Liouville property of infinite Ramanujan graphs

Definitions and mathematical statement

Fix an integer $d\ge3$ and an infinite connected simple $d$-regular graph $G$. Its simple-random-walk operator acts on functions by \[ (Pf)(v)=\frac1d\sum_{w\sim v}f(w). \] On $\ell^2(V(G))$ with counting measure this is a bounded self-adjoint operator. Call $G$ Ramanujan when \[ \rho(G):=\|P\|_{\ell^2\to\ell^2}=\frac{2\sqrt{d-1}}{d}, \] the spectral radius of the infinite $d$-regular tree. Equivalently, $\rho(G)=\limsup_{n\to\infty}P^n(v,v)^{1/n}$; the limsup handles possible periodicity.

A real function $f$ on $V(G)$ is harmonic if $Pf=f$. The graph is Liouville if every bounded harmonic real function is constant. Prove that no such infinite Ramanujan graph is Liouville: there must exist a nonconstant $f$ with $\sup_v|f(v)|<\infty$ and $Pf=f$.

No transitivity or group structure is assumed. The hypothesis is the sharp tree spectral-radius value, not merely positive expansion or a spectral radius strictly below one. The conclusion concerns bounded harmonic functions and therefore differs from existence of an arbitrary unbounded harmonic function.

The degree restriction removes a necessary degenerate exception to the source's shorthand: the two-way infinite path is $2$-regular, has spectral radius one equal to its tree value, and is Liouville. The intended nonamenable Ramanujan question is precisely the range $d\ge3$, for which the displayed tree spectral radius is strictly less than one.

Short English statement

Must every infinite Ramanujan graph of degree at least three admit a nonconstant bounded harmonic function?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 10000047 (AMR-099-0047), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Benjamini - Coarse Geometry and Randomness (Saint-Flour notes), Conjecture 13.1, PDF page 107. Original source indexed by AMR-099-0047. https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=107.
• S3 I. Benjamini, Coarse Geometry and Randomness, primary Saint-Flour notes; the numbered question and its surrounding definitions. https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf.