The method is applied to three specific aperiodic sequences, which represent different types of perturbation, according to a relevance-irrelevance criterion. On the Thue-Morse lattice, for which the modulation is an irrelevant perturbation, the surface magnetization vanishes with a square root singularity, like in the homogeneous lattice. For the period-doubling sequence, the perturbation is marginal and βs is a continuous function of the modulation amplitude. Finally, the Rudin-Shapiro sequence, which corresponds to the relevant case, displays an anomalous surface critical behavior which is analyzed via scaling considerations. Depending on the value of the modulation, the surface magnetization either vanishes with an essential singularity or remains finite at the bulk critical point, i.e., the surface phase transition is of first order.

We consider layered two-dimensional Ising and directed walk models and show that the two problems are inherently related. The information about the zero-field thermodynamical properties of the Ising model is contained into the transfer matrix of the directed walk. For several hierarchical and aperiodic distributions of the couplings, critical exponents for the two problems are obtained exactly through renormalization. cond-mat/9606118 R 2 J RJ RJ J J J J

We study the Brownian motion of a classical particle in one-dimensional inhomogeneous environments where the transition probabilities follow quasiperiodic or aperiodic distributions. Exploiting an exact correspondence with the transverse-field Ising model with inhomogeneous couplings we obtain many new analytical results for the random walk problem. In the absence of global bias the qualitative behavior of the diffusive motion of the particle and the corresponding persistence probability strongly depend on the fluctuation properties of the environment. In environments with bounded fluctuations the particle shows normal diffusive motion and the diffusion constant is simply related to the persistence probability. On the other hand in a medium with unbounded fluctuations the diffusion is ultra-slow, the displacement of the particle grows on logarithmic time scales. For the borderline situation with marginal fluctuations both the diffusion exponent and the persistence exponent are continuously varying functions of the aperiodicity. Extensions of the results to disordered media and to higher dimensions are also discussed.

Abstract. -Two-dimensional layered aperiodic Ising systems are studied in the extreme anisotropic limit where they correspond to quantum Ising chains in a transverse field. The modulation of the couplings follows an aperiodic sequence generated through substitution. According to Luck's criterion, such a perturbation becomes marginal when the wandering exponent of the sequence vanishes. Three marginal sequences are considered: the period-doubling, paperfolding and three-folding sequences. They correspond to bulk perturbations for which the critical temperature is shifted. The surface magnetization is obtained exactly for the three sequences. The scaling dimensions of the local magnetization on both surfaces, xms and xms, vary continuously with the modulation factor. The low-energy excitations of the quantum chains are found to scale as L z with the size L of the system. This is the behaviour expected for a strongly anisotropic system, where z is the ratio of the exponents of the correlation lengths in the two directions. The anisotropy exponent z is here simply equal to xms +xms. The anisotropic scaling behaviour is verified numerically for other surface and bulk critical properties as well.Résumé. -Onétudie des systèmes d'Ising apériodiques en couchesà deux dimensions dans la limite anisotrope extrême où ils correspondentà des chaînes quantiques d'Ising en champ transverse. La modulation des interactions est engendrée par une suite apériodique obtenue par substitution. D'après le critère de Luck, une telle perturbation devient marginale lorsque l'exposant de "divagation" associéà la suite s'annule. Trois suites marginales sont examinées : "doublement de période", "pliage de papier" et "pliage ternaire". Elles correspondentà des perturbations de volume entrainant un changement de température critique. Des expressions exactes de l'aimantation de surface sont obtenues pour les trois suites. Les dimensions anormales de l'aimantation locale sur les deux surfaces, xms et xms, varient continûment avec le facteur de modulation. Les excitations de basseénergie des chaînes quantiques varient en L z avec la taille L du système. C'est là le comportement attendu pour un système fortement anisotrope où z est le rapport des exposants associés aux longueurs de corrélation dans les deux directions. L'exposant d'anisotropie z s'exprime simplement comme la somme des exposants magnétiques xms + xms. Le comportement d'échelle anisotrope est vérifié numériquement pour d'autres propriétés critiques de surface et de volume.(

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