The standard scaling mechanism for matching asymptotic behaviors on the approach to criticality fails for directed percolation, as indicated by transfer-matrix calculations for sizes up to N=15. A formulation is proposed incorporating irrelevant-variable corrections in a new pattern of asymptotic scaling. Numerical data support these theoretical predictions and, among other results, yield a new estimate, \ensuremath{\lambda}=0.37\ifmmode\pm\else\textpm\fi{}0.04, for the leading irrelevant-variable exponent.
The finite-size scaling of the correlation length for directed and undirected systems is reviewed with emphasis on the asymptotic eigenvalue degeneracy associated with long-range order or long-range connectivity. The standard scaling mechanism for matching asymptotic behaviors on approach to criticality applies for Ising and other models. However, numerical evidence suggests that for directed percolation in 2D a modified formulation is appropriate, incorporating irrelevant-variable corrections in a new pattern of asymptotic scaling.
A transfer-matrix simulation scheme for the three-dimensional (d=3) bond percolation is presented. Our scheme is based on Novotny's transfer-matrix formalism, which enables us to consider arbitrary (integral) number of sites N constituting a unit of the transfer-matrix slice even for d=3. Such an arbitrariness allows us to perform systematic finite-size-scaling analysis of the criticality at the percolation threshold. Diagonalizing the transfer matrix for N=4, 5,..., we obtain an estimate for the correlation-length critical exponent v=0.81(5).
The rate of convergence of the transfer matrix finite-size scaling (or phenomenological renormalisation) method is studied. It is shown both heuristically and numerically that the convergence of estimates for exponents, etc. is governed asymptotically by the leading irrelevant-variable scaling exponent. The more rapid apparent convergence rates observed in many practical calculations for two-dimensional lattices of widths up to ten lattice spacings are attributed to cancellation between various correction terms.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.
Finite-size scaling (FSS) for a critical phase transition (
t=0) states that within a window of size
∣t∣∼L−1/ν, the scaling behavior of any observable
Q in a system of linear size
L asymptotically follows a scaling form as
Q(t,L)=LYQQ~(tL1/ν), where
ν is the correlation-length exponent,
YQ is an FSS exponent and
Q~(x) is a function of the scaled distance-to-criticality
x≡tL1/ν. We systematically study the asymptotic scaling behavior of
Q~(∣x∣→∞) for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with
t→0 and
∣x∣→∞. This crossover FSS theory predicts that when the criticality is approached at a slower speed as
∣t∣∼L−λ with
λ<1/ν, the FSS becomes
λ-dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality
t=0 can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.