Take a \(2n \) x \(2n \) chessboard. Cut out four triangles from the angles, each with side \(n-1\): you get an Aztec Diamond (AD). The problem is: can you tile it with \(2\) x \(1\) domino tiles?
This app generates typical random ADs of increasing size (first figure). What do you observe for large \(\, n\)?
This phenomenon is known as the Arctic Circle theorem, and is well explained in this video of Mathologer.
Each AD tiling maps to a Non-Overlapping Paths problem (second figure): draw random paths connecting the two bottom sides of an AD, that do not overlap with each other.