We investigate the Glauber dynamics of the generalized (2+1)-dimensional p-SOS model(<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$1<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>) under a hard floor constraint. This setting induces entropic repulsion: the integer-valued interface height is forced to remain above the wall and consequently rises to a typical height H(p, L) that depends on both the parameter p and the lattice size L. The phenomenon of entropic repulsion has been extensively studied in a variety of random interface models. In the classical SOS model (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>), [<CitationRef CitationID="CR8">8</CitationRef>, <CitationRef CitationID="CR9">9</CitationRef>] derived an exponential lower bound for the mixing time, demonstrating that the Glauber dynamics mixes only after an exponentially long time in the low-temperature regime (large <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>, the inverse temperature). However, beyond this case, no rigorous lower bounds were previously known: even for the widely studied Discrete Gaussian model (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>), the metastable slowdown predicted by the entropic repulsion picture had remained an open problem. On the equilibrium side, [<CitationRef CitationID="CR18">18</CitationRef>] obtained sharp large-deviation principles and precise estimates of typical and maximal heights for all <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$1\le p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$1<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Specifically, we establish an exponential lower bound, showing that the mixing time satisfies <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\tau _{\textrm{mix}}\ge \exp {\left( cL^{1-o(1)}\right) }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>τ</mi> <mtext>mix</mtext> </msub> <mo>≥</mo> <mo>exp</mo> <mfenced close=")" open="("> <mi>c</mi> <msup> <mi>L</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$c>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> depending on p and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>. In addition, we provide a refined metastability analysis, proving that the hitting time of an intermediate level aH(p, L) is at least <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\exp {\left( cL^{a^{d(p)}-o(1)}\right) }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>exp</mo> <mfenced close=")" open="("> <mi>c</mi> <msup> <mi>L</mi> <mrow> <msup> <mi>a</mi> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>-</mo> <mi>o</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$0<a<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>a</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and d(p) is a positive function depending on p. The proof relies on an extension of the classical Peierls-type contour estimates, originally developed for <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$p=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, to the nonlinear p-SOS setting. Taken together, these results demonstrate that entropic repulsion induces uniformly slow mixing across the entire p-SOS family, thereby extending a phenomenon that had previously been established only for <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$p=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.

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

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2026年7月17日

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Seokun Choi

摘要

We investigate the Glauber dynamics of the generalized (2+1)-dimensional p-SOS model( $1<p<\infty $ $1 < p < \infty$ ) 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 ( $p=1$ $p = 1$ ), [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 $ $β$ , the inverse temperature). However, beyond this case, no rigorous lower bounds were previously known: even for the widely studied Discrete Gaussian model ( $p=2$ $p = 2$ ), 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<\infty $ $1 \leq p < \infty$ , 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<p<\infty $ $1 < p < \infty$ . Specifically, we establish an exponential lower bound, showing that the mixing time satisfies $\tau _{\textrm{mix}}\ge \exp {\left( cL^{1-o(1)}\right) }$ $τ_{mix} \geq exp (c, L^{1 - o (1)})$ for some $c>0$ $c > 0$ depending on p and $\beta $ $β$ . In addition, we provide a refined metastability analysis, proving that the hitting time of an intermediate level aH(p, L) is at least $\exp {\left( cL^{a^{d(p)}-o(1)}\right) }$ $exp (c, L^{a^{d (p)} - o (1)})$ , where $0<a<1$ $0 < a < 1$ and d(p) is a positive function depending on p. The proof relies on an extension of the classical Peierls-type contour estimates, originally developed for $p=1$ $p = 1$ , to the nonlinear p-SOS setting. Taken together, these results demonstrate that entropic repulsion induces uniformly slow mixing across the entire p-SOS family, thereby extending a phenomenon that had previously been established only for $p=1$ $p = 1$ .