Rotation-, translation-, scale-, and Markov-invariant random tilings

Definitions and mathematical statement

A planar tiling is a collection of tiles covering $\mathbb R^2$ with pairwise disjoint interiors, in an admissible measurable tiling category. Let $\mathcal T$ be a random tiling. Require its law to be invariant under translations and rotations. A usual translation-mixing condition is \[ \Pr(A\cap\tau_x B)\longrightarrow\Pr(A)\Pr(B) \qquad(|x|\to\infty) \] for events determined by bounded observation windows.

Does there exist one nontrivial tiling law satisfying this invariance and mixing, stationarity under the intended local clustering-and-rescaling rule, and the stated spatial Markov property? If $\mathcal C$ is the clustering rule and $D_r$ the compensating dilation, the stationarity condition would be \[ D_r\mathcal C(\mathcal T)\ \stackrel{d}=\ \mathcal T. \] It is invariance under this combined operation, not automatically invariance under every pure dilation. The rule, scale factor, randomization, and meaning of local are not supplied.

The spatial Markov requirement says a tile should not reveal the complementary tiling along its boundary, but the source gives no conditional sigma-algebras, boundary data, or conditional law. It cannot be replaced without justification by unconditional independence of a tile and its complement. The tile shapes, local finiteness, nontriviality constraints, and precise mixing notion are also not fixed; trivial one-tile/identity-clustering interpretations must be distinguished from the intended construction.

Short English statement

Construct a single random planar tiling law with Euclidean invariance, mixing, clustering/rescaling stationarity, and the intended boundary Markov property. The operation and conditional-independence data are missing.

Sources

• Itai Benjamini, Euclidean vs. Graph Metric (2012), Section 4.1, unnumbered tiling question, PDF page 4, https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=4; preserved copy https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf; accessed 2026-07-29. The supplied provenance reports extracted text checked against rendered pages.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10000070$, source code catalogue entry, Rotation-, translation-, scale-, and Markov-invariant random tilings. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.