Correlation bounds for fields and matroids

Let $G$ be a finite connected graph, and let $T$ be a spanning tree of $G$ chosen uniformly at random. The work of Kirchhoff on electrical networks can be used to show that the events $e_1 \in T$ and $e_2 \in T$ are negatively correlated for any distinct edges $e_1$ and $e_2$. What can be said for such events when the underlying matroid is not necessarily graphic? We use Hodge theory for matroids to bound the correlation between the events $e \in B$, where $B$ is a randomly chosen basis of a matroid. As an application, we prove Mason's conjecture that the number of $k$-element independent sets of a matroid forms an ultra-log-concave sequence in $k$.


INTRODUCTION AND RESULTS
Let G be a finite connected graph, and let w " pw e q be a set of positive weights on the edges e of G. Randomly pick a spanning tree T of G so that the probability of selecting an individual tree t is proportional to the product of the weights of its edges: PpT " tq 9 ź ePt w e .
The work of Kirchhoff on electrical networks can be used to show that, for any distinct edges i and j, the events i P T and j P T are negatively correlated: PpT contains i | T contains jq ď PpT contains iq.
Equivalently, for any distinct edges i and j, we have Ppi P T, j P Tq Ppi R T, j R Tq ď Ppi P T, j R Tq Ppi R T, j P Tq.
Let E be a finite set. A matroid on E is a nonempty collection of subsets of E, called bases of the matroid, that satisfies the exchange property: For any bases b 1 , b 2 and e 1 P b 1 zb 2 , there is e 2 P b 2 zb 1 such that`b 1 ze 1˘Y e 2 is a basis.
An independent set is a subset of a basis, a dependent set is a subset of E that is not independent, a circuit is a minimal dependent set, the rank of a subset of E is the cardinality of any one of its maximal independent subsets, and a flat is a subset of E that is maximal for its rank. The rank of a matroid is the cardinality of any one of its bases. For any unexplained matroid terms and facts, we refer to Oxley's book [Oxl11]. The collection of spanning trees of a connected graph is the best-known example of a matroid.
Let M be a matroid on E, and fix a set of positive weights w " pw e q on the elements e of E. Randomly pick a basis B of the matroid so that the probability of selecting an individual basis b is proportional to the product of the weights of its elements: PpB " bq 9 ź ePb w e .
In this more general setup, for any distinct i and j in E, do we still have the negative correlation Ppi P B, j P Bq Ppi R B, j R Bq ď Ppi P B, j R Bq Ppi R B, j P Bq? The answer is "yes" if the matroid is regular [FM92], if the matroid is representable over F 3 and F 4 [COSW04], if the cardinality of E is at most 7, or if the rank of M is at most 3 [Wag05]. Examples below show that distinct elements of E can define positively correlated events for more general matroids.
Example 1. Let S be the 2-dimensional skeleton of the 5-dimensional simplex. A spanning tree of S is a maximal subset of the twenty triangles in S that does not contain any 2-cycle over F 2 . Choose one such B uniformly at random. Then, for any two disjoint triangles in S, say i " 123 and j " 456 in Figure 1A, we have
Example 2. Let G be the graph in Figure 1B. Consider the collection of all forests in G with exactly six edges, and choose one such B uniformly at random. Then, for the edges labelled i and j in Figure 1B, we have Example 3. Let A 1 , A 2 , A 3 , A 4 be the four finite sets shown in Figure 1C. A system of distinct representatives is a set tx 1 , x 2 , x 3 , x 4 u of size four such that x k P A k for all k. Choose one such B uniformly at random. Then, for the elements labelled i and j in Figure 1C, we have In Section 3, We use the Hodge theory for matroids in [HW17,AHK18] to bound the correlation between the events e P B.
Theorem 5. For any distinct elements i and j in a matroid M of positive rank d, Theorem 5 implies the covariance bound CovpB contains i, B contains jq ă PpB contains iq PpB contains jq.
Compare the notion of approximate independence in [Kah00, Section 4].
