Estimates of critical percolation probabilities for a set of two-dimensional lattices

1972 ◽  
Vol 71 (1) ◽  
pp. 97-106 ◽  
Author(s):  
D. G. Neal
Keyword(s):  
Monte Carlo ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Critical Percolation ◽  
Site Percolation

AbstractThis paper describes new detailed Monte Carlo investigations into bond and site percolation problems on the set of eleven regular and semi-regular (Archimedean) lattices in two dimensions.

scholarly journals Interface Roughening in Two Dimensions

2021 ◽  
Vol 182 (3) ◽  
Author(s):  
Gernot Münster ◽  
Manuel Cañizares Guerrero
Keyword(s):  
Field Theory ◽  
Monte Carlo ◽  
Capillary Wave ◽  
System Size ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Wave Approximation ◽  
Statistical Field ◽  
Statistical Field Theory ◽  
Interface Roughening

AbstractRoughening of interfaces implies the divergence of the interface width w with the system size L. For two-dimensional systems the divergence of $$w^2$$ w 2 is linear in L. In the framework of a detailed capillary wave approximation and of statistical field theory we derive an expression for the asymptotic behaviour of $$w^2$$ w 2 , which differs from results in the literature. It is confirmed by Monte Carlo simulations.

LUTTINGER-LIQUID BEHAVIOUR IN 2D: THE VARIATIONAL APPROACH

1993 ◽  
Vol 07 (03) ◽  
pp. 119-141 ◽  
Author(s):  
CLAUDIUS GROS ◽  
ROSER VALENTÍ
Keyword(s):  
Monte Carlo ◽  
Variational Formulation ◽  
Variational Approach ◽  
Order Parameter ◽  
Luttinger Liquid ◽  
Two Dimensions ◽  
Point Of View ◽  
Two Dimensional ◽  
Liquid Type ◽  
D Wave

We study a variational formulation of the Luttinger-liquid concept in two dimensions. We show that a Luttinger-liquid wavefunction with an algebraic singularity at the Fermiedge is given by a Jastrow-Gutzwiller type wavefunction, which we evaluate by variational Monte Carlo for lattices with up to 38 × 38 = 1444 sites. We therefore find that, from a variational point of view, the concept of a Luttinger liquid is well defined even in 2D. We also find that the Luttinger liquid state is energetically favoured by the projected kinetic energy in the context of the 2D t-J model. We study and find coexistence of d-wave superconductivity and Luttinger-liquid behaviour in two-dimensional projected wavefunctions. We then argue that generally, any two-dimensional d-wave superconductor should be unstable against Luttinger-liquid type correlations along the (quasi-1D) nodes of the d-wave order parameter, at temperatures small compared to the gap.

scholarly journals Diffusion Monte Carlo Study of Electrons in Two-dimensional Layers

1996 ◽  
Vol 49 (1) ◽  
pp. 161 ◽  
Author(s):  
Francesco Rapisarda ◽  
Gaetano Senatore
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Study ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Diffusion Monte Carlo ◽  
Intermediate Coupling ◽  
Particle Spacing ◽  
Large Coupling ◽  
Liquid Phases ◽  
Fixed Node

We investigate the phase diagram of electrons in two dimensions at T = 0 by means of accurate diffusion Monte Carlo simulations within the fixed-node approximation. At variance with previous studies, we find that in an isolated layer Slater-Jastrow nodes yield stability of the fully polarised fluid at intermediate coupling, before freezing into a triangular crystal sets in. We have also studied coupled layers of electrons and of electrons and holes. Preliminary results show that at large coupling, as two layers are brought together from infinity, inter-layer correlation first stabilises the crystalline phase at distances of the order of the in-plane inter-particle spacing. As the distance is further decreased the effect of correlation, as expected, turns into an enhanced screening, which disrupts the crystalline order in favour of liquid phases.

