Intermediate Disorder Regime for Directed Polymers in Dimension<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>
Why this work is in the frame
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Bibliographic record
Abstract
We introduce a new disorder regime for directed polymers in dimension $1+1$ by scaling the inverse temperature $\ensuremath{\beta}$ with the length of the polymer $n$. We scale ${\ensuremath{\beta}}_{n}\ensuremath{\mathrel{:=}}\ensuremath{\beta}{n}^{\ensuremath{-}\ensuremath{\alpha}}$ for $\ensuremath{\alpha}\ensuremath{\ge}0$. This scaling interpolates between the weak disorder ($\ensuremath{\beta}=0$) and strong disorder regimes ($\ensuremath{\beta}>0$). The fluctuation exponents $\ensuremath{\zeta}$ for the polymer end point and $\ensuremath{\chi}$ for the free energy depend on $\ensuremath{\alpha}$ in this regime, with $\ensuremath{\alpha}=0$ corresponding to the Kardar-Parisi-Zhang polymer exponents $\ensuremath{\zeta}=2/3$, $\ensuremath{\chi}=1/3$, and $\ensuremath{\alpha}\ensuremath{\ge}1/4$ corresponding to the simple random walk exponents $\ensuremath{\zeta}=1/2$, $\ensuremath{\chi}=0$. For $\ensuremath{\alpha}\ensuremath{\in}(0,1/4)$ the exponents interpolate linearly between these two extremes. At $\ensuremath{\alpha}=1/4$ we exactly identify the limiting distribution of the free energy and the end point of the polymer.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
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