Spectral Gaps on Discretized Loop Spaces

We study spectral gaps w.r.t. marginals of pinned Wiener measures on spaces of discrete loops (or, more generally, pinned paths) on a compact Riemannian manifold M. The asymptotic behaviour of the spectral gap as the time parameter T of the underlying Brownian bridge goes to 0 is investigated. It turns out that depending on the choice of a Riemannian metric on the base manifold, very different asymptotic behaviours can occur. For example, on discrete loop spaces over sufficiently round ellipsoids the gap grows of order α/T as T ↓ 0. The strictly positive rate α stabilizes as the discretization approaches the continuum limit. On the other extreme, if there exists a closed geodesic γ : S1 → M such that the sectional curvature on γ(S1) is strictly negative, and the loop is pinned close to γ(S1), then the gap decays of order exp (-β/T), and the decay rate β approaches +∞ as the discretization approaches the continuum limit.