Rayleigh Property in Negative Dependence

• Rayleigh property is a negative dependence concept defined by the real stability of multiaffine generating polynomials, leading to inherent negative correlation and association.
• It originates from electrical network theory and matroid analysis, where conductance monotonicity and basis-generating functions illustrate its combinatorial significance.
• Extensions like the strongly Rayleigh property preserve closure under marginalization and conditioning, impacting entropy bounds, paving theorems, and central limit results.

The Rayleigh property encompasses a fundamental class of negative dependence phenomena across combinatorics, probability, statistical physics, and algebraic geometry. It originates in the study of electrical networks, where monotonicity of effective conductance translates into negative correlation among network edges. In modern theory, the property is defined via real stability of multiaffine generating polynomials for discrete measures, with broad implications including closure structures, negative association, paving theorems, entropy bounds, and links to important classes of matroids and point processes. Several extensions and analogous properties, notably the strongly Rayleigh property, further reinforce its central role in the analytic and combinatorial structure of measures on finite sets.

1. Definitional Principles and Polynomial Formulation

Given a finite set $[n] = \{1, \ldots, n\}$ and a probability measure $\mu$ on $2^{[n]}$, the ordinary generating polynomial is

$$P_\mu(z_1, \ldots, z_n) = \sum_{S \subseteq [n]} \mu(S) \prod_{i \in S} z_i.$$

The measure $\mu$ is said to possess the Rayleigh property if $P_\mu$ is real stable, i.e., $P_\mu(z) \neq 0$ whenever $\operatorname{Im} z_i > 0$ for all $i$ (Alishahi et al., 2020, Cossette et al., 24 Apr 2025, Gao et al., 2014). Practically, this real stability implies classic negative correlation inequalities:

$$\operatorname{Cov}_\mu(1_{i \in S}, 1_{j \in S}) = \mu(\{i, j\} \subseteq S) - \mu(i \in S)\mu(j \in S) \leq 0, \quad i \ne j.$$

More generally, Rayleigh measures (or those with the Rayleigh property) satisfy negative association: for increasing functions $\mu$0 and $\mu$1 of disjoint coordinates,

$\mu$2

A strengthening, the strongly Rayleigh property, requires real stability to persist under marginalization, conditioning, and product constructions, yielding full closure under these operations (Gao et al., 2014, Cossette et al., 24 Apr 2025).

2. Origins, Electrical Networks, and Matroidal Generalization

The classical context arises in resistive electrical networks: for a graph $\mu$3, assigning positive conductances $\mu$4 to edges, Rayleigh's law asserts that increasing any $\mu$5 cannot decrease effective conductance between two fixed nodes. Using the matrix-tree theorem, this monotonicity is shown equivalent to negative correlation of edge-inclusion in random spanning trees sampled proportionally to $\mu$6 (Marcott, 2016). The property is robust to network reductions (series/parallel/minor operations).

Choe and Wagner extended this to matroids: a matroid $\mu$7 on $\mu$8 is Rayleigh if its basis-generating polynomial

$\mu$9

satisfies for all $2^{[n]}$0 and $2^{[n]}$1:

$2^{[n]}$2

Thus, the Rayleigh property on matroids interpolates directly between combinatorics and real stability theory (Marcott, 2016, Gao et al., 2014).

3. Strongly Rayleigh Measures and Structural Closure

A random vector $2^{[n]}$3 is strongly Rayleigh if its probability generating function $2^{[n]}$4 is real stable. Strongly Rayleigh measures are closed under:

• permutations of coordinates,
• scaling of variables,
• marginalization and conditioning,
• diagonalization (aggregation of coordinates).

This is the maximal negative dependence property preserved under all natural restrictions and projections (Cossette et al., 24 Apr 2025, Alishahi et al., 2020). Borcea–Brändén–Liggett characterized strongly Rayleigh measures as the closure of determinantal point processes and their conditionings, projections, and external field inserts, showing that negative association under all conditionings is equivalent to real stability.

4. Kernel Polynomial and Determinantal Processes

For strongly Rayleigh point processes, the kernel polynomial unifies DPP theory with broader negatively dependent frameworks:

$2^{[n]}$5

serves as a generalization of the determinantal kernel:

• For DPPs with kernel $2^{[n]}$6 and $2^{[n]}$7,

$2^{[n]}$8

• The kernel polynomial is always real stable and inherits key invariance and restriction properties. Taking partial derivatives or specializations corresponds to conditioning and marginalization in the underlying process (Alishahi et al., 2020).

