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

TL;DR

This paper identifies a universal critical scaling profile for trees and connected subgraphs on complete graphs, with implications for high-dimensional lattice models.

Contribution

It introduces a unified scaling profile for generating functions of trees and connected subgraphs, applicable in a critical window, extending to high-dimensional lattice structures.

Findings

Single scaling profile applies to both trees and connected subgraphs

Profile conjectured to hold for high-dimensional lattice models

Provides insights into finite-size scaling in combinatorial structures

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 $d \ge 8$.