Ursula Porod, Steve Zelditch
Semi-classical Limit for Random Walks
Preprint series: ESI preprints

MSC:

58F06 Geometric quantization (applications of representation theory), See also {22E45, 81S10}

60B15 Probability measures on groups, Fourier transforms, factorization

60J15 Random walks

22E30 Analysis on real and complex Lie groups, See also {33C80,

43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.

Abstract: Let $(G, \mu)$ be a discrete symmetric random walk on a compact Lie group
$G$ with step distribution $\mu$ and let $T_{\mu}$ be the associated transitio
n
operator on
$L^2(G)$. The irreducibles $V_{\rho}$ of the left regular representation of $G$
on
$L^2(G)$ are finite dimensional invariant subspaces for $T_{\mu}$ and the spectr
um
of $T_{\mu}$ is the union of the sub-spectra
$\sigma(T_{\mu}\1_{V_{\rho}})$ on the irreducibles, which consist
of real eigenvalues $\{ \lambda_{\rho 1},...,\lambda_{\rho dim
V_{\rho}}\}$. Our main result is an asymptotic expansion for the spectral measu
res
$$ m_{\rho}^{\mu}(\lambda) := \frac{1}{dim V_{\rho}} \sum_{j=1}^{dim V_{\rho}}
\delta(\lambda - \lambda_{\rho j})$$
along rays of representations in a positive Weyl chamber ${\bf t}^*_+$, i.e. for

sequences of representations $k \rho$, $k\in \N$ with $k\rightarrow \infty$. As
a corollary we obtain some estimates on the spectral radius of the random walk.
We also analyse the fine structure of the spectrum for certain random walks on
$U(n)$ (for which $T_{\mu}$ is essentially a direct sum of Harper operators).
Keywords: random walks on compact Lie groups, geometric quantization, Szegoe limit theorems