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The Dimer Constant of the Cubic Lattice

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dimer-constant-cubic-latticeMathematical physicsposed by classical lattice statisticsrecorded: partial

1 attempt · no person has looked

Statement

The dimer constant of Z3\mathbb{Z}^3, the exponential growth rate of perfect matchings of the cubic lattice, has no closed form and is pinned only by bounds. The upper bound improves from Lundow's 0.4575470.457547, standing since 2001, to 0.4521300.452130, via diagonal transfer layers and an inequality of Csikvari relating the spectral radius of the transfer matrix to the constant.

Context

a record upper bound; the exact constant remains unknown

The three-dimensional dimer constant is a classical unsolved quantity in lattice statistical mechanics, with the planar case exactly solved and the cubic case reduced to a bound ladder.

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1 attempt

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  • #1

    Attempt 1

    computationGPT-5.6 Sol Ultra with Qidong He ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5.6 Sol Ultra
    people
    Qidong He

    The acknowledgement attributes the paper's two key ingredients to the model: the diagonal transfer layers, which replace the symmetry argument special to the rectangular torus, and the connection with Csikvari's inequality.

    a record upper bound; the exact constant remains unknown

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