Subcritical Gaussian multiplicative chaos in the Wiener space: construction, moments and volume decay
Bibliographic record
Abstract
Abstract We construct and study properties of an infinite dimensional analog of Kahane’s theory of Gaussian multiplicative chaos (Kahane in Ann Sci Math Quebec 9(2):105-150, 1985). Namely, if $$H_T(\omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>T</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is a random field defined w.r.t. space-time white noise $$\dot{B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:math> and integrated w.r.t. Brownian paths in $$d\ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , we consider the renormalized exponential $$\mu _{\gamma ,T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:msub> </mml:math> , weighted w.r.t. the Wiener measure $$\mathbb {P}_0(\textrm{d}\omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mtext>d</mml:mtext> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We construct the almost sure limit $$\mu _\gamma = \lim _{T\rightarrow \infty } \mu _{\gamma ,T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>γ</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>lim</mml:mo> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> in the entire weak disorder (subcritical) regime and call it subcritical GMC on the Wiener space . We show that $$\begin{aligned} \mu _\gamma \Big \{\omega : \lim _{T\rightarrow \infty } \frac{H_T(\omega )}{T(\phi \star \phi )(0)} \ne \gamma \Big \}=0 \qquad \text{ almost } \text{ surely, } \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>γ</mml:mi> </mml:msub> <mml:mrow> <mml:mo>{</mml:mo> </mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>:</mml:mo> <mml:munder> <mml:mo>lim</mml:mo> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:munder> <mml:mfrac> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>T</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>⋆</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>)</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mfrac> <mml:mo>≠</mml:mo> <mml:mi>γ</mml:mi> <mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace /> <mml:mspace /> <mml:mtext>almost</mml:mtext> <mml:mspace /> <mml:mspace /> <mml:mtext>surely,</mml:mtext> <mml:mspace /> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> meaning that $$\mu _\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>γ</mml:mi> </mml:msub> </mml:math> is supported almost surely only on $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> - thick paths , and consequently, the normalized version is singular w.r.t. the Wiener measure. We then characterize uniquely the limit $$\mu _\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>γ</mml:mi> </mml:msub> </mml:math> w.r.t. the mollification scheme $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> in the sense of Shamov (J Funct Anal 270:3224–3261, 2016) – we show that the law of $$\dot{B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:math> under the random rooted measure $$\mathbb Q_{\mu _\gamma }(\textrm{d}\dot{B}\textrm{d}\omega )= \mu _\gamma (\textrm{d}\omega ,\dot{B})P(\textrm{d}\dot{B})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>γ</mml:mi> </mml:msub> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mtext>d</mml:mtext> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mtext>d</mml:mtext> <mml:mi>ω</mml:mi>
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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 | 0.001 |
| 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".