Hom shifts form a class of multidimensional shifts of finite type (SFT) where adjacent symbols must be neighbors in a fixed finite undirected simple graph $G$. This talk is about gluing distance in Hom shifts: given two $nxn$ admissible partial blocks, how far do they need to be so that we can glue them together (i.e embed) in a larger admissible block.

The gluing gap measures how far any two square patterns of size $n$ can be glued, which has a clear analogy with gap fo specification property in one-dimensional subshifts. We prove that the gluing gap either depends linearly on $n$ or is dominated by $log(n)$. It is clear that there are Hom shift, where gluing gap is bounded by constant, thus independent of $n$. To support our results, we find a Hom shift with gap $\mathrm{\Theta }(log(n))$, infirming a conjecture formulated by R. Pavlov and M. Schraudner.

This talk is based on a joint work with Silvere Gangloff and Benjamin Hellouin de Menibus