  We give a construction of an infinite stable looptree, which we denote by
$\mathcal{L}_{\alpha}^{\infty}$, and prove that it arises both as a local limit
of the compact stable looptrees of Curien and Kortchemski (2015), and as a
scaling limit of the infinite discrete looptrees of Richier (2017) and
Bj\"ornberg and Stef\'ansson (2015). As a consequence, we are able to prove
various convergence results for volumes of small balls in compact stable
looptrees, explored more deeply in a companion paper. We also establish the
spectral dimension of $\mathcal{L}_{\alpha}^{\infty}$, and show that it agrees
with that of its discrete counterpart. Moreover, we show that Brownian motion
on $\mathcal{L}_{\alpha}^{\infty}$ arises as a scaling limit of random walks on
discrete looptrees, and as a local limit of Brownian motion on compact stable
looptrees, which has similar consequences for the limit of the heat kernel.
