Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions

Recently, we argued [Chin. Phys. Lett. $39$, 080502 (2022)] that the Ising model simultaneously exhibits two upper critical dimensions $(d_c=4, d_p=6)$ in the Fortuin-Kasteleyn (FK) random-cluster representation. In this paper, we perform a systematic study of the FK Ising model on hypercubic lattices with spatial dimensions $d$ from 5 to 7, and on the complete graph. We provide a detailed data analysis of the critical behaviors of a variety of quantities at and near the critical points. Our results clearly show that many quantities exhibit distinct critical phenomena for $4 < d < 6$ and $d\geq 6$, and thus strongly support the argument that $6$ is also an upper critical dimension. Moreover, for each studied dimension, we observe the existence of two configuration sectors, two lengthscales, as well as two scaling windows, and thus, two sets of critical exponents are needed to describe these behaviors. Our finding enriches the understanding of the critical phenomena in the Ising model.Comment: 17 pages, 17 figure

Geometric properties of the complete-graph Ising model in the loop representation

The exact solution of the Ising model on the complete graph (CG) provides an important, though mean-field, insight for the theory of continuous phase transitions. Besides the original spin, the Ising model can be formulated in the Fortuin-Kasteleyn random-cluster and the loop representation, in which many geometric quantities have no correspondence in the spin representations. Using a lifted-worm irreversible algorithm, we study the CG-Ising model in the loop representation, and, based on theoretical and numerical analyses, obtain a number of exact results including volume fractal dimensions and scaling forms. Moreover, by combining with the Loop-Cluster algorithm, we demonstrate how the loop representation can provide an intuitive understanding to the recently observed rich geometric phenomena in the random-cluster representation, including the emergence of two configuration sectors, two length scales and two scaling windows.Comment: 10 pages, 10 figure

Interplay of the complete-graph and Gaussian fixed-point asymptotics in finite-size scaling of percolation above the upper critical dimension

Percolation has two mean-field theories, the Gaussian fixed point (GFP) and the Landau mean-field theory or the complete graph (CG) asymptotics. By large-scale Monte Carlo simulations, we systematically study the interplay of the GFP and CG effects to the finite-size scaling of percolation above the upper critical dimension $d_c = 6$ with periodic, free, and cylindrical boundary conditions. Our results suggest that, with periodic boundaries, the \emph{unwrapped} correlation length scales as $L^{d/6}$ at the critical point, diverging faster than $L$ above $d_c$. As a consequence, the scaling behaviours of macroscopic quantities with respect to the linear system size $L$ follow the CG asymptotics. The distance-dependent properties, such as the short-distance behaviour of the two-point correlation function and the Fourier transformed quantities with non-zero modes, are still controlled by the GFP. With free boundaries, since the correlation length is cutoff by $L$, the finite-size scaling at the critical point is controlled by the GFP. However, some quantities are observed to exhibit the CG aysmptotics at the low-temperature pseudo-critical point, such as the sizes of the two largest clusters. With cylindrical boundaries, due to the interplay of the GFP and CG effects, the correlation length along the axial direction of the cylinder scales as $\xi_L \sim L^{(d-1)/5}$ within the critical window of size $O(L^{-2(d-1)/5})$, distinct from both periodic and free boundaries. A field-theoretical calculation for deriving the scaling of $\xi_L$ is also presented. Moreover, the one-point surface correlation function along the axial direction of the cylinder is observed to scale as ${\tau}^{(1-d)/2}$ for short distance but then enter a plateau of order $L^{-3(d-1)/5}$ before it decays significantly fast.Comment: 12 pages, 11 figure

High-precision Monte Carlo study of directed percolation in (d+1) dimensions

We present a Monte Carlo study of the bond and site directed (oriented) percolation models in $(d+1)$ dimensions on simple-cubic and body-centered-cubic lattices, with $2 \leq d \leq 7$. A dimensionless ratio is defined, and an analysis of its finite-size scaling produces improved estimates of percolation thresholds. We also report improved estimates for the standard critical exponents. In addition, we study the probability distributions of the number of wet sites and radius of gyration, for $1 \leq d \leq 7$.Comment: 11 pages, 21 figure

Improving rainfall erosivity estimates using merged TRMM and gauge data

Soil erosion is a global issue that threatens food security and causes environmental degradation. Management of water erosion requires accurate estimates of the spatial and temporal variations in the erosive power of rainfall (erosivity). Rainfall erosivity can be estimated from rain gauge stations and satellites. However, the time series rainfall data that has a high temporal resolution are often unavailable in many areas of the world. Satellite remote sensing allows provision of the continuous gridded estimates of rainfall, yet it is generally characterized by significant bias. Here we present a methodology that merges daily rain gauge measurements and the Tropical Rainfall Measuring Mission (TRMM) 3B42 data using collocated cokriging (ColCOK) to quantify the spatial distribution of rainfall and thereby to estimate rainfall erosivity across China. This study also used block kriging (BK) and TRMM to estimate rainfall and rainfall erosivity. The methodologies are evaluated based on the individual rain gauge stations. The results from the present study generally indicate that the ColCOK technique, in combination with TRMM and gauge data, provides merged rainfall fields with good agreement with rain gauges and with the best accuracy with rainfall erosivity estimates, when compared with BK gauges and TRMM alone

Crossover from Isotropic to Directed Percolation

We generalize the directed percolation (DP) model by relaxing the strict directionality of DP such that propagation can occur in either direction but with anisotropic probabilities. We denote the probabilities as $p_{\downarrow}= p \cdot p_d$ and $p_{\uparrow}=p \cdot (1-p_d)$, with $p$ representing the average occupation probability and $p_d$ controlling the anisotropy. The Leath-Alexandrowicz method is used to grow a cluster from an active seed site. We call this model with two main growth directions {\em biased directed percolation} (BDP). Standard isotropic percolation (IP) and DP are the two limiting cases of the BDP model, corresponding to $p_d=1/2$ and $p_d=0,1$ respectively. In this work, besides IP and DP, we also consider the $1/2<p_d<1$ region. Extensive Monte Carlo simulations are carried out on the square and the simple-cubic lattices, and the numerical data are analyzed by finite-size scaling. We locate the percolation thresholds of the BDP model for $p_d=0.6$ and 0.8, and determine various critical exponents. These exponents are found to be consistent with those for standard DP. We also determine the renormalization exponent associated with the asymmetric perturbation due to $p_d -1/2 \neq 0$ near IP, and confirm that such an asymmetric scaling field is relevant at IP.Comment: 8 pages, 8 figure