5. Consequences and Theorems: Negative Dependence, Entropy, Paving, and CLT

The real stability of generating polynomials precipitates several deep results:

• Negative Association and Correlation: For any set partition, the correlation of indicators is nonpositive; extends to all increasing events on disjoint coordinate sets (Alishahi et al., 2020, Gao et al., 2014).
• Entropy Bound: For kernel-polynomial roots $2^{[n]}$9, the Shannon entropy satisfies

$$P_\mu(z_1, \ldots, z_n) = \sum_{S \subseteq [n]} \mu(S) \prod_{i \in S} z_i.$$0

Equality holds for DPPs, and the bound arises from hyperbolic majorization arguments (Alishahi et al., 2020).

• Paving Theorems: Generalizing operator paving to real stable polynomials, any strongly Rayleigh process (under mild leading coefficient constraints) may have its ground set partitioned into a bounded number of subsets such that restrictions exhibit only weak correlations (Alishahi et al., 2020).
• CLT: For sequences of strongly Rayleigh measures satisfying mild degree and variance growth conditions, the normalized sums converge to a multivariate normal. The proof employs root location controls inherited from real stable polynomials, combined with the Cramér–Wold device and Berry–Esseen-type arguments (Ghosh et al., 2016).

6. Rayleigh Property in Matroid Theory: Criteria and Sums-of-Squares

A matroid $$P_\mu(z_1, \ldots, z_n) = \sum_{S \subseteq [n]} \mu(S) \prod_{i \in S} z_i.$$1 is strongly Rayleigh if $$P_\mu(z_1, \ldots, z_n) = \sum_{S \subseteq [n]} \mu(S) \prod_{i \in S} z_i.$$2 is real stable. The Grace–Walsh–Szegő theorem reduces stability testing to a block-reduced polynomial. In the "two-orbit" case, stability of $$P_\mu(z_1, \ldots, z_n) = \sum_{S \subseteq [n]} \mu(S) \prod_{i \in S} z_i.$$3 is equivalent to real-rootedness (with nonpositive roots) of a specific univariate polynomial $$P_\mu(z_1, \ldots, z_n) = \sum_{S \subseteq [n]} \mu(S) \prod_{i \in S} z_i.$$4. For rank-2 orbit matroids, the Rayleigh difference can be expressed as a sum of squares precisely when an explicit Gram matrix is positive semidefinite. Structural inequalities (e.g., discriminant nonnegativity, Gram criteria) yield explicit characterization for symmetric matroids (Gao et al., 2014).

Criterion	Matroid Family	Practical Test
Two-orbit Grace-Walsh-Szegő	Highly symmetric	$$P_\mu(z_1, \ldots, z_n) = \sum_{S \subseteq [n]} \mu(S) \prod_{i \in S} z_i.$$5 real-rooted, nonpositive roots
Quadratic case	Low rank/size blocks	Discriminant $$P_\mu(z_1, \ldots, z_n) = \sum_{S \subseteq [n]} \mu(S) \prod_{i \in S} z_i.$$6 for $$P_\mu(z_1, \ldots, z_n) = \sum_{S \subseteq [n]} \mu(S) \prod_{i \in S} z_i.$$7
Rank-2 orbit, sum-of-squares	Line + general points	Gram matrix positive semidefinite

7. Geometric, Physical, and Interpolation Perspectives

In probability simplex geometry, the set of strongly Rayleigh Bernoulli vector laws forms a convex polytope, minimal in supermodular order, whose entropy-maximizing element is a conditional Bernoulli law supported on two adjacent hyperfaces. Interpolation between this extremal law and independence forms a chain in the supermodular order, clarifying the geometry of extremal negative dependence (Cossette et al., 24 Apr 2025).

For Rayleigh dissipation in geometric mechanics, the Rayleigh property defines a class of velocity-dependent dissipative forces, with the dissipation function $$P_\mu(z_1, \ldots, z_n) = \sum_{S \subseteq [n]} \mu(S) \prod_{i \in S} z_i.$$8 yielding forces $$P_\mu(z_1, \ldots, z_n) = \sum_{S \subseteq [n]} \mu(S) \prod_{i \in S} z_i.$$9. Forced Lagrange-d’Alembert equations, energy decay, and reduction theorems are all naturally formulated in terms of Rayleigh data, unifying mechanics on flat and Riemannian spaces (López-Gordón, 2021).