Critical behavior of the massless free field at the depinning transition. (English) Zbl 0992.82011

Bolthausen, Erwin; Velenik, Yvan

Commun. Math. Phys. 223, No. 1, 161-203 (2001).

The paper deals with the following model. To each site \(x\) of the lattice \({\mathbb Z}^d\), \(d\geq 2\), there is attached a classical spin \(\phi_x \in {\mathbb R}\). For boxes \(\Lambda\subset{\mathbb Z}^d \), the probability measures \(\mu_\Lambda\) on \({\mathbb R}^{{\mathbb Z}^d}\) are defined as \[ \mu_{\Lambda} (d\phi) = {1 \over Z_\Lambda} \exp\left[ -(\beta/2)\sum_{x,y}p(x-y) V(\phi_x - \phi_y)\right] \prod_{x\in \Lambda} d\phi_x \prod_{x\in\Lambda^c } \delta_0 (d\phi_x), \quad \beta >0, \] where \(\Lambda^c = {\mathbb Z}^d \setminus \Lambda\) and \(\delta_0\) is the Dirac measure concentrated at zero. The function \(V: {\mathbb R} \rightarrow {\mathbb R}\) is supposed to be even and convex, a typical example is \(V(\xi) = V^\star (\xi) := \xi^2/2\) (a Gaussian interaction). The interaction potential \(p:{\mathbb R} \rightarrow {\mathbb R}_+\) (considered also as a transition probability matrix for a random walk on \({\mathbb Z}^d\)) is supposed to have the following properties: (a) \(\sum_{x\in {\mathbb Z}^d} p(x)=1\); (b) \(p(x) = p(-x)\); (c) for any \(x \in {\mathbb R}^d\), there exists a path \(0 := x_0 , x_1 , \dots , x_n := x\) such that \( p(x_k - x_{k-1}) > 0 \) for all \(k=1,2, \dots , n\). This model possesses a continuous symmetry - it is invariant with respect to the shifts \(\phi \rightarrow \phi+ c\), \(c\in {\mathbb R}\). To break this symmetry, the authors employ two types of pinning. The first one is the perturbation of the above \(\mu_\Lambda\) by the factor \[ \exp\left[b\sum_{x\in \Lambda} {\mathbf I}(|\phi_x |\leq a) \right], \quad a, b > 0, \] where \({\mathbf I}\) is the indicator. The second way is the so called \(\delta\)-pinning, where the measure \(\mu_\Lambda\) is modified as follows \[ \mu^\varepsilon_\Lambda (d\phi) ={1\over Z^\varepsilon_\Lambda}\exp\left[ -(\beta/2)\sum_{x,y}p(x-y) V(\phi_x - \phi_y)\right] \prod_{x\in \Lambda}\left(d\phi_x +\varepsilon \delta_0 (d\phi_x)\right) \prod_{x\in\Lambda^c } \delta_0 (d\phi_x). \] For the latter measure and for the Gaussian interaction \(V = V^\star\), the authors prove that its thermodynamic limit (locally weak) exists in all dimensions. The main results of the paper (Theorems 2.2, 2.3) describe the depinning limit of the variance \(\mu^{\star, \epsilon}(\phi_0^2)\) and of the correlation length \[ m_\varepsilon (x) := - \lim_{k\rightarrow \infty}{1 \over k}\log \mu^{\star, \varepsilon}(\phi_0 \phi_{[kx]}), \] where \([kx]\) is the integer part of \(kx\).

Reviewer: Yuri Kozitsky (Lublin)

MSC:

82B27	Critical phenomena in equilibrium statistical mechanics
81T25	Quantum field theory on lattices
82B24	Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics

Keywords:

asymptotics of the variance; interface; wetting transition