Abstract
We give a construction of an infinite stable looptree, which we denote by $\mathcal{L} ^{\infty }_{\alpha }$, 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 $\mathcal{L} ^{\infty }_{\alpha }$, and show that it agrees with that of its discrete counterpart. Moreover, we show that Brownian motion on $\mathcal{L} ^{\infty }_{\alpha }$ 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.
Citation
Eleanor Archer. "Infinite stable looptrees." Electron. J. Probab. 25 1 - 48, 2020. https://doi.org/10.1214/20-EJP413
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