Phase transition for tree-rooted maps
Phase transition for tree-rooted maps
We introduce a model of tree-rooted planar maps weighted by their number of $2$-connected blocks. We study its enumerative properties and prove that it undergoes a phase transition. We give the distribution of the size of the largest $2$-connected blocks in the three regimes (subcritical, critical and supercritical) and further establish that the scaling limit is the Brownian Continuum Random Tree in the critical and supercritical regimes, with respective rescalings $\sqrt{n/\log(n)}$ and $\sqrt{n}$.
14 pages
- University of Paris France
- Leibniz Association Germany
- Délégation Ile-de-France Villejuif France
- French National Centre for Scientific Research France
- UNIVERSITE GUSTAVE EIFFEL France
Planar maps, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Asymptotic Enumeration, Random trees, Probability (math.PR), 510, 004, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], Mathematics of computing → Enumeration, Mathematics of computing → Generating functions, Mathematics of computing → Stochastic processes, FOS: Mathematics, Mathematics of computing → Probability and statistics, Mathematics - Combinatorics, Combinatorics (math.CO), Mathematics - Probability, Phase transition, ddc: ddc:004
Planar maps, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Asymptotic Enumeration, Random trees, Probability (math.PR), 510, 004, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], Mathematics of computing → Enumeration, Mathematics of computing → Generating functions, Mathematics of computing → Stochastic processes, FOS: Mathematics, Mathematics of computing → Probability and statistics, Mathematics - Combinatorics, Combinatorics (math.CO), Mathematics - Probability, Phase transition, ddc: ddc:004
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