The local weak limit of the minimum spanning tree of the complete graph

TL;DR

This paper proves that the minimum spanning tree of a complete graph with exponential edge weights converges locally to a random infinite tree, with cubic volume growth, confirming physics predictions and contrasting with invasion percolation behavior.

Contribution

It establishes the local weak limit of the MST of complete graphs with exponential weights and describes its structure and volume growth, linking to the PWIT and invasion percolation.

Findings

MST converges locally to a random infinite tree M.

Tree M exhibits cubic volume growth.

Results confirm physics-based volume growth predictions.

Abstract

Assign i.i.d. standard exponential edge weights to the edges of the complete graph K_n, and let M_n be the resulting minimum spanning tree. We show that M_n converges in the local weak sense (also called Aldous-Steele or Benjamini-Schramm convergence), to a random infinite tree M. The tree M may be viewed as the component containing the root in the wired minimum spanning forest of the Poisson-weighted infinite tree (PWIT). We describe a Markov process construction of M starting from the invasion percolation cluster on the PWIT. We then show that M has cubic volume growth, up to lower order fluctuations for which we provide explicit bounds. Our volume growth estimates confirm recent predictions from the physics literature, and contrast with the behaviour of invasion percolation on the PWIT and on regular trees, which exhibit quadratic volume growth.