Walks on the Penrose lattice

Langie, Greet; Igloi, G

doi:10.1088/0305-4470/25/8/018

Cited by 17 publications

(25 citation statements)

References 16 publications

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“…ordinary two-dimensional Ising behaviour. In the same way, random and selfavoiding walks, studied via Monte-Carlo techniques, were shown to keep their usual asymptotic properties [7].…”

Section: Introductionmentioning

confidence: 82%

Surface magnetization of aperiodic ising quantum chains

Turban¹,

Berche²

1993

Z. Physik B - Condensed Matter

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Abstract. We study the surface magnetization of aperiodic Ising quantum chains. Using fermion techniques, exact results are obtained in the critical region for quasiperiodic sequences generated through an irrational number as well as for the automatic binary Thue-Morse sequence and its generalizations modulo p. The surface magnetization exponent keeps its Ising value, βs = 1/2, for all the sequences studied. The critical amplitude of the surface magnetization depends on the strength of the modulation and also on the starting point of the chain along the aperiodic sequence.

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“…ordinary two-dimensional Ising behaviour. In the same way, random and selfavoiding walks, studied via Monte-Carlo techniques, were shown to keep their usual asymptotic properties [7].…”

Section: Introductionmentioning

confidence: 82%

Surface magnetization of aperiodic ising quantum chains

Turban¹,

Berche²

1993

Z. Physik B - Condensed Matter

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“…As a related earlier work, one may mention an investigation of the Brownian motion on the two-dimensional Penrose lattice, where normal diffusive behavior has been found [8].…”

Section: Introductionmentioning

confidence: 93%

Anomalous diffusion in aperiodic environments

Iglói

Turban²,

Rieger

1999

Phys. Rev. E

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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.

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“…In the field of critical phenomena, quasiperiodic or aperiodic systems are quite interesting since they offer the possibility to interpolate between periodic and random systems. The Ising model [7][8][9], the percolation problem [10,11] and the statistics of self-avoiding walks [12] were found to display the same critical behaviour on the two-dimensional (2d) Penrose lattice as on a periodic one. Universal behaviour was also obtained in 3d [13].…”

Section: Introductionmentioning

confidence: 95%

“…One may notice that ν has to keep its unperturbed value in order to have varying exponents. Otherwise the marginality condition φ = 0 in equation (12) would no longer be satisfied in the perturbed system. Thus, as indicated in the appendices, β s is equal to the scaling dimension of the surface magnetization x ms with: These exponents are the extreme anisotropic limits of those obtained analytically on classical 2d systems with a finite value of the anisotropy ratio K 1 /K 2 [48].…”

Section: Introductionmentioning

confidence: 99%

Marginal Anisotropy in Layered Aperiodic Ising Systems

Berche¹,

Berche²,

Turban³

1996

J. Phys. I France

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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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Walks on the Penrose lattice

Cited by 17 publications

References 16 publications

Surface magnetization of aperiodic ising quantum chains

Surface magnetization of aperiodic ising quantum chains

Anomalous diffusion in aperiodic environments

Marginal Anisotropy in Layered Aperiodic Ising Systems

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