2011/12/02 by Svante Janson, Janson, Svante · 5 citations
Mathematics · #05C05 #60C50 #60F05 #60J80 #FOS: Mathematics #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1112.0510
openalex publication_date 2011/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a unified treatment of the limit, as the size tends to infinity, of\nsimply generated random trees, including both the well-known result in the\nstandard case of critical Galton--Watson trees and similar but less well-known\nresults in the other cases (i.e., when no equivalent critical Galton--Watson\ntree exists). There is a well-defined limit in the form of an infinite random\ntree in all cases; for critical Galton--Watson trees this tree is locally\nfinite but for the other cases the random limit has exactly one node of\ninfinite degree.\n The proofs use a well-known connection to a random allocation model that we\ncall balls-in-boxes, and we prove corresponding theorems for this model.\n This survey paper contains many known results from many different sources,\ntogether with some new results.\n