On a Factorization Formula for the Partition Function of Directed Polymers
Bibliographic record
Abstract
Abstract We prove a factorization formula for the point-to-point partition function associated with a model of directed polymers on the space-time lattice $$\mathbb {Z}^{d+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . The polymers are subject to a random potential induced by independent identically distributed random variables and we consider the regime of weak disorder, where polymers behave diffusively. We show that when writing the quotient of the point-to-point partition function and the transition probability for the underlying random walk as the product of two point-to-line partition functions plus an error term, then, for large time intervals [0, t ], the error term is small uniformly over starting points x and endpoints y in the sub-ballistic regime $$\Vert x - y \Vert \le t^{\sigma }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>‖</mml:mo> <mml:mi>x</mml:mi> <mml:mo>-</mml:mo> <mml:mi>y</mml:mi> <mml:mo>‖</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>σ</mml:mi> </mml:msup> </mml:mrow> </mml:math> , where $$\sigma < 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> can be arbitrarily close to 1. This extends a result of Sinai, who proved smallness of the error term in the diffusive regime $$\Vert x - y \Vert \le t^{1/2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>‖</mml:mo> <mml:mi>x</mml:mi> <mml:mo>-</mml:mo> <mml:mi>y</mml:mi> <mml:mo>‖</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . We also derive asymptotics for spatial and temporal correlations of the field of limiting partition 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 | 0.000 | 0.003 |
| 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 | 0.000 |
| 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".