On the extension of the FKG inequality to n functions

Lieb, Elliott H.; Sahi, Siddhartha

doi:10.1063/5.0065285

The 1971 Fortuin–Kasteleyn–Ginibre inequality for two monotone functions on a distributive lattice is well known and has seen many applications in statistical mechanics and other fields of mathematics. In 2008, one of us (Sahi) conjectured an extended version of this inequality for all n > 2 monotone functions on a distributive lattice. Here, we prove the conjecture for two special cases: for monotone functions on the unit square in $R^{k}$ whose upper level sets are k-dimensional rectangles and, more significantly, for arbitrary monotone functions on the unit square in $R^{2}$ ⁠. The general case for $R^{k}, k > 2$ ⁠, remains open.

I. INTRODUCTION

For functions f, g on a probability space (L, μ), their expectation and correlation are defined by

E_{1} (f) = E (f) ≔ \int_{L} f d μ and E_{2} (f, g) = E (f g) - E (f) E (g) .

(1.1)

Now suppose further that L is a distributive lattice⁸ and that the probability measure μ satisfies

μ (a \lor b) μ (a \land b) \geq μ (a) μ (b) .

(1.2)

In this situation, if f, g are positive monotone (decreasing) functions⁹ on L, then one has

E_{1} (f) \geq 0 and E_{2} (f, g) \geq 0 .

(1.3)

The first inequality is obvious, while the second is the celebrated FKG inequality of Fortuin–Kasteleyn–Ginibre¹ that plays an important role in several areas of mathematics/physics. We will refer to a distributive lattice L with probability measure μ satisfying (1.2) as an FKG poset.

In formulating (1.2), we have tacitly assumed that the poset L is a discrete set. However, the FKG inequality also has important continuous versions, which can be proved by a discrete approximation. For example, if Q_k = [0, 1]^k is the unit hypercube in $R^{k}$ equipped with the partial order: x ≥ y if and only if x_i ≥ y_i for all i, then the FKG inequality holds for the Lebesgue measure and, more generally, for any absolutely continuous measure whose density function satisfies (1.2).

In Ref. 5, Sahi introduced a sequence of multilinear functionals E_n(f₁, …, f_n), n = 1, 2, 3, …, generalizing E₁ and E₂ (see Definition 3.1) and proposed the following conjecture:

Conjecture 1.1

(Ref. 5 , Conjecture 5). If f₁, …, f_n are positive monotone functions on an FKG poset, then

E_{n} (f_{1}, \dots, f_{n}) \geq 0 .

(1.4)

Sahi⁵ proved the conjecture for the lattice {0, 1} × {0, 1} and for a certain subclass of positive monotone functions on the general power set lattice {0, 1}^k equipped with a product measure. Since the functionals E_n satisfy the following “branching” property (Ref. 5, Theorem 6)

E_{n} (f_{1}, \dots, f_{n - 1}, 1) = (n - 2) E_{n - 1} (f_{1}, \dots, f_{n - 1}),

(1.5)

the inequalities (1.4) form a hierarchy in the following sense: if C_n denotes the n-function positivity conjecture, then C_n implies C_n−1 for n > 2.

The work of Sahi was inspired by that of Richards⁴ who first had the idea of generalizing the FKG inequality to more than two functions. A natural first candidate for such an inequality is the cumulant (Ursell function) κ_n, but an easy example shows that the inequality already fails for κ₃. Nevertheless, Richards [Ref. 4, Conjecture 2.5] conjectured the existence of such a hierarchy of inequalities, although without an explicit formula for E_n.

Indeed for n = 3, 4, 5, Sahi’s functional E_n coincides with the “conjugate” cumulant $κ_{n}^{'}$ introduced by Richards [Ref. 4, formula (2.2)], although for n ≥ 6 one has $E_{n} \neq κ_{n}^{'}$ ⁠. We note also that Ref. 4 contains two “proofs” of the positivity of $κ_{3}^{'}, κ_{4}^{'}, κ_{5}^{'}$ —one for a discrete lattice and the other for a continuous analog. However, it seems to us that both proofs have essential gaps. Thus, beyond the special cases treated in Ref. 5, Conjecture 1.1 remains a conjecture, even for n = 3, 4, 5.

In this paper, we provide further evidence in support of Conjecture 1.1. We consider the continuous case of the Lebesgue measure on the unit hypercube Q_k = [0, 1]^k in $R^{k}$ ⁠, and we prove the inequalities (1.4) for the following two additional cases:

for arbitrary positive monotone functions on the unit square in $R^{2}$ and
for monotone characteristic functions of k-dimensional rectangles in [0, 1]^k and, by multlinearity of E_n, for functions whose level sets are (not necessarily homothetic) rectangles.

