Albanese maps and off diagonal long time asymptotics for the heat kernel

Motoko Kotani, Toshikazu Sunada

Research output: Contribution to journalArticlepeer-review

32 Citations (Scopus)

Abstract

We discuss long time asymptotic behaviors of the heat kernel on a noncompact Riemannian manifold which admits a discontinuous free action of an abelian isometry group with a compact quotient. A local central limit theorem and the asymptotic power series expansion for the heat kernel as the time parameter goes to infinity are established by employing perturbation arguments on eigenvalues and eigenfunctions of twisted Laplacians. Our ideas and techniques are motivated partly by analogy with Floque-Bloch theory on periodic Schrödinger operators. For the asymptotic expansion, we make careful use of the classical Laplace method. In the course of a discussion, we observe that the notion of Albanese maps associated with the abelian group action is closely related to the asymptotics. A similar idea is available for asymptotics of the transition probability of a random walk on a lattice graph. The results obtained in the present paper refine our previous ones [4]. In the asymptotics, the Euclidean distance associated with the standard realization of the lattice graph, which we call the Albanese distance, plays a crucial role.

Original languageEnglish
Pages (from-to)633-670
Number of pages38
JournalCommunications in Mathematical Physics
Volume209
Issue number3
DOIs
Publication statusPublished - 2000 Jan 1

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Fingerprint Dive into the research topics of 'Albanese maps and off diagonal long time asymptotics for the heat kernel'. Together they form a unique fingerprint.

Cite this