Related papers: Fisher's scaling relation above the upper critical…
In the 1960's, four famous scaling relations were developed which relate the six standard critical exponents describing continuous phase transitions in the thermodynamic limit of statistical physics models. They are well understood at a…
The correlation length plays a pivotal role in finite-size scaling and hyperscaling at continuous phase transitions. Below the upper critical dimension, where the correlation length is proportional to the system length, both finite-size…
We show numerically that correlation length at the critical point in the five-dimensional Ising model varies with system size L as L^{5/4}, rather than proportional to L as in standard finite size scaling (FSS) theory. Our results confirm a…
We present a unified view of finite-size scaling (FSS) in dimension d above the upper critical dimension, for both free and periodic boundary conditions. We find that the modified FSS proposed some time ago to allow for violation of…
For a delta-correlated velocity field, simultaneous correlation functions of a passive scalar satisfy closed equations. We analyze the equation for the four-point function. To describe a solution completely, one has to solve the matching…
Above the upper critical dimension, the breakdown of hyperscaling is associated with dangerous irrelevant variables in the renormalization group formalism at least for systems with periodic boundary conditions. While these have been…
Moving beyond simple associations, researchers need tools to quantify how variables influence each other in space and time. Correlation functions provide a mathematical framework for characterizing these essential dependencies, revealing…
We show by extensive simulations that the whole supercritical phase of the three-dimensional uniform forest model simultaneously exhibits an infinite tree and a rich variety of critical phenomena. Besides typical scalings like algebraically…
We study long-range power-law correlated disorder on square and cubic lattices. In particular, we present high-precision results for the percolation thresholds and the fractal dimension of the largest clusters as function of the correlation…
Percolation is a cornerstone concept in physics, providing crucial insights into critical phenomena and phase transitions. In this study, we adopt a kinetic perspective to reveal the scaling behaviors of higher-order gaps in the largest…
Universal scaling laws of fluctuations (the $\Delta$-scaling laws) can be derived for equilibrium and off-equilibrium systems when combined with the finite-size scaling analysis. In any system in which the second-order critical behavior can…
We propose a scaling relation for critical phenomena in which a symmetry-breaking field is dengerously irrelevant. We confirm its validity on the 6-state clock model in three and four dimensions by numerical simulation. In doing so, we…
We present a unifying, consistent, finite-size-scaling picture for percolation theory bringing it into the framework of a general, renormalization-group-based, scaling scheme for systems above their upper critical dimensions $d_c$.…
The hyperscaling relation and standard finite-size scaling (FSS) are known to break down above the upper critical dimension due to dangerous irrelevant variables. We establish a coherent formalism for FSS at quantum phase transitions above…
In three dimensions, or more generally, below the upper critical dimension, scaling laws for critical phenomena seem well understood, for both infinite and for finite systems. Above the upper critical dimension of four, finite-size scaling…
Fluctuation scaling (FS) and anomalous diffusion have been discussed in different contexts, even though both are often observed in complex systems. To clarify the relationship between these concepts, we investigated approximately three…
The influence of long-range interactions decaying in d dimensions as 1/R^{d+\sigma} on the critical behavior of systems with Fisher's correlation-function exponent for short-range interactions \eta_{SR}<0, is re-examined. Such systems,…
The problem of the effects of compressibility and large-scale anisotropy on anomalous scaling behavior is considered for two models describing passive advection of scalar density and tracer fields. The advecting velocity field is Gaussian,…
The finite-size critical properties of the ${\cal O}(n)$ vector $\phi^4$ model, with long-range interaction decaying algebraically with the interparticle distance $r$ like $r^{-d-\sigma}$, are investigated. The system is confined to a…
We discuss the universal scaling laws of order parameter fluctuations in any system in which the second-order critical behavior can be identified. These scaling laws can be derived rigorously for equilibrium systems when combined with the…