Anticoncentration of random spanning trees in almost regular graphs

TL;DR

This work establishes an anticoncentration phenomenon for random spanning trees in connected almost-regular graphs with large degree: for any fixed $n$-vertex tree $T$, the probability that a uniformly chosen spanning tree is isomorphic to $T$ decays as $e^{-\Omega(n)}$, while such graphs already contain exponentially many non-isomorphic spanning trees. The authors introduce a graph-theoretic balls-into-bins model and prove sharp Poisson-like concentration bounds for the output degree distribution using Talagrand’s inequality and Brégman–Minc inequalities. A key consequence is that almost-regular graphs host exponentially many unlabeled spanning trees, reinforcing a strong anticoncentration property for spanning-tree isomorphism classes. The methods and the new $(X,Y;H)$-model open avenues for broader applications in anticoncentration phenomena and counting problems in random graph models.

Abstract

The celebrated formula of Otter \emph{[Ann. of Math. (2) 49 (1948), 583--599]} asserts that the complete graph contains exponentially many non-isomorphic spanning trees. In this paper, we show that every connected almost regular graph with sufficiently large degree already contains exponentially many non-isomorphic spanning trees. Indeed, we prove a stronger statement: for every fixed $n$-vertex tree $T$, $$ \Pr\bigl[\mathcal{T} \simeq_{\mathrm{iso}} T\bigr] = e^{-Ω(n)}, $$ where $\mathcal{T}$ is a uniformly random spanning tree of a connected $n$-vertex almost regular graph with sufficiently large degree. To prove this, we introduce a graph-theoretic variant of the classical balls--into--bins model, which may be of independent interest.

Theorems & Definitions (23)

• Theorem 1.1: McKay McKay, Alon Alon
• Theorem 1.2: Kostochka Kostochka
• Theorem 1.3: Otter Otter
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Lemma 2.2: Talagrand
• Lemma 2.3
• proof : Proof of Lemma \ref{['lem:expectation']}
• Lemma 2.4