Some estimates of the logarithmic Sobolev constants on manifolds with boundary and an application to the Ising models (Q1612782)

Let \(M\) be a compact \(d\)-dimensional Riemannian manifold with smooth boundary \(\partial M\), \(U\) a potential function on \(M\), and let \(dm^U\) be \(e^{-U}dm\). Then the logarithmic Sobolev inequality for \(a^U\) is \[ \int_Mf^2 \log \left({f^2\over \|f\|^2_{L^2 (m^U)}}\right) dm^U\leq{2\over \alpha} a^U(f,f), \] \[ a^U(f,g)= \int_M(\nabla f\mid\nabla g) dm^U. \] The logarithmic Sobolev constant \(\alpha(U)\) is the best constant in this inequality. In this paper, estimates of logarithmic Sobolev constants of \textit{J. D. Deuschel} and \textit{D. W. Stroock} [J. Funct. Anal. 92, No. 1, 30-48 (1990; Zbl 0705.60066)] for manifolds without boundary is generalized to manifolds with boundary by adding a certain term which includes the second fundamental form. Then considering the Gibbs measures on \(M^{\mathbb{Z}^\nu}\) determined by a finite range and shift-invariant potential, these results are applied for finite dimensional manifolds to the Ising model. The outline of the paper is as follows: In section 2, some estimates from below of the second fundamental form \(A(\theta,\eta)=\frac 12\nabla_N (\theta|\eta)\), \(( \theta\mid N)= (\eta\mid N)=0\), where \(n\) is the inner normal vector on \(\partial M\) by which the Neumann boundary condition is given, are prepared (Lemma 2.4). By using these estimates, estimates of spectral gap constants are given (Prop. 2.6). Here, the spectral gap constant \(C(U)\) is the best constant to hold the spectral gap inequality \[ \bigl\|f-\langle f \rangle_{ m^U} \bigr\|_{L^2 (m^U)}\leq{1\over C}a^U (f,f),\quad \nabla_Nf=0. \] In section 3, it is shown if \(f\) satisfies the Neumann boundary condition and is positive, then \[ \biggl\langle f\bigl\|\text{Hess}(\log f)\bigr \|^2 \biggr \rangle_m\geq {4\over d+2} \;2\bigl \|\text{Hess} f^{1/2} \bigr\|^2+ (\Delta f^{1/2})^2 \biggr\rangle_m \] (Lemma 3.2). By this inequality and Lemma 2.4, several versions of the logarithmic Sobolev inequalities are derived. Then by using the estimates of spectral gap constants in section 2, estimates of logarithmic Sobolev constants are obtained (Prop. 3.3). As an example, taking \(M=\{(x,y,z) \in S^2;z \geq\cos \varphi_0\}\), it is shown if there exists \(1>c>0\) such that \[ \left(1+ {1\over c}\right) |\cos \varphi_0|\int^{\varphi_0 }_{\pi/2} {1\over\sin \varphi} d\varphi\leq 1, \] then \(C(0)\) and \(\alpha (0)\) are estimated as follows: \[ \begin{aligned} C(0) &\geq 2\left(1-{1\over 2 \bigl (1-|\cos \varphi_0|\int^{\varphi_0}_{\pi/2} (1/\sin\varphi) d\varphi \bigr)} \right),\\ \alpha(0) & \geq {C(0)\over 2}+1-{1\over 2\bigl(1-2 \mid\cos\varphi_0 |\int^{\varphi_0}_{\pi/2}(1/ \sin\varphi) d\varphi \bigr)} \end{aligned} \] (section 4). In section 5, the last section, the stochastic Ising model whose spin space is a manifold with boundary is constructed and a sufficient condition for the logarithmic Sobolev inequality is given (Prop. 5.10), together with an example showing that this result is useful (Example 5.1).