Invariance principles for spatial multitype Galton–Watson trees

By Grégory Miermont. 2006

85 citations recorded by the source corpus.

We prove that critical multitype Galton-Watson trees converge after rescaling to the Brownian continuum random tree, under the hypothesis that the offspring distribution is irreducible and has finite covariance matrices. Our study relies on an ancestral decomposition for marked multitype trees, and an induction on the number of types. We then couple the genealogical structure with a spatial motion, whose step distribution may depend on the structure of the tree in a local way, and show that the resulting discrete spatial trees converge once suitably rescaled to the Brownian snake, under some moment assumptions.

Subfields: Finance, Mathematical Physics, Nature and Landscape Conservation

View Paper: https://www.semanticscholar.org/paper/ee06439cdd82c382387a807a4fef3875a34d7cb0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

1. Jean-François Le Gall
2. Grégory Miermont
3. C. Abraham
4. Nicolas Curien
5. Matthias Winkel
6. Jim Pitman
7. Bénédicte Haas
8. Thomas Duquesne

In Mathematical Physics

1. Nicolas Curien
2. Jean-François Le Gall
3. C. Abraham
4. Grégory Miermont
5. Jérémie Bettinelli
6. Laurent M'enard
7. Igor Kortchemski
8. R. Stephenson

In Nature and Landscape Conservation

1. Jean-François Le Gall
2. C. Abraham

Citations in the corpus

• Rescaled bipartite planar maps converge to the Brownian map (2013, methodology): “Both these constants can be calculated using the formulas found in [14].”
• A view from infinity of the uniform infinite planar quadrangulation (2012, background): “One difference is that [31] deals with the so-called height process of the trees rather than the contour process, but the convergence of the latter is indeed a consequence of the convergence of height processes, as discussed in [16, Section 2.”
• The Brownian map is the scaling limit of uniform random plane quadrangulations (2011, background): “ the ones of the present paper, but we expect that the scaling limits of Section6remain valid. Indeed, such generalizations hold for the basic case of trees, by the invariance principles developed in [20,23]. This program seems reasonable to carry out at least in the case where quadrangulations are ”
• Uniqueness and universality of the Brownian map (2011, methodology): “2 can be derived as a special case of Theorems 2 and 4 in [22].”