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2020 Infinite stable looptrees
Eleanor Archer
Electron. J. Probab. 25: 1-48 (2020). DOI: 10.1214/20-EJP413


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.


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Eleanor Archer. "Infinite stable looptrees." Electron. J. Probab. 25 1 - 48, 2020.


Received: 14 March 2019; Accepted: 5 January 2020; Published: 2020
First available in Project Euclid: 29 January 2020

zbMATH: 1445.60032
MathSciNet: MR4059189
Digital Object Identifier: 10.1214/20-EJP413

Primary: 60F17
Secondary: 28A80 , 54E70 , 60K37

Keywords: limit theorem , random stable looptrees , Spectral dimension , Stable Lévy process


Vol.25 • 2020
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