Expected Volume of Intersection of Wiener Sausages and Heat Kernel Norms on Compact Riemannian Manifolds with Boundary
نویسندگان
چکیده
It is well known that Zk,m(t) is related to the virial coefficients of a quantum system of obstacles K at inverse temperature t. For example, G. E. Uhlenbeck [15] calculated the asymptotic behaviour of Zk,m(t) as t → ∞ in the special case where k = 1, m = 3, and K is a ball. I. McGillivray [14] obtained the full asymptotic series as t → ∞ for k = 1, m ≥ 3, and K an arbitrary, non-polar compact set. The special case where k = 1, m = 2, and K a ball was studied for t → ∞ by M. van den Berg and E. Bolthausen [4]. In this paper, we analyse the asymptotic behaviour of Zk,m(t) as t → 0. Let pRm(·, ·; ·) denote the heat kernel for R given by pRm(x, y; t) = (4πt) e /(4t) , and let pRm−K denote the Dirichlet heat kernel for the open set R m −K. By the Feynman-Kac formula, we have that for x ∈ R −K,
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