Asymptotic inference for an Ising lattice

1976 ◽  
Vol 13 (3) ◽  
pp. 486-497 ◽  
Author(s):  
D. K. Pickard
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Confidence Intervals ◽  
Limit Theorems ◽  
Critical Case ◽  
Ising Lattice ◽  
Asymptotic Inference ◽  
Nearest Neighbours ◽  
Sample Correlation

Kaufmann's exact characterization of the partition function for the classical Ising model is used to obtain limit theorems for the sample correlation between nearest neighbours in the non-critical case. This provides a basis for the asymptotic testing and estimation (by confidence intervals) of the correlation between nearest neighbours.

Asymptotic inference for an Ising lattice

1976 ◽  
Vol 13 (03) ◽  
pp. 486-497 ◽  
Author(s):  
D. K. Pickard
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Confidence Intervals ◽  
Limit Theorems ◽  
Critical Case ◽  
Ising Lattice ◽  
Asymptotic Inference ◽  
Nearest Neighbours ◽  
Sample Correlation

Kaufmann's exact characterization of the partition function for the classical Ising model is used to obtain limit theorems for the sample correlation between nearest neighbours in the non-critical case. This provides a basis for the asymptotic testing and estimation (by confidence intervals) of the correlation between nearest neighbours.

Inference for general Ising models

1982 ◽  
Vol 19 (A) ◽  
pp. 345-357 ◽  
Author(s):  
David K. Pickard
Keyword(s):  
Ising Model ◽  
Thermodynamic Parameters ◽  
Limit Theorems ◽  
Ising Models ◽  
Confidence Regions ◽  
Likelihood Inference ◽  
Partition Functions ◽  
Polynomial Equations ◽  
Interaction Energies ◽  
Asymptotic Inference

In previous papers (1976), (1977a), (1979) limit theorems were obtained for the classical Ising model, and these provided the basis for asymptotic inference. The present paper extends these results to more general Ising models. In two and more dimensions, likelihood inference for the thermodynamic parameters (i.e. the interaction energies) is effectively impossible. The problem is that the error in locating critical and/or confidence regions is as large as their diameters. To remedy this requires more accurate characterizations of the partition functions, but these seem unlikely to be forthcoming. Besag's coding estimators for these parameters are inverse hyperbolic tangents of the roots of simultaneous polynomial equations and hence avoid such location errors. However, little is yet known about their sampling characteristics. Finally, likelihood inference for lattice averages (an alternative parametrization) is straightforward from the limit theorems.

Asymptotic inference for an ising lattice. II

1977 ◽  
Vol 9 (3) ◽  
pp. 476-501 ◽  
Author(s):  
D. K. Pickard
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Ising Model ◽  
Central Limit ◽  
Critical Points ◽  
Limit Theorems ◽  
Relative Rate ◽  
Ising Lattice ◽  
The Central Limit Theorem ◽  
Significant Case

In Pickard (1976) limit theorems were obtained for the classical Ising model at non-critical points. These determined the asymptotic distribution of the sample nearest-neighbour correlation, thereby providing a basis for statistical inference by confidence intervals. In this paper, these limit theorems are extended to the statistically significant case of different vertical and horizontal interactions. Results at critical points are also obtained. Critical points clearly have the potential to seriously distort statistical inferences, especially in their immediate neighbourhoods. For our Ising model it turns out that such distortion is relatively minor. Surprisingly, in the two-parameter case the correlation between the sufficient statistics exhibits peculiar asymptotic behaviour resulting in a singular covariance matrix at critical points in the central limit theorem. Finally, at critical points, unusual norming constants are required for the central limit theorem, and our results are much more sensitive to the relative rate at which m, n tend to infinity.

Some generalized order-disorder transformations

1952 ◽  
Vol 48 (1) ◽  
pp. 106-109 ◽  
Author(s):  
R. B. Potts
Keyword(s):  
Fixed Point ◽  
Partition Function ◽  
Transition Point ◽  
Ising Lattice ◽  
Nearest Neighbours ◽  
Inversion Transformation ◽  
Generalized Order

In considering the statistics of the ‘no-field’ square Ising lattice in which each unit is capable of two configurations and only nearest neighbours interact, Kramers and Wannier (3) were able to deduce an inversion transformation under which the partition function of the lattice is invariant when the temperature is transformed from a low to a high (‘inverted’) value. The important property of this inversion transformation is that its fixed point gives the transition point of the lattice.