First, we treat the case of three functions on $R^{2}$ in Sec. II. This introduces several key ideas, including a reduction to a non-linear inequality involving decreasing sequences. In Sec. III, we define E_n for arbitrary n and prove Conjecture 1.1, first for characteristic functions of k-dimensional rectangles and then for general monotone functions on $R^{2}$ ⁠, that is, we extend Sec. II to all n > 3. This requires additional ideas involving the symmetric group S_n and an intricate induction on n. Subsections III A and III B are written in complete generality, and we hope these ideas will help in the eventual resolution of Conjecture 1.1.

Since the FKG inequality has many applications in probability, combinatorics, statistics, and physics, it reasonable to suppose that the generalized inequality will likewise prove to be useful in one or more of these areas. Although we do not have a compelling application in mind, we feel that it is important to find such an application. Indeed, the right application might provide additional insight into Conjecture 1.1 and perhaps even suggest a line of attack.

To end this section, we tantalize the reader with an interesting reformulation of the inequalities E_n ≥ 0 in terms of a formal power series from Ref. 5. First, if F(x) is a positive function on a probability space L, then it is natural to define the geometric mean of F by

G (F) = \exp (E (\log F)) .

(1.6)

Now, suppose F(x, t) is a power series of the form

F (x, t) = 1 - f_{1} (x) t - f_{2} (x) t^{2} - \dots .

(1.7)

Then, $\log (F (x, t))$ is a well defined power series, and formula (1.6) gives

G (F) = \exp (E (\log (F (x, t))) = 1 - c_{1} t - c_{2} t^{2} - \dots,

(1.8)

where the constants c_j are certain algebraic expressions in various $E (f_{i_{1}} f_{i_{2}} \dots f_{i_{p}})$ ⁠.

Conjecture 1.2

(Ref. 5 , Conjecture 4). If the f₁(x), f₂(x), … is a sequence of positive monotone functions on an FKG poset, then c_n ≥ 0 for all n.

It turns out that Conjectures 1.1 and 1.2 are equivalent. One implication has already been established in Sec. III of Ref. 5, and we prove the other direction in the Appendix. We also refer the reader to Refs. 6 and 7 for related inequalities in an algebraic setting.

II. THE INEQUALITY FOR THREE FUNCTIONS

For three functions, the multilinear functional E_n introduced in Ref. 5 is given by

E_{3} (f, g, h) = 2 E (f g h) + E (f) E (g) E (h) - E (f) E (g h) - E (g) E (f h) - E (h) E (f g) .

(2.1)

We note that E₃ is different from the cumulant (Ursell function), which is given by

κ_{3} (f, g, h) = E (f g h) + 2 E (f) E (g) E (h) - E (f) E (g h) - E (g) E (f h) - E (h) E (f g) .

(2.2)

We will consider the functional E₃ for functions on the unit hypercube,

Q_{k} = {[0, 1]}^{k} = {x = (x_{1}, \dots, x_{k}) | 0 \leq x_{i} \leq 1}

(2.3)

equipped with the Lebesgue measure and the usual partial order: x ≤ x′ if and only if $x_{i} \leq x_{i}^{'}$ for all i. We say that a real valued function f on Q_k is monotone (decreasing) if x ≤ x′ implies f(x) ≥ f(x′). We note that the FKG inequality is usually stated for monotonically increasing functions, but this is a somewhat arbitrary choice. Indeed, FKG and our theorems for decreasing functions are equivalent to the corresponding results for increasing functions. For a general FKG poset, this follows by reversing the partial order, and for Q_k, this follows by the change of variables x_i ↦ 1 − x_i. We also note that monotonicity for Q₁ has the usual 1-variable meaning of a decreasing function.

Theorem 2.1.

If f, g, h are positive monotone functions on [0, 1]², then E₃(f, g, h) ≥ 0.

The generalization of Theorem 2.1 to n functions is given in Theorem 3.6. We now reduce Theorem 2.1 to characteristic functions χ_S, S ⊂ Q_k. These are defined by χ_S(x) = 1 if x ∈ S and χ_S(x) = 0 if x ∉ S. We will say S is monotone if χ_S is monotone.

Lemma 2.2.

It suffices to prove Theorem 2.1 for χ_S, χ_T, χ_U for all monotone S, T, U.

