Proof of the equivalence between double scaling limit and finite-size scaling hypothesis

1994 ◽  
Vol 107 (3) ◽  
pp. 507-510
Author(s):  
A. Peterman ◽  
A. Zichichi
Keyword(s):  
Scaling Limit ◽  
Finite Size ◽  
Finite Size Scaling ◽  
Size Scaling ◽  
Scaling Hypothesis ◽  
Double Scaling Limit

scholarly journals Numerical study on a crossing probability for the four-state Potts model: Logarithmic correction to the finite-size scaling

2019 ◽  
Vol 2019 (9) ◽  
Author(s):  
Kimihiko Fukushima ◽  
Kazumitsu Sakai
Keyword(s):  
Fractal Dimension ◽  
Potts Model ◽  
Scaling Limit ◽  
Finite Size ◽  
Logarithmic Correction ◽  
Finite Size Scaling ◽  
Spin Cluster ◽  
Size Scaling ◽  
Aspect Ratios ◽  
Crossing Probability

Abstract A crossing probability for the critical four-state Potts model on an $L\times M$ rectangle on a square lattice is numerically studied. The crossing probability here denotes the probability that spin clusters cross from one side of the boundary to the other. First, by employing a Monte Carlo method, we calculate the fractal dimension of a spin cluster interface with a fluctuating boundary condition. By comparison of the fractal dimension with that of the Schramm–Loewner evolution (SLE), we numerically confirm that the interface can be described by the SLE with $\kappa=4$, as predicted in the scaling limit. Then, we compute the crossing probability of this spin cluster interface for various system sizes and aspect ratios. Furthermore, comparing with the analytical results for the scaling limit, which have been previously obtained by a combination of the SLE and conformal field theory, we numerically find that the crossing probability exhibits a logarithmic correction ${\sim} 1/\log(L M)$ to the finite-size scaling.

scholarly journals Shape dependence of the finite-size scaling limit in a strongly anisotropic $\mathsf{O(\infty)}$ model

2003 ◽  
Vol 34 (2) ◽  
pp. 205-217 ◽  
Author(s):  
S. Caracciolo ◽  
A. Gambassi ◽  
A. Pelissetto ◽  
M. Gubinelli
Keyword(s):  
Scaling Limit ◽  
Finite Size ◽  
Finite Size Scaling ◽  
Size Scaling ◽  
Shape Dependence

Finite-size scaling of O(n) models with singular behaviour

1989 ◽  
Vol 67 (10) ◽  
pp. 952-956 ◽  
Author(s):  
Scott Allen ◽  
R. K. Pathria
Keyword(s):  
Spontaneous Magnetization ◽  
Finite Size ◽  
Spherical Model ◽  
Singular Part ◽  
Continuous Variables ◽  
Finite Size Scaling ◽  
Size Scaling ◽  
Transition Formula ◽  
Scaling Hypothesis ◽  
Analytical Verification

The finite-size scaling hypothesis of Privman and Fisher (Phys. Rev. B, 30, 322 (1984)) is applied to systems with O(n) symmetry [Formula: see text], confined to geometry Ld − d′ × ∞d′ (where d and d′ are continuous variables such that 2 < d′ < d < 4) and subjected to periodic boundary conditions. Predictions, involving amplitudes as well as exponents, are made on the singular part of the specific heat c(s), spontaneous magnetization m0, magnetic susceptibility χ0, and correlation length ξ0 in the region of the second-order phase transition [Formula: see text]. Analytical verification of these predictions is carried out in the case of the spherical model of ferromagnetism (n = ∞), which includes evaluation of the shift in the critical temperature of the system.

scholarly journals True scale-free networks hidden by finite size effects

2020 ◽  
Vol 118 (2) ◽  
pp. e2013825118
Author(s):  
Matteo Serafino ◽  
Giulio Cimini ◽  
Amos Maritan ◽  
Andrea Rinaldo ◽  
Samir Suweis ◽  
...  
Keyword(s):  
Power Law ◽  
Size Effects ◽  
Finite Size ◽  
Statistical Testing ◽  
Finite Size Scaling ◽  
Finite Size Effects ◽  
Scale Free ◽  
Size Scaling ◽  
Naturally Occurring ◽  
Scaling Hypothesis

We analyze about 200 naturally occurring networks with distinct dynamical origins to formally test whether the commonly assumed hypothesis of an underlying scale-free structure is generally viable. This has recently been questioned on the basis of statistical testing of the validity of power law distributions of network degrees. Specifically, we analyze by finite size scaling analysis the datasets of real networks to check whether the purported departures from power law behavior are due to the finiteness of sample size. We find that a large number of the networks follows a finite size scaling hypothesis without any self-tuning. This is the case of biological protein interaction networks, technological computer and hyperlink networks, and informational networks in general. Marked deviations appear in other cases, especially involving infrastructure and transportation but also in social networks. We conclude that underlying scale invariance properties of many naturally occurring networks are extant features often clouded by finite size effects due to the nature of the sample data.

