Year: 2026
Type: preprint
Abstract: We study the graphon variational problem that governs lower tail large deviations of the number of copies of a fixed nonbipartite graph $H$ in a sparse Erdős--Rényi random graph. Our main new insight is that any non-constant minimizer of the variational problem exhibits a discontinuous jump away from the constant graphon. We deduce that the replica-symmetric and symmetry-breaking regimes are separated by a single critical parameter and that the phase transition is first order. In the supercritical regime $p \gg n^{-1/m_2(H)}$, we transfer these results to the conditioned random graph, showing that the edge density is a discontinuous order parameter and that the conditioned graph concentrates in cut distance near the full minimizer set.
Source: arXiv (Cornell University)
Author: Chin, Byron
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223112209