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A phase transition in block-weighted random maps

William Fleurat, Zéphyr Salvy

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 $Θ(n)$. 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 $Θ(\log(n))$, 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 $Θ(n^{2/3})$, the scaling order for distances is $n^{1/3}$, and the scaling limit is the stable tree of parameter $3/2$.

A phase transition in block-weighted random maps

Abstract

We consider the model of random planar maps of size biased by a weight per -connected block, and the closely related model of random planar quadrangulations of size biased by a weight per simple component. We exhibit a phase transition at the critical value . If , a condensation phenomenon occurs: the largest block is of size . Moreover, for quadrangulations we show that the diameter is of order , and the scaling limit is the Brownian sphere. When , the largest block is of size , the scaling order for distances is , and the scaling limit is the Brownian tree. Finally, for , the largest block is of size , the scaling order for distances is , and the scaling limit is the stable tree of parameter .
Paper Structure (64 sections, 33 theorems, 214 equations, 24 figures, 3 tables)

This paper contains 64 sections, 33 theorems, 214 equations, 24 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model.
  3. Results.
  4. Tree decomposition of maps
  5. Maps and their enumeration
  6. 2-connected maps and block decomposition
  7. Block tree of a map and its applications
  8. Block tree of a quadrangulation
  9. Probabilistic consequences
  10. A word on the probabilistic setting
  11. Phase diagram
  12. Study of the size of the largest blocks
  13. Subcritical case
  14. Supercritical case
  15. Critical case
  16. ...and 49 more sections

Key Result

Proposition 1

The number $m_{n}$ of maps of size $n$ is equal to This implies in particular that $\rho_M=1/12$ and $M(\rho_M)<\infty$, where $\rho_M$ denotes the radius of convergence of $M(z)$.

Figures (24)

  • Figure 1: Approximation of the Brownian sphere by a simple quadrangulation of size $50\ 000$, using a generator by Éric Fusy.
  • Figure 2: Approximation of the Brownian tree by a binary tree of size approximately $70\ 000$.
  • Figure 3: Approximation of the stable tree $3/2$ by a tree of size approximately $150\ 000$.
  • Figure 4: Map drawn according to the subcritical model $\mathbb{P}_{n,1}$ of size around $55\ 000$ (see larger version in \ref{['fig:simul-sub-1']}).
  • Figure 5: Map drawn according to the subcritical model $\mathbb{P}_{n,8/5}$ of size around $55\ 000$ (see larger version in \ref{['fig:simul-sub-8/5']}).
  • ...and 19 more figures

Theorems & Definitions (69)

  • Proposition 1: tutte_1963
  • Definition 1
  • Definition 2
  • Proposition 2: tutte_1963
  • Proposition 3: tutte_19632Louigi
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • ...and 59 more