Lifschitz tail and Wiener sausage. I
Consider a random distribution of obstacles in \({\mathbb{R}}^ d\), \(d\geq 1\), given by the collection of translates \(y+C\), where y runs through the support of a Poisson point measure with intensity \(\nu\) dx \((\nu >0)\) and where C is a fixed nonpolar compact set in \({\mathbb{R}}^ d\). Let \(B_ N\) be the ball of radius N centered at zero, and let \(\ell_ N\) be the empirical measure of the increasing sequence of eigenvalues of the Laplace operator -\({1\over 2} \Delta\) with Dirichlet boundary conditions on \(\partial B_ N\) and on the obstacles intersecting \(B_ N\). Then \(\ell_ N\) (normalized by the volume of \(B_ N)\) converges, as \(N\uparrow \infty\), P-a.s. vaguely to a deterministic measure \(\ell\) with Laplace transform L(t) given by \(L(t)=(2\pi t)^{-d/2}E^ t_{0,0}[\exp (-\nu | W^ C_ t|)].\) Here \(E^ t_{0,0}\) denotes the standard Brownian bridge expectation and \(| W^ C_ t|\) denotes the volume of the Wiener sausage of the bridge up to time t modeled on -C. The author shows that \(t^{-d/(d+2)} \log L(t)\) converges as \(t\uparrow \infty\) to \(c(d,\nu):=-\inf \{\nu | U| +\lambda (U)\},\) where the infimum is taken over the class of open bounded subsets U of \({\mathbb{R}}^ d\) with negligible boundary and \(\lambda\) (U) is the principal Dirichlet eigenvalue of -\({1\over 2} \Delta\) in U. This generalizes a result of \textit{M. D. Donsker} and \textit{S. R. S. Varadhan} [Commun. Pure Appl. Math. 28, 279-301 (1975; Zbl 0348.60031) and ibid., 525-565 (1975; Zbl 0333.60077)] where the same result has been obtained in the case where C is a ball. The methods the author develops in this article for the proof of this extension are new and rely on a discretization argument and on a skillful distinction between ``good'' and ``bad'' points of the Poisson measure. By means of a Tauberian theorem the author obtains as a corollary the behaviour of \(\ell ([0,\lambda])\) as \(\lambda \downarrow 0\), the Lifschitz tail behaviour of the density of states. The results concerning similar questions in the hyperbolic case have been proved by different methods by the author [ibid. 42, No.8, 1033-1065 (1989; Zbl 0696.60079) and ibid. 43, No.1, 1-30 (1990; Zbl 0735.60084)]. [For part II see the paper reviewed below.]
scientific article; zbMATH DE number 4211219
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