On quasisymmetric embeddings of the Brownian map and continuum trees
Bibliographic record
Abstract
Abstract The Brownian map is a model of random geometry on the sphere and as such an important object in probability theory and physics. It has been linked to Liouville Quantum Gravity and much research has been devoted to it. One open question asks for a canonical embedding of the Brownian map into the sphere or other, more abstract, metric spaces. Similarly, Liouville Quantum Gravity has been shown to be “equivalent” to the Brownian map but the exact nature of the correspondence (i.e. embedding) is still unknown. In this article we show that any embedding of the Brownian map or continuum random tree into $${{\,\mathrm{\mathbb {R}}\,}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mrow><mml:mspace/><mml:mi>R</mml:mi><mml:mspace/></mml:mrow></mml:mrow><mml:mi>d</mml:mi></mml:msup></mml:math> , $${{\,\mathrm{\mathbb {S}}\,}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mrow><mml:mspace/><mml:mi>S</mml:mi><mml:mspace/></mml:mrow></mml:mrow><mml:mi>d</mml:mi></mml:msup></mml:math> , $${{\,\mathrm{\mathbb {T}}\,}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mrow><mml:mspace/><mml:mi>T</mml:mi><mml:mspace/></mml:mrow></mml:mrow><mml:mi>d</mml:mi></mml:msup></mml:math> , or more generally any doubling metric space, cannot be quasisymmetric. We achieve this with the aid of dimension theory by identifying a metric structure that is invariant under quasisymmetric mappings (such as isometries) and which implies infinite Assouad dimension. We show, using elementary methods, that this structure is almost surely present in the Brownian continuum random tree and the Brownian map. We further show that snowflaking the metric is not sufficient to find an embedding and discuss continuum trees as a tool to studying “fractal functions”.
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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 | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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 | 0.001 |
| Research integrity | 0.001 | 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".