2015/08/24 by Daniel Bump, Bump, Daniel, Persi Diaconis +7
Mathematics · #60B15 #60J10 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1508.05986
openalex publication_date 2015/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In analyzing a simple random walk on the Heisenberg group we encounter the problem of bounding the extreme eigenvalues of an n× n matrix of the form M=C+D where C is a circulant and D a diagonal matrix. The discrete Schrödinger operators are an interesting special case. The Weyl and Horn bounds are not useful here. This paper develops three different approaches to getting good bounds. The first uses the geometry of the eigenspaces of C and D, applying a discrete version of the uncertainty principle. The second shows that, in a useful limit, the matrix M tends to the harmonic oscillator on L2(ℝ) and the known eigenstructure can be transferred back. The third approach is purely probabilistic, extending M to an absorbing Markov chain and using hitting time arguments to bound the Dirichlet eigenvalues. The approaches allow generalization to other walks on other groups.