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Scaling and Inverse Scaling in Anisotropic Bootstrap percolation

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 Publication date 2014
  fields Physics
and research's language is English




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In bootstrap percolation it is known that the critical percolation threshold tends to converge slowly to zero with increasing system size, or, inversely, the critical size diverges fast when the percolation probability goes to zero. To obtain higher-order terms (that is, sharp and sharper thresholds) for the percolation threshold in general is a hard question. In the case of two-dimensional anisotropic models, sometimes correction terms can be obtained from inversion in a relatively simple manner.

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In some systems, the connecting probability (and thus the percolation process) between two sites depends on the geometric distance between them. To understand such process, we propose gravitationally correlated percolation models for link-adding networks on the two-dimensional lattice $G$ with two strategies $S_{rm max}$ and $S_{rm min}$, to add a link $l_{i,j}$ to connect site $i$ and site $j$ with mass $m_i$ and $m_j$, respectively; $m_i$ and $m_j$ are sizes of the clusters which contain site $i$ and site $j$, respectively. The probability to add the link $l_{i,j}$ is related to the generalized gravity $g_{ij} equiv m_i m_j/r_{ij}^d$, where $r_{ij}$ is the geometric distance between $i$ and $j$, and $d$ is an adjustable decaying exponent. In the beginning of the simulation, all sites of $G$ are occupied and there is no link. In the simulation process, two inter-cluster links $l_{i,j}$ and $l_{k,n}$ are randomly chosen and the generalized gravities $g_{ij}$ and $g_{kn}$ are computed. In the strategy $S_{rm max}$, the link with larger generalized gravity is added. In the strategy $S_{rm min}$, the link with smaller generalized gravity is added, which include percolation on the ErdH os-Renyi random graph and the Achlioptas process of explosive percolation as the limiting cases, $d to infty$ and $d to 0$, respectively. Adjustable strategies facilitate or inhibit the network percolation in a generic view. We calculate percolation thresholds $T_c$ and critical exponents $beta$ by numerical simulations. We also obtain various finite-size scaling functions for the node fractions in percolating clusters or arrival of saturation length with different intervening strategies.
In this paper, we study the diffusive limit of solutions to the generalized Langevin equation (GLE) in a periodic potential. Under the assumption of quasi-Markovianity, we obtain sharp longtime equilibration estimates for the GLE using techniques from the theory of hypocoercivity. We then prove asymptotic results for the effective diffusion coefficient in three limiting regimes: the short memory, the overdamped and the underdamped limits. Finally, we employ a recently developed spectral numerical method in order to calculate the effective diffusion coefficient for a wide range of (effective) friction coefficients, confirming our asymptotic results.
The size of large cliff failures may be described in several ways, for instance considering the horizontal eroded area at the cliff top and the maximum local retreat of the coastline. Field studies suggest that, for large failures, the frequencies of these two quantities decrease as power laws of the respective magnitudes, defining two different decay exponents. Moreover, the horizontal area increases as a power law of the maximum local retreat, identifying a third exponent. Such observation suggests that the geometry of cliff failures are statistically similar for different magnitudes. Power laws are familiar in the physics of critical systems. The corresponding exponents satisfy precise relations and are proven to be universal features, common to very different systems. Following the approach typical of statistical physics, we propose a scaling hypothesis resulting in a relation between the three above exponents: there is a precise, mathematical relation between the distributions of magnitudes of erosion events and their geometry. Beyond its theoretical value, such relation could be useful for the validation of field catalogs analysis. Pushing the statistical physics approach further, we develop a numerical model of marine erosion that reproduces the observed failure statistics. Despite the minimality of the model, the exponents resulting from extensive numerical simulations fairly agree with those measured on the field. These results suggest that the mathematical theory of percolation, which lies behind our simple model, can possibly be used as a guide to decipher the physics of rocky coast erosion and could provide precise predictions to the statistics of cliff collapses.
70 - Adam Gamsa , John Cardy 2005
The probability that a point is to one side of a curve in Schramm-Loewner evolution (SLE) can be obtained alternatively using boundary conformal field theory (BCFT). We extend the BCFT approach to treat two curves, forming, for example, the left and right boundaries of a cluster. This proves to correspond to a generalisation to SLE(kappa,rho), with rho=2. We derive the probabilities that a given point lies between two curves or to one side of both. We find analytic solutions for the cases kappa=0,2,4,8/3,8. The result for kappa=6 leads to predictions for the current distribution at the plateau transition in the semiclassical approximation to the quantum Hall effect.
We obtain an exact finite-size expression for the probability that a percolation hull will touch the boundary, on a strip of finite width. Our calculation is based on the q-deformed Knizhnik--Zamolodchikov approach, and the results are expressed in terms of symplectic characters. In the large size limit, we recover the scaling behaviour predicted by Schramms left-passage formula. We also derive a general relation between the left-passage probability in the Fortuin--Kasteleyn cluster model and the magnetisation profile in the open XXZ chain with diagonal, complex boundary terms.
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