In this article, we introduce Brownian motion on stable looptrees using resistance techniques. We prove an invariance principle characterising it as the scaling limit of random walks on discrete looptrees, and prove precise local and global bounds on its heat kernel. We also conduct a detailed investigation of the volume growth properties of stable looptrees, and show that the random volume and heat kernel fluctuations are locally log-logarithmic, and globally logarithmic around leading terms of r α and t −α α+1 respectively. These volume fluctuations are the same order as for the Brownian continuum random tree, but the upper volume fluctuations (and corresponding lower heat kernel fluctuations) are different to those of stable trees.

We give a construction of an infinite stable looptree, which we denote by L ∞ α , 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 Björnberg and Stefánsson (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 L ∞ α , and show that it agrees with that of its discrete counterpart. Moreover, we show that Brownian motion on L ∞ α 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.

In this article, we study a simple random walk on a decorated Galton-Watson tree, obtained from a Galton-Watson tree by replacing each vertex of degree n with an independent copy of a graph Gn and gluing the inserted graphs along the tree structure. We assume that there exist constants d, R ≥ 1, v < ∞ such that the diameter, effective resistance across and volume of Gn respectively grow like nas n → ∞. We also assume that the underlying Galton-Watson tree is critical with offspring tails decaying like cx −α for some constant c and some α ∈ (1, 2). We establish the fractal dimension, spectral dimension, walk dimension and simple random walk displacement exponent for the resulting metric space as functions of α, d, R and v, along with bounds on the fluctuations of these quantities.

No abstract