The lower tail of triangle counts in sparse random graphs
Dong Huang, Shuangping Li
math.PR
Oct 23, 2026 · v1
math.CO
TL;DR
The authors formalized the paper's arguments on the lower-tail graphon variational problem in Lean 4 with Mathlib, with parts written by AI models.
Abstract
Let $T_n$ be the number of triangles in the Erdős-Rényi random graph $G(n,p)$ with $n^{-1/2}\ll p\ll1$ and fix $η\in(0,1)$. We study the large deviation rate of the lower tail event $T_n\leη\mathbb{E}[T_n]$, and the typical structure of $G(n,p)$ conditioned on this event. By earlier work [16,19,21,49], both the rate and the structure are governed by a variational problem over graphons. In this work we solve this variational problem for every $η\in(0,1)$. We show that every minimizer is either a constant graphon or a step graphon with two blocks of equal size. Correspondingly, the conditioned graph is close to an Erdős-Rényi graph or to a symmetric stochastic block model with two blocks. These two regimes are separated by a first order phase transition at the threshold $η_*=0.009217\ldots$. This resolves a conjecture of Zhao [78] and an open problem of Janson and Warnke [39] on triangles. It thereby gives a complete description of the lower tail of triangle counts in this range of $p$.
Problem
The work asks for the large deviation rate and conditional structure of the lower tail event that the triangle count of G(n,p) is at most η times its mean, for n^{-1/2} ≪ p ≪ 1. Earlier work reduced both questions to a variational problem over graphons, which was unsolved and the subject of a conjecture by Zhao and an open problem of Janson and Warnke.
Approach
The authors derive Euler–Lagrange and vertex-reweighting first-order conditions for minimizers and analyze two-block stochastic block model graphons. At and above the threshold they use computer-assisted dual certificates on twelve convex polygons to classify minimizers. Below the threshold they combine local continuation near the two-block graphons, a spectral argument for small targets, and a global continuation argument. The arguments were also formalized in Lean 4 with Mathlib.
Results
Every minimizer is either the constant graphon or a balanced two-block SBM graphon. The two regimes are separated by a first-order phase transition at η_* = 0.009217…, which gives a complete description of the lower tail of triangle counts in this range of p.