Some exact results for the anisotropic triangular lattice

1969 ◽  
Vol 47 (22) ◽  
pp. 2445-2448 ◽  
Author(s):  
R. W. Gibberd
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Triangular Lattice ◽  
Correlation Functions ◽  
Spin Correlation ◽  
Exact Results ◽  
Combinatorial Approach ◽  
Ising Lattice ◽  
Simple Derivation ◽  
One Dimensional

The combinatorial approach to the triangular Ising model is shown to be considerably simplified at a given temperature TD. This enables the partition function and spin-correlation functions to be evaluated without the use of pfaffians, thus providing a simple derivation of a few of the results obtained previously by Stephenson. We confirm the results of Stephenson, that at the temperature TD, the correlation functions have the same structure as the correlations of a one-dimensional Ising lattice. The only new result is a general expression for some of the spin–spin correlation functions when the spins do not lie on the axes of the lattice.

Asymptotic inference for an ising lattice. II

1977 ◽  
Vol 9 (03) ◽  
pp. 476-501 ◽  
Author(s):  
D. K. Pickard
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Ising Model ◽  
Central Limit ◽  
Critical Points ◽  
Limit Theorems ◽  
Relative Rate ◽  
Ising Lattice ◽  
The Central Limit Theorem ◽  
Significant Case

In Pickard (1976) limit theorems were obtained for the classical Ising model at non-critical points. These determined the asymptotic distribution of the sample nearest-neighbour correlation, thereby providing a basis for statistical inference by confidence intervals. In this paper, these limit theorems are extended to the statistically significant case of different vertical and horizontal interactions. Results at critical points are also obtained. Critical points clearly have the potential to seriously distort statistical inferences, especially in their immediate neighbourhoods. For our Ising model it turns out that such distortion is relatively minor. Surprisingly, in the two-parameter case the correlation between the sufficient statistics exhibits peculiar asymptotic behaviour resulting in a singular covariance matrix at critical points in the central limit theorem. Finally, at critical points, unusual norming constants are required for the central limit theorem, and our results are much more sensitive to the relative rate at which m, n tend to infinity.

Inference for general Ising models

1982 ◽  
Vol 19 (A) ◽  
pp. 345-357 ◽  
Author(s):  
David K. Pickard
Keyword(s):  
Ising Model ◽  
Thermodynamic Parameters ◽  
Limit Theorems ◽  
Ising Models ◽  
Confidence Regions ◽  
Likelihood Inference ◽  
Partition Functions ◽  
Polynomial Equations ◽  
Interaction Energies ◽  
Asymptotic Inference

In previous papers (1976), (1977a), (1979) limit theorems were obtained for the classical Ising model, and these provided the basis for asymptotic inference. The present paper extends these results to more general Ising models.In two and more dimensions, likelihood inference for the thermodynamic parameters (i.e. the interaction energies) is effectively impossible. The problem is that the error in locating critical and/or confidence regions is as large as their diameters. To remedy this requires more accurate characterizations of the partition functions, but these seem unlikely to be forthcoming. Besag's coding estimators for these parameters are inverse hyperbolic tangents of the roots of simultaneous polynomial equations and hence avoid such location errors. However, little is yet known about their sampling characteristics. Finally, likelihood inference for lattice averages (an alternative parametrization) is straightforward from the limit theorems.

scholarly journals The bulk, surface and corner free energies of the anisotropic triangular Ising model

2020 ◽  
Vol 476 (2234) ◽  
pp. 20190713
Author(s):  
Rodney J. Baxter
Keyword(s):  
Free Energy ◽  
Ising Model ◽  
Partition Function ◽  
Temperature Series ◽  
Isotropic Case ◽  
Series Expansions ◽  
Exact Results ◽  
Free Energies ◽  
Bulk Surface ◽  
Bulk Free Energy

We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman’s spinor method to calculate low-temperature series expansions for the partition function to high order. From these, we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk-free energy and with Cardy and Peschel’s conformal invariance predictions for the dominant behaviour at criticality. For the isotropic case, they also agree with Vernier and Jacobsen’s conjecture for the 60 ° corners.

scholarly journals Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

2010 ◽  
Vol 2010 ◽  
pp. 1-23 ◽  
Author(s):  
Josef Diblík ◽  
Denys Ya. Khusainov ◽  
Irina V. Grytsay ◽  
Zdenĕk Šmarda
Keyword(s):  
Lyapunov Functions ◽  
Critical Case ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Simple Eigenvalue ◽  
Discrete Equations ◽  
The Matrix ◽  
The Stability ◽  
Important Characterization

Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalueλ=1of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.

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