An element e of a rank d matroid M is a loop if it is contained in no basis of M, a coloop if it is contained in every basis of M, and free if it is not a coloop and every circuit of M containing e has cardinality d`1. For example, the elements labelled j in matroids of Examples 2 and 3 are free. In Section 4, we remove the factor 2 in Theorem 5 when both i and j are free.
Theorem 6. For any distinct free elements i and j in a matroid M of positive rank d, Can we replace the constant 2 in Theorem 5 by a smaller number? To any matroid M, we associate a nonnegative real number αpMq defined by where the supremum is over all distinct non-loop non-coloop elements i and j in M and all sets of positive weights w on the elements of M. When every element of M is either a loop or a coloop, we set αpMq " 0. It is straightforward to check that, if M K is the dual matroid of M and M is a minor of another matroid N, then αpMq " αpM K q and αpMq ď αpNq.
In addition, if M 1 and M 2 have an element that is neither a loop nor a coloop, then αpM 1 ' M 2 q " max αpM 1 q, αpM 2 q, 1 ( .
We define the correlation constant α F of a field F to be the real number where the supremum is over all matroids M representable over F. The correlation constant of matroids, denoted α Mat , is defined in the same way by taking the supremum over all matroids.
As we can place any number of new elements in parallel to existing elements in any matroid, the values of α F and α Mat remain unchanged if we only consider matroids with constant weights.
In Section 5, We construct explicit examples to produce a lower bound of α F for any field F.
Theorem 7. The correlation constant of any field F satisfies 8 What is the correlation constant of F 2 ? What is the correlation constant of C? Does α F depend on F? What is the correlation constant α Mat ? The first question may be the most tractable one, as the only minor-minimal binary matroid with αpMq larger than 1 is the matroid represented over This matroid, labelled S 8 in Oxley's list [Oxl11,Appendix], was first found by Seymour and Welsh to have positively correlated pair of elements [SW75]. See [CW06] for a proof of the assertion on S 8 . We conjecture, although without much evidence, that the correlation constant of F 2 is 8 7 . We know no matroid M with αpMq larger than 8 7 . The initial motivation for our paper comes from the work of Mason [Mas72], who offered the following three conjectures of increasing strength. Several other authors studied correlations in matroid theory partly in pursuit of these conjectures [SW75, Wag08, BBL09, KN10, KN11].
Conjecture 8. For any n-element matroid N and any positive integer k, (1) I k pNq 2 ě I k´1 pNqI k`1 pNq, where I k pNq is the number of k-element independent sets of N.
Conjecture 8 (2) follows from the special case of Theorem 6 when the weight w is constant.
The implication is based on two standard constructions [Oxl11, Chapter 7]. First, we use the truncation of N to reduce Conjecture 8 (2) to the case k " d´1, where d is the rank of N. Next, we construct the free extension M of N by adding two new free elements i and j. If we pick a basis B of M uniformly at random, then Now Conjecture 8 (2) for N is Theorem 6 for i and j in M.
Conjecture 8 (2) implies an entropy bound that cannot be deduced from Conjecture 8 (1). Recall that the Shannon entropy HpXq of a discrete random variable X is, by definition, where the logarithm is in base 2 and the sum is over all values of X with nonzero probability. For a rank d matroid M, let I M be the size of an independent set drawn uniformly at random from the collection of all independent sets of M. For any d, uniform matroids of rank d show that inf where the infimum is over all matroids of rank d. We show that, asymptotically, the entropy of I M is at most half of the obvious upper bound log d given by Jensen's inequality.
Corollary 10. Uniform random independent sets of matroids satisfy where the supremum is over all matroids of rank d.
Corollary 10 is based on a result of Johnson [Joh07, Theorem 2.5], who showed that the Poisson distribution maximizes entropy in the class of ultra log-concave distributions. Recall that a random variable X taking its values in N is said to have the Poisson distribution with parameter λ if Combined with Conjecture 8 (2), Johnson's result implies that where Ppλq is the Poisson distribution with parameter λ " EpI M q. Using known bounds for the entropy of Poisson distributions from information theory [CT06, Theorem 8.6.5], we get In general, an upper bound of the entropy of a random variable X implies a concentration of X [Juk11, Chapter 22]. The above bound of HpI M q, for example, gives the following.
