Critical scaling profile for trees and connected subgraphs on the complete graph

2024/12/07 by Liu, Yucheng, Slade, Gordon
#05C30 #60K35 #82B41 #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2412.05503

Abstract

We analyse generating functions for trees and for connected subgraphs on the complete graph, and identify a single scaling profile which applies for both generating functions in a critical window. Our motivation comes from the analysis of the finite-size scaling of lattice trees and lattice animals on a high-dimensional discrete torus, for which we conjecture that the identical profile applies in dimensions d ≥ 8.

Related

• Scaling limit of trees with vertices of fixed degrees and heights
• The growth constants of lattice trees and lattice animals in high dimensions
• Spread-out limit of the critical points for lattice trees and lattice animals in dimensions d>8
• Scaling limits and universality: Critical percolation on weighted graphs converging to an L³ graphon
• Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models