Recently, I ended up having to use the notion of an ultralimit of metric spaces and realized that I do not have a good intuition for the "limit object" this construction creates. Given a sequence of metric spaces $(X_n,d_n)$, in what way does the ultralimit resemble $X_n$'s?
Let me explain what kind of answers I am looking for by pointing to a specific example. A special case of the ultralimit construction is the construction of the asymptotic cone of a metric spaces, in which one takes the ultralimit of the metric spaces $(X,d(\cdot,\cdot)/n)$ for a fixed metric space $(X,d(\cdot,\cdot))$ and some fixed non-principal ultrafilter on $\mathbb{N}$.
One may think of the asymptotic cone as the space which codes the recurring configurations of the space $(X,d(\cdot,\cdot))$ if an observer was to "zoom out" from this space. As $n$ gets bigger, points that are not close will be seen closer in $(X,d(\cdot,\cdot)/n)$ and, in the limit case, various "local" configurations will disappear. In a sense, the asymptotic cone somehow codes information regarding the "global" geometry of the underlying space.
Is there some intuitive way to describe what the ultralimit does in general? Let me ask a very concrete question. What is an ultralimit of the sequence of spaces $(C_n,d/n)$ where $C_n$ is the cyclic graph with $n$ vertices and $d$ gives the length of the shortest path between two vertices? In what sense can such ultralimits be "obtained" from these cyclic graphs?
I also added the reference request tag to the post since I would also appreciate being directed to a survey or book that covers basic facts about these ultralimits.