| 1. |
- Lundow, Per Håkan, et al.
(författare)
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EXACT AND APPROXIMATE COMPRESSION OF TRANSFER MATRICES FOR GRAPH HOMOMORPHISMS
- 2008
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Ingår i: LMS Journal of Computation and Mathematics. - : Cambridge University Press. - 1461-1570. ; 11, s. 1-14
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Tidskriftsartikel (refereegranskat)abstract
- The aim of this paper is to extend the previous work on transfer matrix compression in the case of graph homomorphisms. For H-homomorphisms of lattice-like graphs we demonstrate how the automorphisms of H, as well as those of the underlying lattice, can be used to reduce the size of the relevant transfer matrices. As applications of this method we give currently best known bounds for the number of 4- and 5-colourings of the square grid, and the number of 3- and 4-colourings of the three-dimensional cubic lattice. Finally, we also discuss approximate compression of transfer matrices.
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| 2. |
- Lundow, Per Håkan, et al.
(författare)
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The p,q-binomial distribution applied to the 5d Ising model
- 2013
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Ingår i: Philosophical Magazine. - : Informa UK Limited. - 1478-6435 .- 1478-6443. ; 93:14, s. 1755-1770
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Tidskriftsartikel (refereegranskat)abstract
- The leading order form of the magnetization distribution is well-known for the 5d Ising model close to . Its corrections-to-scaling are not known though. Since we have earlier established that this distribution is extremely well-fitted by a -binomial distribution, we report considerably longer series expansions for its moments in terms of three parameters, providing new details on the scaling behaviour of the Ising distribution and its moments near . As applications, we give for example the scaling formulas for the ratios , and the full distribution at .
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| 3. |
- Friedland, S., et al.
(författare)
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On the Validations of the Asymptotic Matching Conjectures
- 2008
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Ingår i: Journal of statistical physics. - : Springer. - 0022-4715 .- 1572-9613. ; 133:3, s. 513-533
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we review the asymptotic matching conjectures for r-regular bipartite graphs, and their connections in estimating the monomer-dimer entropies in d-dimensional integer lattice and Bethe lattices. We prove new rigorous upper and lower bounds for the monomer-dimer entropies, which support these conjectures. We describe a general construction of infinite families of r-regular tori graphs and give algorithms for computing the monomer-dimer entropy of density p, for any p is an element of[0,1], for these graphs. Finally we use tori graphs to test the asymptotic matching conjectures for certain infinite r-regular bipartite graphs.
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| 4. |
- Lundow, Per Håkan, et al.
(författare)
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Broken-cycle-free subgraphs and the log-concavity conjecture for chromatic polynomials
- 2006
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Ingår i: Experimental Mathematics. - : Informa UK Limited. - 1058-6458 .- 1944-950X. ; 15:3, s. 343-353
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Tidskriftsartikel (refereegranskat)abstract
- This paper concerns the coefficients of the chromatic polynomial of a graph. We first report on a computational verification of the strict log-concavity conjecture for chromatic polynomials for all graphs on at most 11 vertices, as well as for certain cubic graphs. In the second part of the paper we give a number of conjectures and theorems regarding the behavior of the coefficients of the chromatic polynomial, in part motivated by our computations. Here our focus is on epsilon(G), the average size of a broken-cycle-free subgraph of the graph G, whose behavior under edge deletion and contraction is studied.
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| 5. |
- Lundow, Per Håkan
(författare)
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Compression of transfer matrices
- 2001
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Ingår i: Discrete Mathematics. - 0012-365X .- 1872-681X. ; 231:03-jan, s. 321-329
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Tidskriftsartikel (refereegranskat)abstract
- We present a method for reducing the size of transfer matrices by exploiting symmetry. For example, the transfer matrix for enumeration of matchings in the graph C-4 x C-4 x P-n can be reduced from order 65536 to 402 simply due to the 384 automorphisms of C-4 x C-4. The matrix for enumeration of perfect matchings can be still further reduced to order 93, all in a straightforward and mechanical way. As an application we report an improved upper bound for the three-dimensional dimer problem.
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| 6. |
- Lundow, Per Håkan, et al.
(författare)
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On the p, q-binomial distribution and the Ising model
- 2010
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Ingår i: Philosophical Magazine. - : Informa UK Limited. - 1478-6435 .- 1478-6443. ; 90:24, s. 3313-3353
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Tidskriftsartikel (refereegranskat)abstract
- We employ p, q-binomial coefficients, a generalisation of the binomial coefficients, to describe the magnetisation distributions of the Ising model. For the complete graph this distribution corresponds exactly to the limit case p = q. We apply our investigation to the simple d-dimensional lattices for d = 1, 2, 3, 4, 5 and fit p, q-binomial distributions to our data, some of which are exact but most are sampled. For d = 1 and d = 5, the magnetisation distributions are remarkably well-fitted by p,q-binomial distributions. For d = 4 we are only slightly less successful, while for d = 2, 3 we see some deviations (with exceptions!) between the p, q-binomial and the Ising distribution. However, at certain temperatures near Tc the statistical moments of the fitted distribution agree with the moments of the sampled data within the precision of sampling. We begin the paper by giving results of the behaviour of the p, q-distribution and its moment growth exponents given a certain parameterisation of p, q. Since the moment exponents are known for the Ising model (or at least approximately for d = 3) we can predict how p, q should behave and compare this to our measured p, q. The results speak in favour of the p, q-binomial distribution's correctness regarding its general behaviour in comparison to the Ising model. The full extent to which they correctly model the Ising distribution, however, is not settled.
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| 7. |
- Åhag, Per, et al.
(författare)
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On a generalised Lambert W branch transition function arising from p,q-binomial coefficients
- 2024
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Ingår i: Applied Mathematics and Computation. - : Elsevier. - 0096-3003 .- 1873-5649. ; 462
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Tidskriftsartikel (refereegranskat)abstract
- With only a complete solution in dimension one and partially solved in dimension two, the Lenz-Ising model of magnetism is one of the most studied models in theoretical physics. An approach to solving this model in the high-dimensional case (d>4) is by modelling the magnetisation distribution with p,q-binomial coefficients. The connection between the parameters p,q and the distribution peaks is obtained with a transition function ω which generalises the mapping of Lambert W function branches W0 and W−1 to each other. We give explicit formulas for the branches for special cases. Furthermore, we find derivatives, integrals, parametrizations, series expansions, and asymptotic behaviours.
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