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Article Dans Une Revue Electronic Journal of Probability Année : 2024

A phase transition in block-weighted random maps

Résumé

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 $3/2$.
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Dates et versions

hal-04491449 , version 1 (06-03-2024)

Identifiants

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William Fleurat, Zéphyr Salvy. A phase transition in block-weighted random maps. Electronic Journal of Probability, 2024, 29, ⟨10.1214/24-EJP1089⟩. ⟨hal-04491449⟩
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