Some Questions — Question 17

Definitions and mathematical statement

Let $G$ be a group with a finite symmetric generating set $S$ of cardinality $d$, excluding the identity, and let $\mathcal C$ be its undirected Cayley graph. Its adjacency operator on $\ell^2(G)$ is \[ (Af)(g)=\sum_{s\in S}f(gs). \] This is a bounded self-adjoint operator of norm at most $d$. Its spectral measure at the identity is the probability measure $\mu$ on $[-d,d]$ characterized by \[ \int t^j\,d\mu(t)=\langle A^j\delta_1,\delta_1\rangle \quad(j\geq0). \] These moments count closed walks based at the identity, and determine the measure uniquely on the compact interval.

For a finite $d$-regular simple graph $H$, let $\mu_H=|V(H)|^{-1}\sum_i\delta_{\lambda_i}$, where the adjacency eigenvalues are counted with multiplicity. Must there be finite $d$-regular graphs $H_n$ with $\mu_{H_n}$ converging weakly to $\mu$? Weak convergence means convergence of integrals of every continuous function on $[-d,d]$; equivalently here, all moments converge.

The approximating graphs need not be Cayley graphs, finite quotients of $G$, or connected. They may be chosen independently at each stage. One may arrange their orders to tend to infinity by taking disjoint unions without changing spectral measures. Using normalized random-walk operators divides every eigenvalue by $d$ and gives an equivalent formulation. The question asks only for spectral approximation, not convergence of all rooted neighborhood statistics.

Short English statement

Is every Cayley-graph spectral measure a weak limit of spectral measures of finite regular graphs of the same degree?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1200017 (AMR-011-0017), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Abert - Some questions (pdf) (2010), Question 17. Original source indexed by AMR-011-0017. https://www.renyi.hu/~abert/questions.pdf.