Bernoulli Graphings: Spectral & Algorithmic Analysis

• Bernoulli graphings are measure-preserving Borel graph structures that encode i.i.d. labelings on infinite regular graphs, serving as the infinite analogue to finite Ramanujan graphs.
• They are constructed from quotienting labeled d-regular trees and possess sharp spectral properties, including an optimal spectral bound of 2√(d-1) for zero-mean functions.
• These graphings underpin analyses of correlation decay in factor-of-i.i.d. processes and inform randomized local algorithms on large girth graphs, bridging probabilistic and combinatorial methods.

A Bernoulli graphing is a measure-preserving Borel graph structure constructed to encode the local statistics of i.i.d. labelings on infinite or large regular graphs. Bernoulli graphings provide a canonical infinite-extent analogue of finite Ramanujan graphs and serve as the precise analytic object for studying the correlation decay in factor-of-i.i.d. processes and randomized local algorithms. These graphings offer a bridge between probabilistic processes on infinite structures (such as $d$-regular trees) and measurable combinatorics, linking expansion, spectral theory, and algorithmic limits.

1. Formal Construction of Bernoulli Graphings

Let $T_d$ denote the infinite rooted $d$-regular tree with distinguished root $o$. The construction involves several measure-theoretic and algebraic ingredients:

• Underlying Probability Space: The product space $Y = [0,1]^{V(T_d)}$ carries the Lebesgue product measure $\mu$, encoding all possible i.i.d. labelings of $T_d$.
• Symmetry Quotient: The space $X = Y / \mathrm{Aut}(T_d,o)$ identifies labelings that agree up to root-preserving automorphisms. The resulting measure $\nu$ is the push-forward of $\mu$ to $T_d$0.
• Graph Structure: Two points $T_d$1 are adjacent in the Bernoulli graphing $T_d$2 if one can be obtained from the other by shifting the root of a labeled tree to an adjacent vertex.
• Adjacency and Markov Operators: For $T_d$3, the adjacency operator $T_d$4 is defined as

$T_d$5

The normalized operator $T_d$6 is a self-adjoint contraction.

This construction generalizes beyond $T_d$7, allowing a Bernoulli graphing to be defined for any unimodular random rooted graph $T_d$8 by distributing i.i.d. Uniform$T_d$9 labels and building a Borel graph structure as above (Backhausz et al., 2013, Jardón--Sánchez et al., 15 Oct 2025, Bencs et al., 2021).

2. Spectral Properties and Ramanujan Graphings

The spectral radius of the adjacency operator $d$0 (restricted to the subspace of zero-mean functions, $d$1) is a key invariant. The following fundamental facts hold:

• For any $d$2-regular graphing on an atomless probability space, $d$3.
• For the Bernoulli graphing $d$4, $d$5, and the normalized $d$6 has spectral radius $d$7.

A $d$8-regular graphing $d$9 is termed Ramanujan if $o$0. The Bernoulli graphing $o$1 realizes the minimal possible spectral radius, making it the canonical infinite Ramanujan graphing (Backhausz et al., 2013). In the case of general unimodular random graphs, the spectrum splits into "structured" and "random" parts; the random part is always contained in $o$2 where $o$3 is the local spectral radius of $o$4 (Jardón--Sánchez et al., 15 Oct 2025). This fact generalizes the Alon-Boppana phenomenon for expander graphs.

3. Correlation Decay in Factor-of-I.I.D. Processes

Every $o$5 defines a factor-of-i.i.d. process on $o$6 via

$o$7

For vertices $o$8 at distance $o$9,

$Y = [0,1]^{V(T_d)}$0

where $Y = [0,1]^{V(T_d)}$1 are combinatorial coefficients from the non-backtracking walk decomposition.

The optimal correlation decay theorem states that for any such process with zero mean and finite variance,

$Y = [0,1]^{V(T_d)}$2

and this is sharp up to constants. The closure of all possible correlation sequences arises as convex combinations of polynomials in the spectral parameter $Y = [0,1]^{V(T_d)}$3 integrated against measures on $Y = [0,1]^{V(T_d)}$4:

$Y = [0,1]^{V(T_d)}$5

This elucidates the precise analytic connection between the spectral theory of $Y = [0,1]^{V(T_d)}$6 and the decay of correlations in local processes (Backhausz et al., 2013).

