Negatively Correlated Random Variables and Mason’s Conjecture for Independent Sets in Matroids

Abstract

Mason's Conjecture asserts that for an m-element rank r matroid M the sequence I k / m k : 0 ≤ k ≤ r is logarithmically concave, in which I k is the number of independent k-sets of M. A related conjecture in probability theory implies these inequalities provided that the set of independent sets of M satisfies a strong negative correlation property we call the Rayleigh condition. This condition is known to hold for the set of bases of a regular matroid. We show that if ω is a weight function on a set system Q that satisfies the Rayleigh condition then Q is a convex delta-matroid and ω is logarithmically submodular. Thus, the hypothesis of the probabilistic conjecture leads inevitably to matroid theory. We also show that two-sums of matroids preserve the Rayleigh condition in four distinct senses, and hence that the Potts model of an iterated twosum of uniform matroids satisfies the Rayleigh condition. Numerous conjectures and auxiliary results are included.

Negative dependence and the geometry of polynomials

“…Indeed, we show that strongly Rayleigh measures enjoy all virtues of negative dependence, including the strongest form of negative association (CNA+). In particular, this allows us to prove several conjectures made by Liggett [55], Pemantle [65], and Wagner [72], respectively, and to recover and extend Lyons' main results [57] on negative association and stochastic domination for determinantal probability measures induced by positive contractions. Moreover, we define a partial order on the set of strongly Rayleigh measures (by means of the notion of proper position for multivariate stable polynomials studied in [7,8,9,15]) and use it to settle Pemantle's questions and conjectures on stochastic domination for truncations of "negatively dependent" measures [65].…”

“…This extends Pemantle's corresponding theorem for symmetric measures and confirms his conjecture on strong negative association (CNA+) in the almost exchangeable case. The results in §6 are also useful in §7, as they allow us to show that the examples we construct there provide counterexamples to the series of conjectures on ultra log-concave rank sequences proposed in [65,72] to which we already alluded. We note that Corollary 6.6 in §6 was subsequently proved by different methods in [48], where it was additionally shown that almost exchangeable measures satisfy the (strong) Feder-Mihail property.…”

“…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.…”

“…give rise to so-called ultra log-concave rank sequences (see §2.1 below for the definition). Subsequently, in [72] one of these conjectures was coined "The Big Conjecture" since it would imply the validity of Mason's long-standing conjecture in enumerative graph/matroid theory [62] for a large class of matroids. In §7 we construct the first counterexamples in the literature to all of these conjectures except Mason's, which once again confirms the delicate nature of negative dependence.…”

“…[55,65,72], and we clarify the connections between these as well as new classes of probability measures that we introduce below. We then formulate the problems and conjectures made in [55,65,72] that we solve in the next sections.…”