Corollary 11. For any matroid M of rank d, there is k such that Clearly, Corollaries 10 and 11 cannot be deduced from Conjecture 8 (1) alone.

HODGE THEORY FOR MATROIDS
We review the results of [HW17] and [AHK18] that will be used to prove Theorems 5 and 6. For our purposes, we may assume that matroids do not have any loops. In the rest of this paper, we fix a positive integer n and work with loopless matroids on finite sets E " t1, . . . , nu and E " t0, 1, . . . , nu. x F , one for each pair of distinct elements e 1 and e 2 of E, and the quadratic monomials one for each pair of incomparable nonempty proper flats F 1 and F 2 of M. We denote the degree q component of ApMq by A q pMq.
Definition 12. A real-valued function c on 2 E is said to be strictly submodular if c ∅ " 0, c E " 0, and, for any pair of incomparable subsets I 1 , I 2 Ď E, we have A strictly submodular function c defines an element Lpcq " Note that strictly submodular functions on 2 E exist. For example, we have the function c I " pnumber of elements in Iqpnumber of elements not in Iq.
We may now state the hard Lefschetz theorem and the Hodge-Riemann relations for matroids [AHK18, Theorem 1.4]. The function "deg" in Theorem 13 is the isomorphism A d pMq » R constructed in [AHK18,Section 5.3]. This isomorphism is uniquely determined by its property degpx F 1 x F 2¨¨¨xF d q " 1 for any chain of nonempty proper flats F 1 Ĺ F 2 Ĺ¨¨¨Ĺ F d in M.
Theorem 13. Let L be an element of A 1 pMq attached to a strictly submodular function on 2 E .
(1) (Hard Lefschetz theorem) For every nonnegative integer q ď d 2 , the multiplication by L defines an isomorphism (2) (Hodge-Riemann relations) For every nonnegative integer q ď d 2 , the multiplication by L defines a symmetric bilinear form A q pMqˆA q pMq ÝÑ R, pη 1 , η 2 q Þ ÝÑ p´1q q degpη 1 η 2 L d´2q q that is positive definite on the kernel of L d´2q`1 .
Theorems 5 and 6, as well as other applications of the Hodge-Riemann relations in combinatorics surveyed in [Huh18], only use the special case q ď 1. It will be interesting to find applications of the Hodge-Riemann relations for q ą 1.

PROOF OF THEOREM 5
Let M be a rank d loopless matroid on E. Let M be the matroid on E obtained from M by adding 0 as a coloop, the direct sum of M and the rank 1 matroid on t0u. For every e in E, we define an element y e " In what follows, relations of the above kind will be called xy-relations. The xy-relations imply that, for example, y e¨ye is zero for any e in E.

Lemma 14.
For any dependent set J of M, we have ź ePJ y e " 0.
Proof. We may suppose that J is a circuit of M. Choose a maximal independent set I of M in J, an element f in I, and an element g in JzI. Since pIzf q Y g is a basis of J, the set of flats of M containing pIzf q Y 0 and not containing f is equal to the set of flats of M containing pIzf q Y 0 and not containing g. Therefore, by the xy-relations, we have ź ePI y e " y f ź ePIzf y e " y g ź ePIzf y e .
If B is a basis of M, F k is the only flat of M containing 0, 1, . . . , k´1, not containing k, and comparable to F k`1 . Thus the xy-relations imply that If Then cpeq ∅ " 0, cpeq E " 0, and, for any subsets I 1 , I 2 of E, we have The submodular inequality of Lemma 16 is straightforward to check. In fact, we have 1 if 0 is in I 1 zI 2 and e is in I 2 zI 1 , 1 if 0 is in I 2 zI 1 and e is in I 1 zI 2 , 0 if otherwise.
