Convergence Rates of Random Discrete Model Curves Approaching SLE Curves in the Scaling Limit
Bibliographic record
Abstract
Recently, A. Kempannien and S. Smirnov provided a framework for showing convergence of discrete model interfaces to the corresponding SLE curves. They show that given a uniform bound on specific crossing probabilities one can deduce that the interface has subsequential scaling limits that can be described almost surely by L\"owner evolutions. This leads to the natural question to investigate the rate of convergence to the corresponding SLE curves. F. Johansson Viklund has developed a framework for obtaining a power-law convergence rate to an SLE curve from a power-law convergence rate for the driving function provided some additional geometric information along with an estimate on the growth of the derivative of the SLE map. This framework is applied to the case of the loop-erased random walk. In this thesis, we show that if your interface satisfies the uniform annulus condition proposed by Kempannien and Smirnov then one can deduce the geometric information required to apply Viklund's framework. As an application, we apply the framework to the critical percolation interface. The first step in this direction for critical percolation was done by I. Binder, L. Chayes and H.K. Lei where they proved that the convergence rate of the Cardy-Smirnov observable is polynomial in the size of the lattice. It relies on a careful analysis of the boundary behaviour of conformal maps and their discrete analytic approximations as well as a Percolation construction of the {\it Harris systems}. Further, we exploit the toolbox developed by D. Chelkak for discrete complex analysis on isoradial graphs to show polynomial rate of convergence for the discrete martingale observables for harmonic explorer and the FK Ising model to the corresponding continuum objects. Then, we apply the framework developed above to gain a polynomial convergence rate for the corresponding curves.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.026 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), 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".