335 research outputs found

    Directed percolation near a wall

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    Series expansion methods are used to study directed bond percolation clusters on the square lattice whose lateral growth is restricted by a wall parallel to the growth direction. The percolation threshold pcp_c is found to be the same as that for the bulk. However the values of the critical exponents for the percolation probability and mean cluster size are quite different from those for the bulk and are estimated by β1=0.7338±0.0001\beta_1 = 0.7338 \pm 0.0001 and γ1=1.8207±0.0004\gamma_1 = 1.8207 \pm 0.0004 respectively. On the other hand the exponent Δ1=β1+γ1\Delta_1=\beta_1 +\gamma_1 characterising the scale of the cluster size distribution is found to be unchanged by the presence of the wall. The parallel connectedness length, which is the scale for the cluster length distribution, has an exponent which we estimate to be ν1=1.7337±0.0004\nu_{1\parallel} = 1.7337 \pm 0.0004 and is also unchanged. The exponent τ1\tau_1 of the mean cluster length is related to β1\beta_1 and ν1\nu_{1\parallel} by the scaling relation ν1=β1+τ1\nu_{1\parallel} = \beta_1 + \tau_1 and using the above estimates yields τ1=1\tau_1 = 1 to within the accuracy of our results. We conjecture that this value of τ1\tau_1 is exact and further support for the conjecture is provided by the direct series expansion estimate τ1=1.0002±0.0003\tau_1= 1.0002 \pm 0.0003.Comment: 12pages LaTeX, ioplppt.sty, to appear in J. Phys.

    Low-density series expansions for directed percolation III. Some two-dimensional lattices

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    We use very efficient algorithms to calculate low-density series for bond and site percolation on the directed triangular, honeycomb, kagom\'e, and (4.82)(4.8^2) lattices. Analysis of the series yields accurate estimates of the critical point pcp_c and various critical exponents. The exponent estimates differ only in the 5th5^{th} digit, thus providing strong numerical evidence for the expected universality of the critical exponents for directed percolation problems. In addition we also study the non-physical singularities of the series.Comment: 20 pages, 8 figure

    The exact evaluation of the corner-to-corner resistance of an M x N resistor network: Asymptotic expansion

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    We study the corner-to-corner resistance of an M x N resistor network with resistors r and s in the two spatial directions, and obtain an asymptotic expansion of its exact expression for large M and N. For M = N, r = s =1, our result is R_{NxN} = (4/pi) log N + 0.077318 + 0.266070/N^2 - 0.534779/N^4 + O(1/N^6).Comment: 12 pages, re-arranged section

    Phase diagram of a dilute ferromagnet model with antiferromagnetic next-nearest-neighbor interactions

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    We have studied the spin ordering of a dilute classical Heisenberg model with spin concentration xx, and with ferromagnetic nearest-neighbor interaction J1J_1 and antiferromagnetic next-nearest-neighbor interaction J2J_2. Magnetic phases at absolute zero temperature T=0T = 0 are determined examining the stiffness of the ground state, and those at finite temperatures T0T \neq 0 are determined calculating the Binder parameter gLg_L and the spin correlation length ξL\xi_L. Three ordered phases appear in the xTx-T phase diagram: (i) the ferromagnetic (FM) phase; (ii) the spin glass (SG) phase; and (iii) the mixed (M) phase of the FM and the SG. Near below the ferromagnetic threshold xFx_{\rm F}, a reentrant SG transition occurs. That is, as the temperature is decreased from a high temperature, the FM phase, the M phase and the SG phase appear successively. The magnetization which grows in the FM phase disappears in the SG phase. The SG phase is suggested to be characterized by ferromagnetic clusters. We conclude, hence, that this model could reproduce experimental phase diagrams of dilute ferromagnets Fex_xAu1x_{1-x} and Eux_xSr1x_{1-x}S.Comment: 9 pages, 23 figure

    Critical frontier of the Potts and percolation models in triangular-type and kagome-type lattices I: Closed-form expressions

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    We consider the Potts model and the related bond, site, and mixed site-bond percolation problems on triangular-type and kagome-type lattices, and derive closed-form expressions for the critical frontier. For triangular-type lattices the critical frontier is known, usually derived from a duality consideration in conjunction with the assumption of a unique transition. Our analysis, however, is rigorous and based on an established result without the need of a uniqueness assumption, thus firmly establishing all derived results. For kagome-type lattices the exact critical frontier is not known. We derive a closed-form expression for the Potts critical frontier by making use of a homogeneity assumption. The closed-form expression is new, and we apply it to a host of problems including site, bond, and mixed site-bond percolation on various lattices. It yields exact thresholds for site percolation on kagome, martini, and other lattices, and is highly accurate numerically in other applications when compared to numerical determination.Comment: 22 pages, 13 figure

    Nonlinear and spin-glass susceptibilities of three site-diluted systems

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    The nonlinear magnetic χ3\chi_{3} and spin-glass χsg\chi_{sg} susceptibilities in zero applied field are obtained, from tempered Monte Carlo simulations, for three different spin glasses (SGs) of Ising spins with quenched site disorder. We find that the relation T3χ3=χsg2/3-T^3\chi_3=\chi_{sg}-2/3 (TT is the temperature), which holds for Edwards-Anderson SGs, is approximately fulfilled in canonical-like SGs. For nearest neighbor antiferromagnetic interactions, on a 0.4 fraction of all sites in fcc lattices, as well as for spatially disordered Ising dipolar (DID) systems, T3χ3-T^3\chi_3 and χsg\chi_{sg} appear to diverge in the same manner at the critical temperature TsgT_{sg}. However, T3χ3-T^3\chi_3 is smaller than χsg \chi_{sg} by over two orders of magnitude in the diluted fcc system. In DID systems, T3χ3/χsg-T^3\chi_3/\chi_{sg} is very sensitive to the systems aspect ratio. Whereas near TsgT_{sg}, χsg\chi_{sg} varies by approximately a factor of 2 as system shape varies from cubic to long-thin-needle shapes, χ3\chi_3 sweeps over some four decades.Comment: 7 pages, 7 figure

    Simple Asymmetric Exclusion Model and Lattice Paths: Bijections and Involutions

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    We study the combinatorics of the change of basis of three representations of the stationary state algebra of the two parameter simple asymmetric exclusion process. Each of the representations considered correspond to a different set of weighted lattice paths which, when summed over, give the stationary state probability distribution. We show that all three sets of paths are combinatorially related via sequences of bijections and sign reversing involutions.Comment: 28 page

    Directed Percolation with a Wall or Edge

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    We examine the effects of introducing a wall or edge into a directed percolation process. Scaling ansatzes are presented for the density and survival probability of a cluster in these geometries, and we make the connection to surface critical phenomena and field theory. The results of previous numerical work for a wall can thus be interpreted in terms of surface exponents satisfying scaling relations generalising those for ordinary directed percolation. New exponents for edge directed percolation are also introduced. They are calculated in mean-field theory and measured numerically in 2+1 dimensions.Comment: 14 pages, submitted to J. Phys.

    Absorbing boundaries in the conserved Manna model

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    The conserved Manna model with a planar absorbing boundary is studied in various space dimensions. We present a heuristic argument that allows one to compute the surface critical exponent in one dimension analytically. Moreover, we discuss the mean field limit that is expected to be valid in d>4 space dimensions and demonstrate how the corresponding partial differential equations can be solved.Comment: 8 pages, 4 figures; v1 was changed by replacing the co-authors name "L\"ubeck" with "Lubeck" (metadata only
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