Exponential intersection tails in three-dimensional slabs

Definitions and mathematical statement

Let $f,g:\mathbb N\to\mathbb Z$ and retain the literal set in the source: \[ S=\{(n,f(n),g(n)):n\in\mathbb N\}\subset\mathbb Z^3. \] A usual exponential-intersection-tails property asks for a probability law $\mu$ on infinite self-avoiding nearest-neighbor paths from a fixed vertex such that, for independent paths $\gamma,\gamma'$ with law $\mu$, there are constants $C<\infty$ and $a>0$ satisfying \[ \Pr\left(|\operatorname{range}(\gamma)\cap\operatorname{range}(\gamma')|\geq m\right) \leq Ce^{-am}\qquad(m\geq1). \] The constants do not depend on $m$. The source does not specify this path/root convention or whether intersections count vertices or edges, so this is explicit standard context.

Characterize the functions $f,g$ for which the intended path law exists. There is a serious geometric ambiguity: the displayed $S$ has exactly one vertex at each first-coordinate level, whereas the title describes three-dimensional slabs. In its induced nearest-neighbor graph, two distinct vertices can be adjacent only at consecutive levels where both $f$ and $g$ are unchanged. Thus this literal set is not a thick slab and cannot silently be treated as one. The intended subset/adjacency, path class, and starting-point convention must be checked before deriving a slab-growth criterion.

Short English statement

Determine when the source's subset supports a law of infinite paths with exponentially decaying pairwise intersection tails. The displayed one-point-per-level set conflicts with the slab description, and path/adjacency conventions are missing.

Sources

• Itai Benjamini, Coarse Geometry and Randomness, Saint-Flour notes, Open Problem 4.30, PDF page 37, https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=37; preserved copy https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf; accessed 2026-07-29. The supplied provenance reports extracted text checked against rendered pages.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10000022$, source code catalogue entry, Exponential intersection tails in three-dimensional slabs. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.