Random tree-weighted graphs
Louigi Addario-Berry, Jordan Barrett

TL;DR
This paper proves that random spanning trees in degree-constrained graphs converge to the Brownian continuum random tree, using a new coalescent process and Poisson approximations for graph features.
Contribution
It introduces a novel coalescent process to analyze random trees in degree-constrained graphs and establishes their convergence to the Brownian CRT.
Findings
Spanning trees in degree-constrained graphs converge to the Brownian CRT.
A new coalescent process effectively constructs random trees with fixed degree sequences.
Poisson approximation for loops and multiple edges in superimposed random graphs.
Abstract
For each , let be a sequence of positive integers with even sum . Let be uniformly distributed over the set of simple graphs with degree sequence , endowed with a spanning tree and rooted along an oriented edge of which is not an edge of . Under a finite variance assumption on degrees in , we show that, after rescaling, converges in distribution to the Brownian continuum random tree as . Our main tool is a new version of Pitman's additive coalescent (https://doi.org/10.1006/jcta.1998.2919), which can be used to build both random trees with a fixed degree sequence, and random tree-weighted graphs with a fixed degree sequence. As an input to the proof, we also derive a Poisson approximation theorem for the number of…
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Taxonomy
TopicsStochastic processes and statistical mechanics · Limits and Structures in Graph Theory · Markov Chains and Monte Carlo Methods
