<p>We investigate the existence of generalised densities for the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Phi ^4_d$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Φ</mi> <mi>d</mi> <mn>4</mn> </msubsup> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(d=1,2,3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> measures, in finite volume, through the lens of Onsager-Machlup (OM) functionals. The latter are rigorously defined for measures on metric spaces as limiting ratios of small ball probabilities. In one dimension, we show that the standard OM functional of the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Phi ^4_1$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Φ</mi> <mn>1</mn> <mn>4</mn> </msubsup> </math></EquationSource> </InlineEquation> measure coincides with the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Phi ^4$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Φ</mi> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation> action as expected. In two dimensions, we show that OM functionals of the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$P(\Phi )_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> measures agree with the corresponding actions, by considering “enhanced" distances, defined with respect to Wick powers of the Gaussian Free Field, which are analogous to rough path metrics. In dimension 3, two natural generalisations of the OM functional are proved to be degenerate. Finally, we recover the <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Phi ^4_3$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Φ</mi> <mn>3</mn> <mn>4</mn> </msubsup> </math></EquationSource> </InlineEquation> action, under appropriate regularity conditions, by considering joint small radius-large frequency limits.</p>

On the Onsager-Machlup Functional of the $\Phi ^4$-Measure

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2026年3月31日

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Ioannis Gasteratos,
Zachary Selk

摘要

We investigate the existence of generalised densities for the $\Phi ^4_d$ $Φ_{d}^{4}$ $(d=1,2,3)$ $(d = 1, 2, 3)$ measures, in finite volume, through the lens of Onsager-Machlup (OM) functionals. The latter are rigorously defined for measures on metric spaces as limiting ratios of small ball probabilities. In one dimension, we show that the standard OM functional of the $\Phi ^4_1$ $Φ_{1}^{4}$ measure coincides with the $\Phi ^4$ $Φ^{4}$ action as expected. In two dimensions, we show that OM functionals of the $P(\Phi )_2$ $P {(Φ)}_{2}$ measures agree with the corresponding actions, by considering “enhanced" distances, defined with respect to Wick powers of the Gaussian Free Field, which are analogous to rough path metrics. In dimension 3, two natural generalisations of the OM functional are proved to be degenerate. Finally, we recover the $\Phi ^4_3$ $Φ_{3}^{4}$ action, under appropriate regularity conditions, by considering joint small radius-large frequency limits.

On the Onsager-Machlup Functional of the \(\Phi ^4\)-Measure

摘要