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We investigate the short-time expansion of the heat kernel of a Laplace typeoperator on a compact Riemannian manifold and show that the lowest order termof this expansion is given by the Fredholm determinant of the Hessian of theenergy functional on a space of finite energy paths. This is the asymptoticbehavior to be expected from formally expressing the heat kernel as a pathintegral and then (again formally) using Laplace's method on the integral. Wealso relate this to the zeta determinant of the Jacobi operator, which isanother way to assign a determinant to the Hessian of the energy functional. Weconsider both the near-diagonal asymptotics as well as the behavior at the cutlocus.

Heat Kernel Asymptotics, Path Integrals and Infinite-Dimensional Determinants