Rayleigh Matroids

Abstract

Motivated by a property of linear resistive electrical networks, we introduce the class of Rayleigh matroids. These form a subclass of the balanced matroids defined by Feder and Mihail [9] in 1992. We prove a variety of results relating Rayleigh matroids to other well-known classes -in particular, we show that a binary matroid is Rayleigh if and only if it does not contain S 8 as a minor. This has the consequence that a binary matroid is balanced if and only if it is Rayleigh, and provides the first complete proof in print that S 8 is the only minor-minimal binary non-balanced matroid, as claimed in [9]. We also give an example of a balanced matroid which is not Rayleigh. † Research supported by the Natural Sciences and Engineering Research Council of Canada under operating grant OGP0105392. Proposition 3.4. Let M be a matroid with ground set E, and let I, J be disjoint subsets of E. If M is Rayleigh and y > 0 then M I M J M IJ M. Proof. The inequality is trivial if either I or J is dependent, so assume that both I and J are independent in M. We first prove the result for I = {e 1 } and J = {f 1 , . . . , f k }. Notice that the Rayleigh difference of {e, f} in M may also be expressed as ∆M{e, f} = M e M f − M ef M. Thus, the Rayleigh condition is that if y > 0 then M e M f M ef M. Since every (contraction) minor

Negative dependence and the geometry of polynomials

“…There are many important examples of negatively dependent "repelling" random variables in probability theory, combinatorics, stochastic processes and statistical mechanics: uniform random spanning tree measures [16], symmetric exclusion processes [52,55], random cluster models (with q < 1) [33,42,65], balanced and Rayleigh matroids [20,29,68,70,72], competing urns models [26], etc; see, e.g., [45,65] and the references therein for a discussion of some of these examples and several others. To add to this list, in §3 we show that both the inequalities characterizing multi-affine real stable polynomials [8,9,15] and Hadamard-FischerKotelyansky type inequalities in matrix theory [28,40,39] may in fact be viewed as natural manifestations of negative dependence properties.…”

“…The notion of a Rayleigh polynomial was introduced in [72, §3.1], the terminology being motivated by its similarity with the Rayleigh monotonicity property of the (Kirchhoff) effective conductance of linear resistive electrical networks; see [20,73]. It was first considered for uniform measures on the set of bases of a matroid [20] as a strengthening of weaker notions studied in [29,69].…”

“…It was first considered for uniform measures on the set of bases of a matroid [20] as a strengthening of weaker notions studied in [29,69]. The following properties of Rayleigh polynomials are consequences of Definition 2.5.…”

“…The term is taken from [20], where a matroid was defined to be strongly Rayleigh if (5) holds for its bases-generating polynomial. However, it was not known then that condition (5) is equivalent to stability.…”

Real stable polynomials and matroids: optimization and counting

“…The first question leads us directly to the notion of Rayleigh and strongly Rayleigh matroids introduced in [CW06]. As shown in [Bra07] the class of matroids M such that h(z) is real stable (for B being the set of bases of M) is precisely equal to the class of matroids enjoying the strongly Rayleigh property.…”