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Mathematics > Probability
Title: A phase transition in block-weighted random maps
(Submitted on 3 Feb 2023 (v1), last revised 5 Feb 2024 (this version, v4))
Abstract: We consider the model of random planar maps of size $n$ biased by a weight $u>0$ per $2$-connected block, and the closely related model of random planar quadrangulations of size $n$ biased by a weight $u>0$ per simple component. We exhibit a phase transition at the critical value $u_C=9/5$. If $u<u_C$, a condensation phenomenon occurs: the largest block is of size $\Theta(n)$. Moreover, for quadrangulations we show that the diameter is of order $n^{1/4}$, and the scaling limit is the Brownian sphere. When $u > u_C$, the largest block is of size $\Theta(\log(n))$, the scaling order for distances is $n^{1/2}$, and the scaling limit is the Brownian tree. Finally, for $u=u_C$, the largest block is of size $\Theta(n^{2/3})$, the scaling order for distances is $n^{1/3}$, and the scaling limit is the stable tree of parameter $3/2$.
Submission history
From: Zéphyr Salvy [view email][v1] Fri, 3 Feb 2023 13:19:51 GMT (233kb,D)
[v2] Tue, 11 Apr 2023 12:45:57 GMT (12324kb,D)
[v3] Sat, 21 Oct 2023 08:28:26 GMT (13415kb,D)
[v4] Mon, 5 Feb 2024 14:03:28 GMT (13415kb,D)
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