Proof.

Any positive f can be written as an integral over the characteristic functions of its upper level sets. Thus, $f (x) = \int_{0}^{\infty} ξ_{s} (x) d s$ ⁠, with ξ_s(x) = 1 if f(x) > s and 0 otherwise (see the “layer cake principle” in Ref. 2). If f is monotone, then ξ_s is monotone for every s. Since E₃ is multi-linear in f, g, h, this reduces Theorem 2.1 to the case of monotone characteristic functions.■

We now describe a further reduction of Theorem 2.1 to a discrete family of characteristic functions. Let $A = A (m)$ be the set of decreasing m-tuples of integers, each between 0 and m,

A (m) ≔ {a \in Z^{m} | m \geq a_{1} \geq \dots \geq a_{m} \geq 0} .

(2.4)

For each $a \in A$ ⁠, we define a monotone subset S_a of Q₂ = [0, 1]² as follows. Divide Q₂ uniformly into m² little squares, write D_i,j for the square with top right vertex (i/m, j/m), and set

S_{a} = ⋃_{j \leq a_{i}} D_{i, j}, χ_{a} = χ_{S_{a}} .

(2.5)

Then, S_a is a monotone subset of Q₂, and, conversely, any monotone union of D_i,j is of this form.

Lemma 2.3.

It suffices to prove Theorem 2.1 for χ_a, χ_b, χ_c ; $a, b, c \in A (m)$ ; for all m.

Proof.

By Lemma 2.2, it suffices to consider monotone characteristic functions χ_S, χ_T, χ_U. Divide Q₂ uniformly into m² little squares D_i,j as before, and let S^m, T^m, U^m be the unions of the D_i,j contained in S, T, U, respectively; then, these are monotone subsets of Q₂ of the form (2.5). Moreover, $χ_{S^{m}}, χ_{S^{m}} χ_{T^{m}}$ ⁠, etc., converge to χ_S, χ_Sχ_T, etc., in L¹ as m → ∞. Thus, if $E_{3} (χ_{S^{m}}, χ_{T^{m}}, χ_{U^{m}}) \geq 0$ ⁠, then we get $E_{3} (χ_{S}, χ_{T}, χ_{U}) = \lim_{m \to \infty} E_{3} (χ_{S^{m}}, χ_{T^{m}}, χ_{U^{m}}) \geq 0$ ⁠.■

A. Proof of the three function inequality in two dimensions

We now prove Theorem 2.1 for χ_a, χ_b, χ_c, which suffices by Lemma 2.3. To simplify the notation, we work directly with a, b, c and we define the product ab, expectation $E (a)$ ⁠, etc., as follows:

{(a b)}_{i} = \min {a_{i}, b_{i}},

(2.6)

E_{1} (a) = E (a) = (a_{1} + \dots + a_{m}) / m^{2},

(2.7)

E_{2} (a, b) = E (a b) - E (a) E (b),

(2.8)

E_{3} (a, b, c) = 2 E (a b c) + E (a) E (b) E (c) - E (a) E (b c) - E (b) E (a c) - E (c) E (a b) .

(2.9)

Then, we have χ_ab = χ_aχ_b, $E (a) = E (χ_{a})$ ⁠, E₂(a, b) = E₂(χ_a, χ_b), E₃(a, b, c) = E₃(χ_a, χ_b, χ_c).

In particular, by the FKG inequality, we obtain the following lemma:

Lemma 2.4.

For all a, b in $A$ ⁠, we have E₂(a, b) ≥ 0.■

To study E₃(a, b, c), we consider certain perturbations of a. We say that $a \in A$ has a descent at i if a_i > a_i+1, and in this case, we can define three new sequences a⁻ = a^−,i, a⁺ = a^+,i, a^⋆ = a^⋆,i, also in $A$ ⁠, in which the following changes, and only these, are made to a:

a_{i}^{-} = a_{i + 1}, a_{i + 1}^{+} = a_{i}, a_{i + 1}^{⋆} = a_{i + 1} + 1 .

(2.10)

Lemma 2.5.

If a has a descent at i, but b does not, then we have $E (a^{+} b) + E (a^{-} b) = 2 E (a b)$ .

Proof.

Let b_i = b_i+1 = β, say, then we have

{(a^{+} b)}_{i} = {(a^{+} b)}_{i + 1} = \min {a_{i}, β} = {(a b)}_{i},

(2.11)

{(a^{-} b)}_{i} = {(a^{-} b)}_{i + 1} = \min {a_{i + 1}, β} = {(a b)}_{i + 1} .

