Fluctuation exponent of the KPZ/stochastic Burgers equation
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Bibliographic record
Abstract
We consider the stochastic heat equation <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript t Baseline upper Z equals partial-differential Subscript x Superscript 2 Baseline upper Z minus upper Z ModifyingAbove upper W With dot"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>Z</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mi>Z</mml:mi> <mml:mo> − </mml:mo> <mml:mi>Z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>W</mml:mi> <mml:mo> ˙ </mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial _tZ= \partial _x^2 Z - Z \dot W</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> on the real line, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper W With dot"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>W</mml:mi> <mml:mo> ˙ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is space-time white noise. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h left-parenthesis t comma x right-parenthesis equals minus log upper Z left-parenthesis t comma x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo> − </mml:mo> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>Z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h(t,x)=-\operatorname {log} Z(t,x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is interpreted as a solution of the KPZ equation, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis t comma x right-parenthesis equals partial-differential Subscript x Baseline h left-parenthesis t comma x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u(t,x)=\partial _x h(t,x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a solution of the stochastic Burgers equation. We take <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z left-parenthesis 0 comma x right-parenthesis equals exp left-brace upper B left-parenthesis x right-parenthesis right-brace"> <mml:semantics> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>exp</mml:mi> <mml:mo> </mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Z(0,x)=\exp \{B(x)\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">B(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a two-sided Brownian motion, corresponding to the stationary solution of the stochastic Burgers equation. We show that there exist <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than c 1 less-than-or-equal-to c 2 greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> ≤ </mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>></mml:mo> <mml:mi mathvariant="normal"> ∞
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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 it