We are ready to prove Theorem 5. The equality holds in Theorem 5 when d " 1. Suppose from now on that d ě 2. Let w " pw e q be the given set of positive weights on E. For distinct elements i and j in E, define where the sum is over all elements of E other than i and j. Lemma 15 shows that where B ij is the set of bases of M not containing i and not containing j. Similarly, where B j i is the set of bases of M containing i and not containing j, and where B ij is the set of bases of M containing i and containing j. Theorem 5 obviously holds if B ij or B ij is empty. We suppose from now on that B ij and B ij are nonempty.
Let L be any element of A 1 pMq attached to a strictly submodular function on 2 E . By Lemma 16, Theorem 13 applies to the element L ij` L for any positive real number . By the Hodge-Riemann relations for q ď 1, any matrix representing the symmetric bilinear form ust have exactly one positive eigenvalue. Thus, by continuity, any matrix representing the symmetric bilinear form A 1 pMqˆA 1 pMq ÝÑ R, pa 1 , a 2 q Þ ÝÑ deg´η 1 η 2 L d´2 ijh as at most one positive eigenvalue. Now consider the symmetric matrix Cauchy's eigenvalue interlacing theorem shows that H ij has at most one positive eigenvalue as well. On the other hand, H ij has at least one positive eigenvalue, because its lower-right diagonal entry is positive. A straightforward computation reveals that the determinant of H ij is a positive multiple of The determinant must be nonnegative by the condition on the eigenvalues of H ij , and hence This completes the proof of Theorem 5.

PROOF OF THEOREM 6
Let i and j be distinct free elements in a rank d matroid Y, and let Z be the deletion of i and j from Y. We prove Theorem 6 for i and j in Y. When d " 1, no basis of Y contains both i and j, and the equality holds in Theorem 6. Suppose from now on that d ě 2.
Write B ij , B i j , B j i , B ij for the set of bases containing and/or not containing i, j, and I m for the collection of m-element independent sets. Since i and j are free, we have natural bijections If the rank of Z is less than d, Theorem 6 clearly holds, as the left-hand side of the inequality is zero. If the rank of Z is d, Theorem 6 follows from the following version of Corollary 9 applied to Z.
Proposition 17. For any matroid M of rank d ě 2 and any set of positive weights w " pw e q, where I m " I m pMq is the collection of m-element independent sets of M.
The proof of Proposition 17 is similar to that of Theorem 5. We define an element Proof. We prove by descending induction on m. The case m " d is Lemma 15, and the case of dependent I is Lemma 14. For the induction step, suppose without loss of generality that t1, . . . , du is a basis of M. It is enough to show that y 1¨¨¨ym´1¯ym α d´m "´y 1¨¨¨ym´1¯α d´m`1 .
By the xy-relations, the difference of the right-hand side and the left-hand side iś where the sum is over all proper flats G of M containing 0, 1, . . . , m. For any such G, we claim To see this, use the linear relations in ApMq to write where the k-th sum is over all proper flats F m`k of M containing m`k. Since t1, . . . , du is a basis of M, no proper flat of M contains t0, 1, . . . , du, and hence the right-hand side is zero by the quadratic relations in ApMq.
We are ready to prove Proposition 17. Define another element where the sum is over all elements e in E. By Lemma 18, for any nonnegative ineteger m ď d, where I m is the collection of m-element independent sets of M.
Let L be any element of A 1 pMq attached to a strictly submodular function on 2 E . By Lemma 16, Theorem 13 applies to the element L 0` L for any positive real number . By the Hodge-Riemann relations for q ď 1, any matrix representing the symmetric bilinear form A 1 pMqˆA 1 pMq ÝÑ R, pη 1 , η 2 q Þ ÝÑ deg´η 1 η 2`L0` L˘d´2m ust have exactly one positive eigenvalue. Thus any matrix representing the symmetric bilinear form A 1 pMqˆA 1 pMq ÝÑ R, pa 1 , a 2 q Þ ÝÑ deg´η 1 η 2 L d´2 0h as at most one positive eigenvalue. Now consider the symmetric matrix Cauchy's eigenvalue interlacing theorem shows that H 0 has at most one positive eigenvalue. On the other hand, H 0 has at least one positive eigenvalue, because its lower-right diagonal entry is positive. The determinant of H 0 is a positive multiple of which must be nonpositive by the condition on the eigenvalues of H 0 .

