Directed Polymers in the Intermediate Disorder Regime and the Seppäläinen-Johansson Model
Bibliographic record
Abstract
In this thesis, we study two discrete models of random growth in the Kardar--Parisi--Zhang (KPZ) universality class: the directed polymer and the Sepp\"al\"ainen--Johansson first-passage percolation model. The directed polymer was introduced by Huse and Henley as a model for the domain wall in a ferromagnetic Ising model with random bond impurities. This model depends on a parameter $\beta$, the inverse temperature. We consider the intermediate disorder regime, which consists in taking $\beta$ to depend on the length of the polymer $2n$, with $\beta=n^{-\alpha}$ for some $\alpha>0$. In this regime, there is a critical phase transition that happens at $\alpha=\frac{1}{4}$. When $\alpha>\frac{1}{4}$, the fluctuations of the free energy are of order $n^{(1-4\alpha)/4}$ and converge to a Gaussian. For $\alpha<\frac{1}{4}$, it was conjectured that the polymer should fall back in the KPZ regime, and that the fluctuations should instead be of order $n^{(1-4\alpha)/3}$, and converge after rescaling to the Tracy--Widom GUE distribution. We prove this conjecture for $\frac{1}{8}<\alpha<\frac{1}{4}$ for arbitrary i.i.d weights with exponential moments. The Sepp\"al\"ainen--Johansson model was introduced by Sepp\"al\"ainen as a simplified version of first-passage percolation where he was able to explicitly compute the limiting shape for Bernoulli weights. The behaviour of the fluctuations for this process were later studied by Johansson. We consider a generalization of this model, involving two families of i.i.d random variables $\{\xi_{ij}\}$ and $\{\eta_{ij}\}$ corresponding to the weights of the horizontal and vertical edges respectively. We obtain an explicit formula for the limiting shape of the first-passage distance expressed in terms of the corresponding limit shapes of the two sets of weights for the Sepp\"al\"ainen--Johansson model. We also study the limiting fluctuations of this model when at least one of the sets of weights is Bernoulli distributed, and we prove that these converge to the Airy$_2$ process.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
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) | 0.001 | 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 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".