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 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.

Phase Transitions, Critical Phenomena, and Finite-Size Scaling

2019 ◽  
pp. 111-176
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Phase Transition ◽  
Phase Transitions ◽  
Critical Phenomena ◽  
Quantum Phase Transitions ◽  
Finite Size ◽  
System Size ◽  
Thermal Fluctuations ◽  
Finite Size Scaling ◽  
Size Scaling

Interacting many-particle systems may undergo phase transitions and exhibit critical phenomena in the limit of infinite system size, while the precursors of these phenomena are studied in the theory of finite-size scaling. After surveying the basic notions of phases, phase diagrams, and phase transitions, this chapter focuses on critical behaviour at a second-order phase transition. The Landau-Ginzburg theory and the concept of scaling prepare readers for an elementary introduction to the concepts of the renormalization group, followed by an introduction into the field of quantum phase transitions where quantum fluctuations take over the role of thermal fluctuations.

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

Field theory in a finite geometry: Finite size scaling

2019 ◽  
pp. 335-360
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Correlation Length ◽  
Zero Mode ◽  
Finite Size ◽  
System Size ◽  
Finite Geometry ◽  
Effective Field ◽  
Finite Size Scaling ◽  
Size Scaling ◽  
Statistical Systems

Computer simulations of critical statistical systems or quantum field theory models are performed with systems where sizes are finite. In transfer matrix calculations, all sizes but one are also finite. In systems where the correlation length is large, it is thus important to understand how the infinite size limit is reached. This problem is investigated in Chapter 19. RG equations allow proving the properties of universality and of finite size scaling. When the correlation length is larger than the linear system size, a phenomenon of dimensional reduction is observed. With periodic boundary conditions, fields have a zero mode. A local expansion generates an effective field theory for the zero mode.

scholarly journals FINITE SIZE SCALING ANALYSIS OF THE ANDERSON TRANSITION

2010 ◽  
Vol 24 (12n13) ◽  
pp. 1841-1854 ◽  
Author(s):  
B. Kramer ◽  
A. MacKinnon ◽  
T. Ohtsuki ◽  
K. Slevin
Keyword(s):  
Finite Size ◽  
Scaling Analysis ◽  
Finite Size Scaling ◽  
Corrections To Scaling ◽  
Anderson Transition ◽  
The Past ◽  
Size Scaling ◽  
Symmetry Classes ◽  
Comparison With Experiment ◽  
Future Work

This chapter describes the progress made during the past three decades in the finite size scaling analysis of the critical phenomena of the Anderson transition. The scaling theory of localization and the Anderson model of localization are briefly sketched. The finite size scaling method is described. Recent results for the critical exponents of the different symmetry classes are summarised. The importance of corrections to scaling are emphasised. A comparison with experiment is made, and a direction for future work is suggested.

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.

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