scholarly journals Complexity of Ising Polynomials

2012 ◽  
Vol 21 (5) ◽  
pp. 743-772 ◽  
Author(s):  
TOMER KOTEK
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
External Field ◽  
Polynomial Time ◽  
Tutte Polynomial ◽  
Study Phase ◽  
Exceptional Set ◽  
Exponential Time ◽  
Low Dimension ◽  
Special Case

This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weights. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomialZ(G;x,y,z). This polynomial was studied with respect to its approximability by Goldberg, Jerrum and Paterson.Z(G;x,y,z) generalizes a bivariate polynomialZ(G;t,y), which was studied in by Andrén and Markström.We consider the complexity ofZ(Gt,y) andZ(G;x,y,z) in comparison to that of the Tutte polynomial, which is well known to be closely related to the Potts model in the absence of an external field. We show thatZ(G;x,y,z) is #P-hard to evaluate at all points in3, except those in an exceptional set of low dimension, even when restricted to simple graphs which are bipartite and planar. A counting version of the Exponential Time Hypothesis, #ETH, was introduced by Dell, Husfeldt and Wahlén in order to study the complexity of the Tutte polynomial. In analogy to their results, we give under #ETHa dichotomy theorem stating that evaluations ofZ(G;t,y) either take exponential time in the number of vertices ofGto compute, or can be done in polynomial time. Finally, we give an algorithm for computingZ(G;x,y,z) in polynomial time on graphs of bounded clique-width, which is not known in the case of the Tutte polynomial.

The Computational Complexity of the Tutte Plane: the Bipartite Case

1992 ◽  
Vol 1 (2) ◽  
pp. 181-187 ◽  
Author(s):  
D. L. Vertigan ◽  
D. J. A. Welsh
Keyword(s):  
Computational Complexity ◽  
Ising Model ◽  
Partition Function ◽  
Polynomial Time ◽  
Spanning Trees ◽  
Tutte Polynomial ◽  
Wide Range ◽  
Number Of Spanning Trees ◽  
Bipartite Case ◽  
Complete Characterisation

Along different curves and at different points of the (x, y)-plane the Tutte polynomial evaluates a wide range of quantities. Some of these, such as the number of spanning trees of a graph and the partition function of the planar Ising model, can be computed in polynomial time, others are # P-hard. Here we give a complete characterisation of which points and curves are easy/hard in the bipartite case.

Knots, matroids and the Ising model

1993 ◽  
Vol 113 (1) ◽  
pp. 107-139 ◽  
Author(s):  
W. Schwärzler ◽  
D. J. A. Welsh
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Knot Theory ◽  
Tutte Polynomial ◽  
Signed Graphs ◽  
Kauffman Bracket ◽  
Link Diagrams ◽  
Bracket Polynomial

AbstractA polynomial is defined on signed matroids which contains as specializations the Kauffman bracket polynomial of knot theory, the Tutte polynomial of a matroid, the partition function of the anisotropic Ising model, the Kauffman–Murasugi polynomials of signed graphs. It leads to generalizations of a theorem of Lickorish and Thistlethwaite showing that adequate link diagrams do not represent the unknot. We also investigate semi-adequacy and the span of the bracket polynomial in this wider context.

The Umbral Transfer-Matrix Method. IV. Counting Self-Avoiding Polygons and Walks

10.37236/1572 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Doron Zeilberger
Keyword(s):  
Transfer Matrix ◽  
Cross Section ◽  
Polynomial Time ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Vertical Cross Section ◽  
Trivial Case ◽  
Exponential Time ◽  
Self Avoiding Walks ◽  
Special Case

This is the fourth installment of the five-part saga on the Umbral Transfer-Matrix method, based on Gian-Carlo Rota's seminal notion of the umbra. In this article we describe the Maple packages USAP, USAW, and MAYLIS. USAP automatically constructs, for any specific $r$, an Umbral Scheme for enumerating, according to perimeter, the number of self-avoiding polygons with $\leq 2r$ horizontal edges per vertical cross-section. The much more complicated USAW does the analogous thing for self-avoiding walks. Such Umbral Schemes enable counting these classes of self-avoiding polygons and walks in polynomial time as opposed to the exponential time that is required by naive counting. Finally MAYLIS is targeted to the special case of enumerating classes of saps with at most two horizontal edges per vertical cross-section (equivalently column-convex polyominoes by perimeter), and related classes. In this computationally trivial case we can actually automatically solve the equations that were automatically generated by USAP. As an example, we give the first fully computer-generated proof of the celebrated Delest-Viennot result that the number of convex polyominoes with perimeter $2n+8$ equals $(2n+11)4^n-4(2n+1)!/n!^2$.

scholarly journals Two-dimensional Ising Model on a 4?6?12 Lattice

1985 ◽  
Vol 38 (2) ◽  
pp. 227
Author(s):  
KY Lin ◽  
WN Huang
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Two Dimensional ◽  
Special Case

We have considered a two-dimensional Ising model on a 4-6-12 lattice. The partition function is evaluated exactly by the method of Pfaffian. The Ising model on a ruby lattice is a special case of our model.

scholarly journals Polynomial-time approximation algorithms for the antiferromagnetic Ising model on line graphs

2021 ◽  
pp. 1-17
Author(s):  
Martin Dyer ◽  
Marc Heinrich ◽  
Mark Jerrum ◽  
Haiko Müller
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Polynomial Time ◽  
Line Graph ◽  
Glauber Dynamics ◽  
Monte Carlo Algorithm ◽  
Line Graphs ◽  
Time Approximation ◽  
On Line ◽  
Antiferromagnetic Ising Model

Abstract We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the ‘winding’ technology devised by McQuillan [CoRR abs/1301.2880 (2013)] and developed by Huang, Lu and Zhang [Proc. 27th Symp. on Disc. Algorithms (SODA16), 514–527]. We show that exact computation of the partition function is #P-hard, even for line graphs, indicating that an approximation algorithm is the best that can be expected. We also show that Glauber dynamics for the Ising model is rapidly mixing on line graphs, an example being the kagome lattice.

Ground states for the ising model with an external field on the Cayley tree

2018 ◽  
Vol 2018 (3) ◽  
pp. 147-155
Author(s):  
M.M. Rakhmatullaev ◽  
M.A. Rasulova
Keyword(s):  
Ising Model ◽  
External Field ◽  
Cayley Tree ◽  
Ground States

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.

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