Hard squares with negative activity

Fendley, Paul; Schoutens, Kareljan; Eerten, Hendrik van

doi:10.1088/0305-4470/38/2/002

Cited by 34 publications

(62 citation statements)

References 13 publications

(15 reference statements)

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“…3 Two graphs, and the independence complex of the top one. This complex has reduced Euler characteristic 1, and is homotopic to a 0-dimensional sphere (two points).…”

Section: Fig 2 the Rectangular Graphs R(m N ) Defined In Sectionmentioning

confidence: 99%

“…In Sections 3 to 5, we apply this general machinery to determine the homotopy type of the independence complex of several subgraphs of the square grid: the tilted rectangles of Figures 2 and 5 (Section 3), the parallelograms of Figure 9 (Section 5), and variations on them (Section 7.1). All these graphs have open boundary conditions (as opposed to the toric boundary conditions of [3,8]). However, in Section 4, we identify two sides of the rectangles of Figure 2 to obtain rectangles with cylindric boundary conditions.…”

Section: Fig 2 the Rectangular Graphs R(m N ) Defined In Sectionmentioning

confidence: 99%

“…This is motivated by the work of Fendley et al who conjectured that the eigenvalues of various transfer matrices related to the enumeration of independent sets are roots of unity [3]. One first observation is that finding the whole spectrum (λ 1 , .…”

Section: T(c I C I+1 )mentioning

confidence: 99%

“…It is conjectured in [3] that all the eigenvalues of O K are roots of unity. Let Z C (K, N ) denote the alternating number of independent sets on the cylinder C (K, N ).…”

Section: Theoremmentioning

confidence: 99%

“…In 2004, Fendley, Schoutens and van Eerten [3] published a series of remarkable conjectures on the partition function of the hard-square model specialized at u = −1. For instance, they observed that for an M × N -grid, taken with toric boundary conditions, the partition function at u = −1 seemed to be equal to 1 as soon as M and N were coprime.…”

Section: Introductionmentioning

confidence: 99%

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On the independence complex of square grids

2007

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The enumeration of independent sets of regular graphs is of interest in statistical mechanics, as it corresponds to the solution of hard-particle models. In 2004, it was conjectured by Fendley et al., that for some rectangular grids, with toric boundary conditions, the alternating number of independent sets is extremely simple. More precisely, under a coprimality condition on the sides of the rectangle, the number of independent sets of even and odd cardinality always differ by 1. In physics terms, this means looking at the hard-particle model on these grids at activity −1. This conjecture was recently proved by Jonsson.Here we produce other families of grid graphs, with open or cylindric boundary conditions, for which similar properties hold without any size restriction: the number of independent sets of even and odd cardinality always differ by 0, ±1, or, in the cylindric case, by some power of 2.We show that these results reflect a stronger property of the independence complexes of our graphs. We determine the homotopy type of these complexes using Forman's discrete Morse theory. We find that these complexes are either contractible, or homotopic to a sphere, or, in the cylindric case, to a wedge of spheres.Finally, we use our enumerative results to determine the spectra of certain transfer matrices describing the hard-particle model on our graphs at activity −1. These

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“…3 Two graphs, and the independence complex of the top one. This complex has reduced Euler characteristic 1, and is homotopic to a 0-dimensional sphere (two points).…”

Section: Fig 2 the Rectangular Graphs R(m N ) Defined In Sectionmentioning

confidence: 99%

Section: Fig 2 the Rectangular Graphs R(m N ) Defined In Sectionmentioning

confidence: 99%

Section: T(c I C I+1 )mentioning

confidence: 99%

“…It is conjectured in [3] that all the eigenvalues of O K are roots of unity. Let Z C (K, N ) denote the alternating number of independent sets on the cylinder C (K, N ).…”

Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the independence complex of square grids

2007

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Lattice Supersymmetry from the Ground Up

Fendley

Schoutens

2009

New Trends in Mathematical Physics

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Hard Squares for z = –1

Baxter

2011

Ann. Comb.

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The hard square model in statistical mechanics has been investigated for the case when the activity z is −1. For cyclic boundary conditions, the characteristic polynomial of the transfer matrix has an intriguingly simple structure, all the eigenvalues x being zero, roots of unity, or solutions of x 3 = 4cos 2 (πm/N). Here we tabulate the results for lattices of up to 12 columns with cyclic or free boundary conditions and the two obvious orientations. We remark that they are all unexpectedly simple and that for the rotated lattice with free or fixed boundary conditions there are obvious likely generalizations to any lattice size.

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Hard squares with negative activity

Cited by 34 publications

References 13 publications

On the independence complex of square grids

On the independence complex of square grids

Lattice Supersymmetry from the Ground Up

Hard Squares for z = –1

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