Percolation thresholds and fractal dimensions for square and cubic lattices with long-range correlated defects

Zierenberg, Johannes; Fricke, Niklas; Marenz, Martin; Spitzner, F. Paul; Blavatska, V.; Janke, Wolfhard

doi:10.1103/physreve.96.062125

Cited by 43 publications

(42 citation statements)

References 47 publications

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“…At ν=10 5 , the cluster grows in a fractal percolation-like pattern (figure 2(a)) with its dimension saturating at D∼1.9 for increasing cluster size ( figure 3(a)). Note that this is in practical agreement with the theoretical value of D= 91/48≈1.8958 [50,52]. At ν=10, the cluster grows in a compact circular fashion ( figure 2(b)) with a cluster dimension of D=2 ( figure 3(b)).…”

Section: Simulation Resultssupporting

confidence: 88%

“…To understand how a particular growth pattern is formed, computer growth models are often employed to simulate the underlying growth process. Well-known examples include the models of diffusion-limited aggregation (DLA) [35][36][37][38][39] (and relevant diffusion-based models [40,41]), cluster-cluster aggregation (CCA) [42] and percolation [43][44][45][46][47][48][49][50][51][52] for the growth of fractal-like clusters, as well as Eden's model [53,54], including its noise-reduced [55,56], off-lattice [57] and anisotropy-corrected [58] variants, for the growth of compact-like structures. Such growth models, which do not involve any detailed molecular mechanism, provides a unified description of growth processes across different experimental systems.…”

Section: Introductionmentioning

confidence: 99%

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A dynamical model for fractal and compact growth in supercooled systems

Chan

Dierking

2020

J. Phys. Commun.

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A dynamical model that can exhibit both fractal percolation growth and compact circular growth is presented. At any given cluster size, the dimension of a cluster growing on a two-dimensional square lattice depends on the ratio between the rates of two probabilistic processes, namely (i) the aggregation of lattice sites into the growing cluster and (ii) the relaxation of lattice sites into those available for potential aggregation. The proposed model approaches the limit of two-dimensional invasion percolation if the aggregation process is much faster than the relaxation process, and it approaches Eden's model for compact circular growth if the relaxation process is much faster than the aggregation process. Experimental examples of the fractal-growth regime include the percolation-like growth of bent-core smectics and calamitic smectics, where such fractal growth is attributed to the slow relaxation of molecules in a viscous supercooled medium.

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Section: Simulation Resultssupporting

confidence: 88%

Section: Introductionmentioning

confidence: 99%

A dynamical model for fractal and compact growth in supercooled systems

Chan

Dierking

2020

J. Phys. Commun.

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“…Using this approach, we simulated the system sizes L ∈ {8, 12,16,24,32,48,64,96,128,192 We first discuss the ratio of correlation length and system size, ξ/L. This is known to be universal for a given choice of boundary conditions and aspect ratio.…”

mentioning

confidence: 99%

Universality from disorder in the random-bond Blume-Capel model

et al. 2018

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Using high-precision Monte Carlo simulations and finite-size scaling we study the effect of quenched disorder in the exchange couplings on the Blume-Capel model on the square lattice. The first-order transition for large crystal-field coupling is softened to become continuous, with a divergent correlation length. An analysis of the scaling of the correlation length as well as the susceptibility and specific heat reveals that it belongs to the universality class of the Ising model with additional logarithmic corrections which is also observed for the Ising model itself if coupled to weak disorder. While the leading scaling behavior of the disordered system is therefore identical between the second-order and first-order segments of the phase diagram of the pure model, the finite-size scaling in the ex-first-order regime is affected by strong transient effects with a crossover length scale L^{*}≈32 for the chosen parameters.

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“…The above prediction was supported by many numerical works, see for instance [3,4,12,25,26]. There are no theoretical predition for D f in the range −3/4 < H < 0.…”

Section: Introductionmentioning

confidence: 84%

Topological effects and conformal invariance in long-range correlated random surfaces

et al. 2020

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We consider discrete random fractal surfaces with negative Hurst exponent H<0H<0. A random colouring of the lattice is provided by activating the sites at which the surface height is greater than a given level hh. The set of activated sites is usually denoted as the excursion set. The connected components of this set, the level clusters, define a one-parameter (HH) family of percolation models with long-range correlation in the site occupation. The level clusters percolate at a finite value h=h_ch=hc and for H\leq-\frac{3}{4}H≤−34 the phase transition is expected to remain in the same universality class of the pure (i.e. uncorrelated) percolation. For -\frac{3}{4}<H< 0−34<H<0 instead, there is a line of critical points with continously varying exponents. The universality class of these points, in particular concerning the conformal invariance of the level clusters, is poorly understood. By combining the Conformal Field Theory and the numerical approach, we provide new insights on these phases. We focus on the connectivity function, defined as the probability that two sites belong to the same level cluster. In our simulations, the surfaces are defined on a lattice torus of size M\times NM×N. We show that the topological effects on the connectivity function make manifest the conformal invariance for all the critical line H<0H<0. In particular, exploiting the anisotropy of the rectangular torus (M\neq NM≠N), we directly test the presence of the two components of the traceless stress-energy tensor. Moreover, we probe the spectrum and the structure constants of the underlying Conformal Field Theory. Finally, we observed that the corrections to the scaling clearly point out a breaking of integrability moving from the pure percolation point to the long-range correlated one.

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Percolation thresholds and fractal dimensions for square and cubic lattices with long-range correlated defects

Cited by 43 publications

References 47 publications

A dynamical model for fractal and compact growth in supercooled systems

A dynamical model for fractal and compact growth in supercooled systems

Universality from disorder in the random-bond Blume-Capel model

Topological effects and conformal invariance in long-range correlated random surfaces

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