scholarly journals Anisotropic scaling of the two-dimensional Ising model I: the torus

2019 ◽  
Vol 7 (3) ◽  
Author(s):  
Hendrik Hobrecht ◽  
Fred Hucht
Keyword(s):  
Free Energy ◽  
Ising Model ◽  
Boundary Conditions ◽  
Scaling Limit ◽  
Finite Size ◽  
Square Lattice ◽  
Two Dimensional ◽  
Finite Size Scaling ◽  
Size Scaling ◽  
Anisotropic Scaling

We present detailed calculations for the partition function and the free energy of the finite two-dimensional square lattice Ising model with periodic and antiperiodic boundary conditions, variable aspect ratio, and anisotropic couplings, as well as for the corresponding universal free energy finite-size scaling functions. Therefore, we review the dimer mapping, as well as the interplay between its topology and the different types of boundary conditions. As a central result, we show how both the finite system as well as the scaling form decay into contributions for the bulk, a characteristic finite-size part, and – if present – the surface tension, which emerges due to at least one antiperiodic boundary in the system. For the scaling limit we extend the proper finite-size scaling theory to the anisotropic case and show how this anisotropy can be absorbed into suitable scaling variables.

scholarly journals SIMPLICIAL QUANTUM GRAVITY ON A RANDOMLY TRIANGULATED SPHERE

1999 ◽  
Vol 14 (24) ◽  
pp. 3885-3903 ◽  
Author(s):  
CHRISTIAN HOLM ◽  
WOLFHARD JANKE
Keyword(s):  
Quantum Gravity ◽  
Scaling Limit ◽  
Finite Size ◽  
System Size ◽  
Finite Size Scaling ◽  
Corrections To Scaling ◽  
Regge Calculus ◽  
Expectation Value ◽  
Size Scaling ◽  
Scaling Analyses

We study 2D quantum gravity on spherical topologies employing the Regge calculus approach with the dl/l measure. Instead of the normally used fixed nonregular triangulations we study random triangulations which are generated by the standard Voronoi–Delaunay procedure. For each system size we average the results over four different realizations of the random lattices. We compare both types of triangulations quantitatively and investigate how the difference in the expectation value of the squared curvature, R2, for fixed and random triangulations depends on the lattice size and the surface area A. We try to measure the string susceptibility exponents through finite-size scaling analyses of the expectation value of an added R2 interaction term, using two conceptually quite different procedures. The approach, where an ultraviolet cutoff is held fixed in the scaling limit, is found to be plagued with inconsistencies, as has already previously been pointed out by us. In a conceptually different approach, where the area A is held fixed, these problems are not present. We find the string susceptibility exponent γ′ str in rough agreement with theoretical predictions for the sphere, whereas the estimate for γ str appears to be too negative. However, our results are hampered by the presence of severe finite-size corrections to scaling, which lead to systematic uncertainties well above our statistical errors. We feel that the present methods of estimating the string susceptibilities by finite-size scaling studies are not accurate enough to serve as testing grounds to decide the success or failure of quantum Regge calculus.

scholarly journals Continuous Demixing Transition of Binary Liquids: Finite‐Size Scaling from the Analysis of Sub‐Systems

2021 ◽  
pp. 2000235
Author(s):  
Yogyata Pathania ◽  
Dipanjan Chakraborty ◽  
Felix Höfling
Keyword(s):  
Finite Size ◽  
Finite Size Scaling ◽  
Binary Liquids ◽  
Size Scaling

scholarly journals A NEW APPROACH TO DYNAMIC FINITE-SIZE SCALING

2003 ◽  
Vol 14 (07) ◽  
pp. 945-954 ◽  
Author(s):  
MEHMET DİLAVER ◽  
SEMRA GÜNDÜÇ ◽  
MERAL AYDIN ◽  
YİĞİT GÜNDÜÇ
Keyword(s):  
Three Dimensional ◽  
Finite Size ◽  
Taylor Series Expansion ◽  
Scaling Behavior ◽  
Dynamic Scaling ◽  
Finite Size Scaling ◽  
New Approach ◽  
Size Scaling ◽  
Dynamic Exponent ◽  
Good Agreement

In this work we have considered the Taylor series expansion of the dynamic scaling relation of the magnetization with respect to small initial magnetization values in order to study the dynamic scaling behavior of two- and three-dimensional Ising models. We have used the literature values of the critical exponents and of the new dynamic exponent x0 to observe the dynamic finite-size scaling behavior of the time evolution of the magnetization during early stages of the Monte Carlo simulation. For the three-dimensional Ising model we have also presented that this method opens the possibility of calculating z and x0 separately. Our results show good agreement with the literature values. Measurements done on lattices with different sizes seem to give very good scaling.

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