Rescaled ultralimits and asymptotic cones

Fix a free ultrafilter $\omega$ on the library's natural numbers $\mathbb{N}$, which contain $0$, in the sense of Ultrafilter. A set in $\omega$ is called large. For a real sequence, ${\lim }_{\omega }{a}_n=a$ means that $\{n:\mathrm{\mid }{a}_n-a\mathrm{\mid }<\epsilon \}$ is large for every $\epsilon >0$.

Let $(X_n,d_n,e_n)$ be pointed metric spaces (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric) and ${\lambda }_n>0$. Put

$\mathcal{B}=\{(x_n)\in {\prod }_n{X}_n:{\sup }_n{\lambda }_n{d}_n(x_n,e_n)<\mathrm{\infty }\},D(x,y)={\lim }_{\omega }{\lambda }_n{d}_n(x_n,y_n)\mathrm{.}$

Declare $x\sim y$ when $D(x,y)=0$. The rescaled ultralimit is $\mathcal{B}\mathrm{/}\text{\hspace{0.17em}⁣}\sim$, with distance $d_{\omega }([x],[y])=D(x,y)$ and basepoint $[e]$. These formulas are provisional until the two results in justified_by establish existence of the real limit and the quotient metric.

For one fixed space $X_n=X$ and positive scales ${\lambda }_n\to 0$ in the ordinary sense, write ${\mathrm{Cone}}_{\omega }(X,e,\lambda )$ and call it an asymptotic cone. Basepoints $e_n$ may vary arbitrarily. A sequence bounded only on a large set is interpreted by replacing its other coordinates with $e_n$; the quotient-metric lemma proves independence of that replacement.