Random planar maps and growth-fragmentations
Let \(T^{(p)}\) be a random Boltzmann triangulation of the \(p\)-gon. The main object of interest in the present paper is the sequence of length (or perimeters) of the cycles of \(B_r(T^{(p)})\) ranked in decreasing order, which is denoted by \(L^{(p)}(r)= (L_1^{(p)}, L_2^{(p)},\dots)\). The main result of the paper is a functional invariance principle that describes the scaling limit of the process \(L^{(p)}\). \(X(t)\) is the size of a typical particle or cell at age \(t\). The rest of the present paper is organized as follows. In Section 2, the notion of general branching peeling explorations are introduced. Using results from [the first and third authors, Ann. Appl. Probab. 26, No. 4, 2556--2595 (2016; Zbl 1352.60103)] a scaling limit for the length of the locally largest cycle in large Boltzmann triangulations is obtained. In Section 3, the authors then use the peeling by layers, which is specific peeling algorithm that is the key in this work, to get the scaling limit of the locally largest cycle at given heights. Section 4 introduces the basics on cell systems and their scaling limits in order to prove the following result: Theorem 1. \((\frac{1}{p} L^{(p)}(r \sqrt{p}); r\ge 0) \to(X(\frac{2 t_{\Delta^1}}{a_{\Delta^1}}\times r); r\ge 0)\), as \(p\to \infty\), where convergence holds in distribution in the space of càdlàg process taking values in \(l_3^{\downarrow}\) equipped with the Skorokhod \(J_1\) topology (\(t_{\Delta^1}= \frac{\sqrt{3}}{8\sqrt{\pi}}\), and \(a_{\Delta^1}=\frac{1}{2\sqrt{3}}\)). Finally, Section 5 is devoted to the proof of a result which completes Theorem 1 by a more geometric point of view.
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