Exponentially slow Mixing arising from Entropic Repulsion in p-SOS model

Abstract We investigate the Glauber dynamics of the generalized (2+1)-dimensional p -SOS model( $$1&lt;p&lt;\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> ) under a hard floor constraint. This setting induces entropic repulsion: the integer-valued interface height is forced to remain above the wall and consequently rises to a typical height H ( p , L ) that depends on both the parameter p and the lattice size L . The phenomenon of entropic repulsion has been extensively studied in a variety of random interface models. In the classical SOS model ( $$p=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ), [8, 9] derived an exponential lower bound for the mixing time, demonstrating that the Glauber dynamics mixes only after an exponentially long time in the low-temperature regime (large $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> , the inverse temperature). However, beyond this case, no rigorous lower bounds were previously known: even for the widely studied Discrete Gaussian model ( $$p=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> ), the metastable slowdown predicted by the entropic repulsion picture had remained an open problem. On the equilibrium side, [18] obtained sharp large-deviation principles and precise estimates of typical and maximal heights for all $$1\le p&lt;\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , but the dynamical consequences of these results had not been established. Our main contribution is to close this gap by proving that exponentially slow(stretched-exponential) mixing arising from entropic repulsion persists throughout the regime $$1&lt;p&lt;\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . Specifically, we establish an exponential lower bound, showing that the mixing time satisfies $$\tau _{\textrm{mix}}\ge \exp {\left( cL^{1-o(1)}\right) }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>τ</mml:mi> <mml:mtext>mix</mml:mtext> </mml:msub> <mml:mo>≥</mml:mo> <mml:mo>exp</mml:mo> <mml:mfenced> <mml:mi>c</mml:mi> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mfenced> </mml:mrow> </mml:math> for some $$c&gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> depending on p and $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> . In addition, we provide a refined metastability analysis, proving that the hitting time of an intermediate level aH ( p , L ) is at least $$\exp {\left( cL^{a^{d(p)}-o(1)}\right) }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>exp</mml:mo> <mml:mfenced> <mml:mi>c</mml:mi> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:msup> <mml:mi>a</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mo>-</mml:mo> <mml:mi>o</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:msup>