Spectral Theory for Borel PMP Graphs

We initiate a systematic study of spectral theory for bounded-degree Borel pmp graphs. Specifically, we study spectral properties of the associated adjacency and Laplacian operators. We start with proving a spectral characterization of approximate measurable bipartiteness. Next, we adapt classical theorems of Wilf and Hoffman to give novel upper and lower bounds on the approximate measurable chromatic number. Using similar techniques, we then show that the approximate measurable chromatic number of a pmp graph generated by $n$ bounded-to-one functions is at most $2n + 1$. Next, concerning matchings, we introduce a measurable version of Tutte’s condition and show that a spectral assumption analogous to the one from a classical theorem of Brouwer and Haemers implies this measurable Tutte condition. Finally, we show that the spectrum is continuous under local-global convergence.

💡 Research Summary

This paper initiates a systematic study of spectral theory for bounded‑degree Borel probability‑measure‑preserving (pmp) graphs, extending many classical finite‑graph results to the measurable infinite setting. The authors define the adjacency operator (T_G) and the Laplacian operator (L_G = D_G - T_G) on (L^2(X)), where ((X,\mu)) is a standard probability space and (D_G) is the degree function. They first prove that both operators are bounded and self‑adjoint, using Borel transport maps to give a self‑contained proof of these facts.

The first major result (Theorem A) gives a spectral characterization of approximate measurable bipartiteness. If a Borel pmp graph is approximately bipartite, then the spectrum of (T_G) is symmetric about zero. Conversely, for a regular ergodic graph with a spectral gap, the presence of (-d) (where (d) is the degree) in the spectrum forces approximate bipartiteness. The proof adapts the finite‑graph argument but must handle approximate eigenfunctions and the error set introduced by the “almost” bipartition, requiring delicate control via Borel transports.

The second line of results concerns bounds on the approximate measurable chromatic number (\chi_{\mathrm{appr}}^\mu(G)). Theorem B (Wilf‑type bound) shows (\chi_{\mathrm{appr}}^\mu(G) \le \lfloor M(T_G)\rfloor + 1), where (M(T_G)) is the spectral radius of the adjacency operator. This bound can be substantially tighter than the classical degree‑plus‑one bound and improves on Brooks’ theorem in many non‑regular cases. The proof constructs a decreasing sequence of vertex sets based on average degree (\le M(T_G)) and then applies a list‑coloring algorithm that colors a large proportion of the graph while controlling the “bad” set.

Theorem C addresses graphs generated by a finite collection of bounded‑to‑one Borel functions. For a pmp graph generated by (n) such functions, the authors prove (\chi_{\mathrm{appr}}^\mu(G) \le 2n+1). This matches the optimal bound known for commuting functions and improves earlier quadratic bounds. The argument blends the backward list‑coloring technique from Theorem B with Borel transport constructions, showing that the combinatorial structure imposed by the generating functions limits the chromatic complexity.

Theorem D (Hoffman‑type bound) provides a lower‑spectral bound: (\chi_{\mathrm{appr}}^\mu(G) \le \lceil 1 - M(T_G)/m(T_G) \rceil), where (M(T_G)) and (m(T_G)) are the maximum and minimum spectral values of the adjacency operator. For regular graphs the proof follows the classical route via the Laplacian, but for non‑regular graphs the authors develop a block‑decomposition of (T_G) relative to an approximate coloring and use a novel inequality (Lemma 6.6) relating the spectrum of the whole operator to the spectra of its diagonal blocks.

In the matching section, the authors introduce a measurable analogue of Tutte’s condition, called the strict measurable Tutte condition. Theorem E shows that for an ergodic regular pmp graph, if the Laplacian satisfies (2m_L \ge M_L) (where (m_L) and (M_L) are the infimum and supremum of the Rayleigh quotient on the orthogonal complement of constants), then the strict measurable Tutte condition holds, guaranteeing a measurable perfect matching. This result mirrors the classical Brouwer–Haemers spectral condition for perfect matchings but requires substantial new ideas to handle measurability and the lack of a direct measurable Tutte theorem.

Finally, Section 8 discusses local‑global convergence of bounded‑degree graphs. By combining recent results on graph limits, the authors prove that the spectrum of a graph that arises as a local‑global limit of finite graphs varies continuously under this convergence. Thus, spectral data are preserved in the limit, linking the theory of graphings, local‑global limits, and operator spectra.

Overall, the paper provides a comprehensive bridge between spectral graph theory and descriptive combinatorics, delivering new measurable analogues of classic theorems, optimal chromatic bounds for function‑generated graphs, a measurable Tutte condition, and continuity of spectra under graph limits. These contributions significantly advance the toolkit available for tackling combinatorial problems on infinite, definable structures.