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Recurrence of planar graph limits

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We will show that any distributional limit of finite planar graphs in which the degree of the root decays exponentially is almost surely recurrent. As a corollary, we obtain that the uniform infinite planar triangulation/quadrangulation (UIPT/UIPQ) are almost surely recurrent, resolving conjectures of Angel, Benjamini and Schramm. Joint work with Asaf Nachmias.

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