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

Abstract We consider the $$\phi ^4_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ϕ</mml:mi> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> </mml:math> measure in an interval of length $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> , defined by a symmetric double-well potential W and inverse temperature $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> . Our results concern its asymptotic behavior in the joint limit $$\beta , \ell \rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , both in the subcritical regime $$\ell \ll \textrm{e}^{\beta C_W}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>≪</mml:mo> <mml:msup> <mml:mtext>e</mml:mtext> <mml:mrow> <mml:mi>β</mml:mi> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>W</mml:mi> </mml:msub> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> and in the supercritical regime $$\ell \gg \textrm{e}^{\beta C_W}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>≫</mml:mo> <mml:msup> <mml:mtext>e</mml:mtext> <mml:mrow> <mml:mi>β</mml:mi> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>W</mml:mi> </mml:msub> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , where $$C_W$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>W</mml:mi> </mml:msub> </mml:math> 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.