4. Role in Local Algorithms and Large-Girth Graphs

Randomized local algorithms on large-girth, $Y = [0,1]^{V(T_d)}$7-regular finite graphs yield, within each local ball, the same joint distribution as a factor-of-i.i.d. process on $Y = [0,1]^{V(T_d)}$8. Therefore, the correlation decay bounds for $Y = [0,1]^{V(T_d)}$9 transfer directly: for any $\mu$0-local algorithm on a finite $\mu$1-regular graph of girth $\mu$2, the correlation at distance $\mu$3 is bounded by $\mu$4 (Backhausz et al., 2013).

This provides sharp, algorithm-independent decay rates for correlations in outputs of local algorithms as separation grows. The infinite Bernoulli graphing thus acts as an "envelope" object describing all possible limits of local processes on large expander graphs.

5. Extensions to General Unimodular Random Graphs and Skeleton Spectrum

For ergodic unimodular random graphs $\mu$5, the associated Bernoulli graphing $\mu$6 is constructed analogously by distributing i.i.d. labels and considering the quotient up to root automorphisms. The $\mu$7 space on $\mu$8 admits an orthogonal decomposition into:

• Structured subspace $\mu$9: Functions depending only on $T_d$0, whose spectrum is the skeleton operator (Markov chain on unlabeled rooted graphs by root shifts).
• Random subspace $T_d$1: Functions orthogonal to $T_d$2, capturing the random-label effect.

The main spectral result is that the random spectrum $T_d$3 is contained within $T_d$4—the interval determined by the local spectral radius of $T_d$5. In particular, for unimodular quasi-transitive quasi-trees, equality holds: $T_d$6 (Jardón--Sánchez et al., 15 Oct 2025). This establishes a precise measurable analogue to the Alon–Boppana and Ramanujan bounds for finite graphs.

6. Spectral Gap, Expansion, and Applications to Factor-of-I.I.D. Constructions

If $T_d$7 is unimodular, quasi-transitive, and non-amenable, the Bernoulli graphing $T_d$8 has a positive spectral gap: the Markov operator $T_d$9 satisfies

$X = Y / \mathrm{Aut}(T_d,o)$0

where $X = Y / \mathrm{Aut}(T_d,o)$1 is the spectral radius of the simple random walk on $X = Y / \mathrm{Aut}(T_d,o)$2 and $X = Y / \mathrm{Aut}(T_d,o)$3 that on the orbit chain. Positive spectral gap implies measurable expansion (Cheeger constant), enabling Borel matchings and other measurable combinatorial structures (Bencs et al., 2021).

Notably, every non-amenable, quasi-transitive, unimodular graph $X = Y / \mathrm{Aut}(T_d,o)$4 of even degrees admits a factor-of-i.i.d. balanced orientation, and if $X = Y / \mathrm{Aut}(T_d,o)$5 is regular and bipartite, a factor-of-i.i.d. perfect matching also exists (Bencs et al., 2021).

7. Connections, Limitations, and Open Problems

Bernoulli graphings unify and extend several aspects of spectral and probabilistic graph theory, providing an exact infinite-graph analogue to the theory of expander graphs and Ramanujan spectra. The main limitations arise in the presence of non-expanding factors: there exist ergodic unimodular random graphs whose Bernoulli graphings lack spectral gap (e.g., certain decorated trees admitting factors from non-expanding group actions), demonstrating that strong expansion requires the absence of such obstructions (Jardón--Sánchez et al., 15 Oct 2025).

Open questions include whether Bernoulli graphings of more general random graphs (such as configuration-model sequences or Galton–Watson trees) are always relatively Ramanujan, whether more general unimodular random graphs satisfy $X = Y / \mathrm{Aut}(T_d,o)$6, and how new eigenvalue results for random lifts inform the measurable theory beyond quasi-trees (Jardón--Sánchez et al., 15 Oct 2025).

Summary Table: Key Properties of Bernoulli Graphings

Construction Ingredient	Spectral Bound	Application
$X = Y / \mathrm{Aut}(T_d,o)$7 from $X = Y / \mathrm{Aut}(T_d,o)$8	$X = Y / \mathrm{Aut}(T_d,o)$9	Optimal correlation decay
General $\nu$0 unimodular	$\nu$1	Measurable local algorithm limits
Non-amenable $\nu$2	Positive spectral gap	Factor-of-i.i.d. combinatorics

Bernoulli graphings thus serve as a central framework for the analysis of spectral, combinatorial, and algorithmic phenomena on infinite networks and their finite approximations.