Bipodal optimizers in the upper-tail variational problem for regular subgraph densities

Abstract

Let $H$ be a fixed $d$-regular graph with $d\ge2$, and let $t(H,\cdot)$ denote its homomorphism density. We study the upper-tail event $t(H,G(n,p))\ge r^{|E(H)|}$ for fixed $0<p<r<1$ in a dense Erdős--Rényi random graph $G(n,p)$. Near the Lubetzky--Zhao replica-symmetric phase boundary and away from the exceptional target density $(d-1)/d$, we prove that the optimizer of the Chatterjee--Varadhan variational problem on the symmetry-breaking side is bipodal (two-block) and unique up to relabeling. Its block parameters depend analytically on $(p,r)$. To treat the exceptional boundary point, where the nonexceptional theory degenerates, we construct an analytic curve approaching that point from the symmetry-breaking side along which the unique optimizers are nonconstant rank-one bipodal graphons. In both settings, we derive asymptotic expansions of the edge-density deficit and the rate function that governs the exponential decay of the upper-tail probability. Moreover, the conditioned random graph converges in cut distance to the corresponding bipodal optimizer as $n\to\infty$. As $(p,r)$ approaches the phase boundary from the symmetry-breaking side, the optimizers converge to their constant limits through two distinct mechanisms. For each fixed nonexceptional target density, one block shrinks to zero measure, giving convergence in $L^1$ but not in $L^\infty$. Along the exceptional curve, both blocks remain macroscopic: their sizes tend to $1/2$ and all three block densities tend to $(d-1)/d$, yielding convergence in $L^\infty$.