Bounds for the cubic-lattice dimer constant
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What is the cubic-lattice dimer constant ℓ₃, the exponential growth rate governing perfect matchings on the three-dimensional cubic lattice? Transfer-layer calculations give rigorous upper bounds from finite systems.
Why it matters and what counts as progress
Why it matters: Tighter bounds quantify the entropy of cubic-lattice perfect matchings where an exact three-dimensional solution is unavailable.
What counts as progress: Lower a certified upper bound, raise a lower bound or prove convergence and error estimates for a transfer-layer family.
How the problem developed
- 1966 · Existence theorem [1]
John M. Hammersley Established existence of the dimer entropy limit for the cubic lattice, making the limiting constant well defined.
- 2001 · Upper bound [1]
Peter Lundow Obtained the earlier rigorous upper bound 0.457547 using transfer-matrix methods.
- 11 Aug 2026 · Upper bound [1]
Qidong He Improved the rigorous upper bound for the cubic-lattice dimer constant to 0.452130.
AI contribution
A new upper bound on the dimer constant of ℤ³
A diagonal transfer-layer family gives an upper bound for the cubic-lattice dimer constant. Its (m,n)=(5,5), p=2 exact finite computation yields ℓ₃≤0.452130.
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Publication dates and editorial changes.
AI-suggested transfer method improves the cubic-lattice dimer bound