On the Validation of the Monte Carlo Technique in Simulation of Grain Growth in Small, Two-Dimensional Systems

2007 ◽  
Vol 558-559 ◽  
pp. 1087-1092
Author(s):  
Ola Hunderi ◽  
Knut Marthinsen ◽  
Nils Ryum
Keyword(s):  
Monte Carlo ◽  
Grain Boundary ◽  
Grain Growth ◽  
Computer Simulations ◽  
Two Dimensions ◽  
Theoretical Treatment ◽  
Two Dimensional ◽  
Simulation Techniques ◽  
Boundary Curvature ◽  
Kinetics Of

The kinetics of grain growth in real systems is influenced by several unknown factors, making a theoretical treatment very difficult. Idealized grain growth, assuming all grain boundaries to have the same energy and mobility (mobility M = k/ρ, where k is a constant and ρ is grain boundary curvature) can be treated theoretically, but the results obtained can only be compared to numerical grain growth simulations, as ideal grain growth scarcely exists in nature. The validity of the simulation techniques thus becomes of great importance. In the present investigation computer simulations of grain growth in two dimensions using Monte Carlo simulations and the grain boundary tracking technique have been investigated and compared in small grain systems, making it possible to follow the evolution of each grain in the system.

Universal Ratio χ in Two-Dimensional Square Random-Site Percolation

1998 ◽  
Vol 09 (01) ◽  
pp. 147-155 ◽  
Author(s):  
Krzysztof Malarz ◽  
Ana Maria Vidales
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Small Distance ◽  
Square Lattice ◽  
Two Dimensional ◽  
Site Percolation ◽  
Random Site ◽  
Percolation Phenomena ◽  
Simulation Results ◽  
Universal Ratio

The percolation phenomena on two-dimensional square lattice is considered. The quotient χ of connectivity length ξ> above percolation threshold pc and ξ< below pc at the same small distance Δp is discussed. The results of two different algorithms and programs and agreement with theoretical/mathematical predications is presented, in contrast to previous contradictory Monte Carlo simulation results.

ON LOGARITHMIC CORRECTIONS IN TWO-DIMENSIONAL PERCOLATION

1993 ◽  
Vol 07 (26) ◽  
pp. 1661-1665
Author(s):  
M. MARSILI ◽  
G. JUG
Keyword(s):  
Asymptotic Behavior ◽  
Critical Exponent ◽  
Transfer Matrix ◽  
Finite Size ◽  
Two Dimensions ◽  
Matrix Analysis ◽  
Two Dimensional ◽  
Site Percolation ◽  
Logarithmic Corrections ◽  
Transfer Matrix Analysis

The possibility of unusual leading logarithmic corrections to the asymptotic behavior of the percolation connectedness length ξ in two dimensions is explored through a finite-size transfer-matrix analysis on strips of widths L≤12. It is found that, for both square-site and triangular-site percolation problems, no such corrections arise and the accepted exact value of the critical exponent ν is recovered.

scholarly journals Dynamic Monte Carlo Study of the Two-Dimensional Quantum XY Model

1998 ◽  
Vol 12 (29n30) ◽  
pp. 1237-1243 ◽  
Author(s):  
H. P. Ying ◽  
H. J. Luo ◽  
L. Schülke ◽  
B. Zheng
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Study ◽  
Two Dimensions ◽  
Xy Model ◽  
Dynamic Scaling ◽  
Two Dimensional ◽  
Time Dynamic ◽  
Dynamic Monte Carlo ◽  
Short Time ◽  
Dynamical Exponents

We present a dynamic Monte Carlo study of the spin-1/2 quantum XY model in two-dimensions at the Kosterlitz–Thouless phase transition temperature. The short-time dynamic scaling behaviour is found and the dynamical exponents θ, z and the static exponent η are determined.

scholarly journals A conformal bootstrap approach to critical percolation in two dimensions

2016 ◽  
Vol 1 (1) ◽  
Author(s):  
Marco Picco ◽  
Sylvain Ribault ◽  
Raoul Santachiara
Keyword(s):  
Monte Carlo ◽  
Potts Model ◽  
Virasoro Algebra ◽  
Two Dimensions ◽  
Critical Percolation ◽  
Conformal Bootstrap ◽  
Bootstrap Approach

We study four-point functions of critical percolation in two dimensions, and more generally of the Potts model. We propose an exact ansatz for the spectrum: an infinite, discrete and non-diagonal combination of representations of the Virasoro algebra. Based on this ansatz, we compute four-point functions using a numerical conformal bootstrap approach. The results agree with Monte-Carlo computations of connectivities of random clusters.

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