<p>We consider the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\phi ^4_1$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>ϕ</mi> <mn>1</mn> <mn>4</mn> </msubsup> </math></EquationSource> </InlineEquation> measure in an interval of length <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>, defined by a symmetric double-well potential <i>W</i> and inverse temperature <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>. Our results concern its asymptotic behavior in the joint limit <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\beta , \ell \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>,</mo> <mi>ℓ</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, both in the subcritical regime <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\ell \ll \textrm{e}^{\beta C_W}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>≪</mo> <msup> <mtext>e</mtext> <mrow> <mi>β</mi> <msub> <mi>C</mi> <mi>W</mi> </msub> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and in the supercritical regime <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\ell \gg \textrm{e}^{\beta C_W}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>≫</mo> <msup> <mtext>e</mtext> <mrow> <mi>β</mi> <msub> <mi>C</mi> <mi>W</mi> </msub> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$C_W$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>W</mi> </msub> </math></EquationSource> </InlineEquation> denotes the surface tension. In the former case, in which the measure concentrates on the pure phases, we prove the corresponding large deviation principle. The associated rate function is the Modica–Mortola functional modified to take into account the entropy of the locations of the interfaces. Furthermore, we provide the sharp asymptotics of the probability of having a given number of transitions between the two pure phases. In the supercritical regime, the measure no longer concentrates and we show that the interfaces are asymptotically distributed according to a Poisson point process.</p>

Asymptotics of the $\phi ^4_1$ Measure in the Sharp Interface Limit

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2025年9月19日

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Lorenzo Bertini,
Paolo Buttà,
Giacomo Di Gesù

摘要

We consider the $\phi ^4_1$ $ϕ_{1}^{4}$ measure in an interval of length $\ell $ $ℓ$ , defined by a symmetric double-well potential W and inverse temperature $\beta $ $β$ . Our results concern its asymptotic behavior in the joint limit $\beta , \ell \rightarrow \infty $ $β, ℓ \to \infty$ , both in the subcritical regime $\ell \ll \textrm{e}^{\beta C_W}$ $ℓ ≪ e^{β C_{W}}$ and in the supercritical regime $\ell \gg \textrm{e}^{\beta C_W}$ $ℓ ≫ e^{β C_{W}}$ , where $C_W$ $C_{W}$ denotes the surface tension. In the former case, in which the measure concentrates on the pure phases, we prove the corresponding large deviation principle. The associated rate function is the Modica–Mortola functional modified to take into account the entropy of the locations of the interfaces. Furthermore, we provide the sharp asymptotics of the probability of having a given number of transitions between the two pure phases. In the supercritical regime, the measure no longer concentrates and we show that the interfaces are asymptotically distributed according to a Poisson point process.

Asymptotics of the \(\phi ^4_1\) Measure in the Sharp Interface Limit

摘要