Upper bounds on the dimer constant of the cubic lattice
the best proved upper bound on the dimer constant of
- steps
- 2
- by AI
- 0
- since
- 1998
The line is the frontier over time, falling as the bound comes down: lower is better here. Filled dots are steps that moved it; muted dots are results that did not. Orange dots are catalog entries, results with AI in the loop. Hollow dots are candidates under review and never move the line.Grey dots along the top edge are results from before the quantity had a number, placed at the worst end because they have no value on this axis. Dots that would overlap are nudged sideways a few pixels. Hover a dot for its value and attribution.
About this frontier
The dimer constant is the exponential growth rate of the number of perfect matchings of boxes in ; Hammersley proved in 1966 that the limit exists and is independent of the box sequence. In two dimensions the analogous constant has the closed form of Kasteleyn, Fisher and Temperley, with Catalan's constant. In three dimensions no closed form is known and is pinned only between bounds. The rigorous lower bound has been since Minc drew it in 1978 from what is now Schrijver's theorem, while non-rigorous estimates put the truth near . This frontier tracks the upper bound, which has moved three times in sixty years.
Every step, newest first
| Date | Value | Who | Model | Status | Source |
|---|---|---|---|---|---|
| 30 Jul 2026 | Diagonal transfer layers on x1+x2+x3 = h rather than coordinate planes, so every edge joins consecutive layers and m, n need no parity restriction; Csikvari's 2017 inequality between a bipartite matching polynomial and its 2-lifts does the comparison, and a Collatz-Wielandt variant gives monotone computable bounds. Closes about half the gap Lundow left. Unreviewed preprint: a candidate. | Qidong He, with AI assistance | GPT-5.6 Sol Ultra | candidate unreviewed | entry |
| 2001 | best Ciucu's scheme pushed to (m,n) = (4,6), which needs both dimensions even because the comparison uses even powers. Stood for twenty-five years; Friedland-Peled and Lundow-Markstrom discuss why further upscaling is prohibitive. Value quoted from He's 2026 paper as the previous best. | Per Hakan Lundow | – | historical | source ↗ |
| 1998 | Duke Mathematical Journal 94 (1998), 1-11. The transfer-matrix scheme every later bound uses: coordinate-plane cross sections, the constant bounded by the spectral radius of a 2^(mn) matrix, compressed by symmetry. Ciucu reached (m,n) = (4,4). The value is from Finch's October 1998 survey, which records the bounds of the day as 0.419989 and 0.463107. | Mihai Ciucu | – | historical | source ↗ |
Lundow's 0.457547 and He's 0.452130 are both stated in He's 2026 paper, read here, which credits Ciucu with the method only. Ciucu's numeric value is not there; it comes from Finch's survey of constants in statistical mechanics (arXiv:math/9810155, October 1998), read here, recording the rigorous bounds of the day as 0.419989 and 0.463107 and citing Ciucu for the upper. The lower bound and the Nagle and Beichl-Sullivan estimates are as in He's introduction. Chen, Golin and Yong's 2019 claim to beat Lundow is not a row: He's footnote records that they gave no number.