Shattering in the Ising p-spin glass model
Bibliographic record
Abstract
Abstract We study the Ising p -spin glass model for large p . We show that for any inverse temperature $$\sqrt{\ln 2}<\beta <\sqrt{2\ln 2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msqrt> <mml:mrow> <mml:mo>ln</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msqrt> <mml:mo><</mml:mo> <mml:mi>β</mml:mi> <mml:mo><</mml:mo> <mml:msqrt> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>ln</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math> and any large p , the model exhibits shattering : w.h.p. as $$n\rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , there exists exponentially many well-separated clusters such that (a) each cluster has exponentially small Gibbs mass, and (b) the clusters collectively contain all but a vanishing fraction of Gibbs mass. Moreover, these clusters consist of configurations with energy near $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> . Range of temperatures for which shattering occurs is within the replica symmetric region. To the best of our knowledge, this is the first shattering result regarding the Ising p -spin glass models. Furthermore, we show that for any $$\gamma >0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and any large enough p , the model exhibits an intricate geometrical property known as the multi Overlap Gap Property above the energy value $$\gamma \sqrt{2\ln 2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:msqrt> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>ln</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math> . Our proofs are elementary, and in particular based on simple applications of the first and the second moment methods.
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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 | 0.000 |
| 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 | 0.001 |
| 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".