PROOF OF THEOREM 7
The upper bound follows from Theorem 5. We construct explicit vector configurations over F to show the lower bound 8 7 . Fix a prime number p and an integer d ě 2. Consider the d-dimensional vector space F d p over the field with p elements, and let e 1 , e 2 , . . . , e d be the standard basis vectors of F d p .
Definition 19. Let M d p be the rank d matroid represented by the vectors e 1 , e 2`¨¨¨`ed , and 1e 1`ed , 2e 1`ed ,¨¨¨pe 1`ed .
We write i for the vector e 1 and j for the vector e 2`¨¨¨`ed .
The matroid M 4 2 is isomorphic to the matroid S 8 mentioned in the introduction. For any d, the matroid M d 2 is the self-dual matroid obtained from the binary spike Z d in Oxley's list by deleting any element other than the tip [Oxl11, Appendix]. 2 For any p, the matroid M d p has a spike-like structure in that it has a "tip" i and "legs" (1) The contraction M{i{j is the uniform matroid U d´2,d´1 with each element replaced by p parallel copies. Any basis of the contraction is disjoint from one of the parallel classes and contains exactly one point from each one of the remaining parallel classes. Therefore, 1˙d´2 .
(2) The deletion Mzizj is represented by the p-point lines L 2 ze 1 , . . ., L d ze 1 in F d p . Any basis of the deletion must contain exactly two points from one of the lines and one point from each one of the remaining lines. Therefore, (3) The contraction-deletion M{izj is the boolean matroid U d´1,d´1 with each element replaced by p parallel copies. Any basis of the contraction-deletion contains exactly one element from 2 According to Geelen [Gee08], "it all goes wrong for spikes." The spike Z d was first used by Seymour to demonstrate that an independence oracle algorithm for testing whether a matroid is binary cannot run in polynomial time relative to the size of the ground set [Sey81]. each parallel class. Therefore, It remains to compute the number of bases of M not containing i and containing j. There are two types of such bases, corresponding to the two terms in the right-hand side of |B i j pMq| " pp d´1´pd´2 q`pd´1qpd´2qˆp 2˙ˆp 1˙d´3 . shows that there are exactly pp d´1´pd´2 q such bases. A basis of the second type contains exactly two points from one of the lines, no point from another, and one point from each one of the remaining lines. It is clear that any basis in B i j must be one of the two types. Combining the four numbers, we obtain a ratio that depends only on d and not on p: The maximum of the ratio is 8 7 , achieved when d " 5. This proves Theorem 7 when the field F has characteristic p.
For fields of characteristic zero, let i and j be distinct elements of a finite set A 1 . Let A 2 , . . . , A d be a family of pm`1q-element subsets of A 1 zi whose union is A 1 zi and whose pairwise intersection is tju. We extend the transversal matroid construction in Example 3 as follows.
Definition 20. The matroid N d m is the transversal matroid of the family A 1 , A 2 , . . . , A d .
The matroid N 6 2 is isomorphic to the truncated graphic matroid in Example 2. By definition, bases of N " N d m are the systems of distinct representatives of A 1 , A 2 , . . . , A d . For m " p, the matroids M d p and N d m share three of the four minors obtained by deleting and/or contracting i, j. For any m, we have |B ij pNq| " pd´1qˆm 1˙d´2 , |B ij pNq| " pd´1qˆm 2˙ˆm 1˙d´2 , |B j i pNq| "ˆm 1˙d´1 .
There are two types of bases of N not containing i and containing j, corresponding to the two terms in the right-hand side of |B i j pNq| " m d´1`p d´1qpd´2qˆm 2˙ˆm 1˙d´3 . E-mail address: wang@math.wisc.edu