Towards a theory of negative dependence

Abstract

A molecular theory for the rank dependence of orientational relaxation in Brownian dipolar lattice P͑X 1 ϭ0,X 2 ϭ0,X 3 ϭ0 ͒ϭ16, 1379 P͑X 1 ϭ0,X 2 ϭ0,X 3 ϭ1 ͒ϭ8, P͑X 1 ϭ0,X 2 ϭ1,X 3 ϭ0 ͒ϭ8, P͑X 1 ϭ0,X 2 ϭ1,X 3 ϭ1 ͒ϭ8, P͑X 1 ϭ1,X 2 ϭ0,X 3 ϭ0 ͒ϭ12ϩ⑀,When 0р⑀р0.8 then this measure satisfies CNA and hence JNRD and h-NLC. However, when ⑀Ͼ0, then applying the external field ͑, 1, 1͒ for any positive Ͻ⑀/(1Ϫ⑀) yields a measure in which X 2 and X 3 are positively correlated, thus violating h-NLC and hence JNRD and CNA. This shows the first three vertical implications in Diag. 1 are strict.Example 2: Suppose nϭ3, and the probabilities for the various possible atoms are in the proportions P͑X 1 ϭ0,X 2 ϭ0,X 3 ϭ0 ͒ϭ0, P͑X 1 ϭ0,X 2 ϭ0,X 3 ϭ1 ͒ϭ1, P͑X 1 ϭ0,X 2 ϭ1,X 3 ϭ0 ͒ϭ1, P͑X 1 ϭ0,X 2 ϭ1,X 3 ϭ1 ͒ϭ10⑀, P͑X 1 ϭ1,X 2 ϭ0,X 3 ϭ0 ͒ϭ1, P͑X 1 ϭ1,X 2 ϭ0,X 3 ϭ1 ͒ϭ1, P͑X 1 ϭ1,X 2 ϭ1,X 3 ϭ0 ͒ϭ10⑀, P͑X 1 ϭ1,X 2 ϭ1,X 3 ϭ1 ͒ϭ⑀.

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].…”

“…In particular, the NLC inequality fails to imply negative association (see Example 2.1), and there is no known localto-global tool comparable to the FKG theorem in the theory of positive association. Thus, no corresponding theory of negatively dependent events exists as yet, but, as explained in [65], there certainly is a need for such a theory. In op.…”

“…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.…”

“…, while the usual definition of negative association for a probability measure µ on 2 [n] is the "negative" analog of positive association [65], namely the inequality F dµ Gdµ ≥ F Gdµ for any pair of increasing functions F, G on 2…”

A Randomized Rounding Approach to the Traveling Salesman Problem

“…In the analysis of this algorithm for asymmetric TSP [4], Asadpour et al use the negative correlation between the edges of random spanning trees to obtain concentration results on the distribution of edges across a cut. For this work, we have to use even stronger virtues of negative dependence [34]. In particular, we use the fact that the distribution of spanning trees belongs to a more general class of measures called Strongly Rayleigh.…”