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

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 $$kx$$.

### 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
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