Lower tails for triangles inside the critical window
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Bibliographic record
Abstract
We study the probability that the random graph $G(n,p)$ is triangle-free. When $p =o(n^{-1/2})$ or $p = ω(n^{-1/2})$ the asymptotics of the logarithm of this probability are known via Janson's inequality in the former case and via regularity or hypergraph container methods in the latter case. We prove for the first time an asymptotic formula for the logarithm of this probability when $p = c n^{-1/2}$ for $c$ a sufficiently small constant. More generally, we study lower-tail large deviations for triangles in random graphs: the probability that $G(n,p)$ has at most $η$ times its expected number of triangles, when $p = c n^{-1/2}$ for $c$ and $η\in [0,1)$ constant. Our results apply for all $c$ if $η\ge .4993$ and for $c$ small enough otherwise. For $η$ small (including the case of triangle-freeness), we prove that a phase transition occurs as $c$ varies, in the sense of a non-analyticity of the rate function, while for $η\ge .4993$ we prove that no phase transition occurs. On the other hand for the random graph $G(n,m)$, with $m = b n^{3/2}$, we show that a phase transition occurs in the lower-tail problem for triangles as $b$ varies for \emph{every} $η\in [0,1)$. Our method involves ingredients from algorithms and statistical physics including the cluster expansion and concentration inequalities for contractive Markov chains.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it