(2.12)

Since the three sequences a⁺b, a⁻b, and ab coincide except at i, i + 1, the result follows.■

Proposition 2.6.

If a has a descent at i, but b and c do not, then

E_{3} (a^{+}, b, c) + E_{3} (a^{-}, b, c) = 2 E_{3} (a, b, c) .

(2.13)

Proof.

Each term of () has a unique factor involving a, which is of the form

E (a d)

⁠, where d = 1, b, c, bc is a sequence in

A

that does not have a descent at i. By Lemma 2.5, we get

E (a^{+} d) + E (a^{-} d) = 2 E (a d) .

(2.14)

The result now follows from formula ().■

Lemma 2.7.

If a, b have a descent at i and b_i+1 ≤ a_i+1, then a^⋆b = ab.

Proof.

Evidently,

{(a^{⋆} b)}_{j} = {(a b)}_{j}

for j ≠ i + 1, and since b_i+1 ≤ a_i+1, we also have

{(a^{⋆} b)}_{i + 1} = b_{i + 1} = {(a b)}_{i + 1} .

(2.15)

Thus, we get a^⋆b = ab, as claimed.■

Proposition 2.8.

If a and b have a descent at i and b_i+1 ≤ a_i+1, then we have a^⋆b = ab and

E_{3} (a^{⋆}, b, c) \leq E_{3} (a, b, c) for all c .

(2.16)

Proof.

By Lemma 2.7, we get

E (a^{⋆} b) = E (a b), E (a^{⋆} b c) = E (a b c)

⁠, and it follows that

E_{3} (a, b, c) - E_{3} (a^{⋆}, b, c) = E_{2} (b, c) [E (a^{⋆}) - E (a)] + E (b) [E (a^{⋆} c) - E (a c)] .

(2.17)

Evidently, we have

E (a^{⋆}) \geq E (a)

and

E (a^{⋆} c) \geq E (a c)

⁠, and by the FKG inequality, we also have E₂(b, c) ≥ 0. Thus, all terms on the right-hand side of () are positive, which proves the result.■

Theorem 2.9.

For all a, b, c in $A$ ⁠, we have E₃(a, b, c) ≥ 0.

Proof.

Let $U$ be the set of triples (a, b, c) in $A$ for which E₃(a, b, c) attains its minimum, and let $V$ be the subset of $U$ for which the quantity $E (a) + E (b) + E (c)$ attains its maximum.

We claim that if

(a, b, c) \in V

⁠, then a, b, c are constant sequences. If this is not the case, then a, say, has a descent at some i. If b, c do not have a descent at i, then by Proposition 2.6 we get

E_{3} (a, b, c) = (E_{3} (a^{+}, b, c) + E_{3} (a^{-}, b, c)) / 2 .

By minimality, E₃(a, b, c) ≤ E₃(a^±, b, c), which forces E₃(a, b, c) = E₃(a^±, b, c). Replacing a by a⁺, we reach a contradiction since

E (a^{+}) > E (a)

⁠.

If b, say, also has a descent at i, then by symmetry we may assume b_i+1 ≤ a_i+1. Then, by Proposition 2.8, E₃(a^⋆, b, c) ≤ E₃(a, b, c), and we again reach a contradiction since $E (a^{⋆}) > E (a)$ ⁠.

Now, we may assume a, b, c are constant sequences and, by symmetry, further assume that

a \equiv m α, b \equiv m β, c \equiv m γ, 0 \leq α \leq β \leq γ \leq 1,

and it follows that E₃(a, b, c) = 2α + αβγ − (αβ + αβ + αγ) = α(1 − β) (2 − γ) ≥ 0.■

This proves Theorem 2.1 for χ_a, χ_b, χ_c and, thus, by Lemma 2.3, in general.

III. THE INEQUALITY FOR n FUNCTIONS

A. The definition of E_n

In this subsection and Subsection III B, we work with arbitrary functions on a probability space. We start by recalling the definition of the multilinear functional E_n(f₁, …, f_n) from Ref. 5. This involves the decomposition of a permutation σ in the symmetric group S_n as a product of disjoint cycles,

σ = (i_{1}, \dots, i_{p}) (j_{1}, \dots, j_{q}) \dots .

(3.1)

For σ as in (3.1), we write C_σ for the number of cycles in σ and we set

E_{σ} (f^{1}, \dots, f^{n}) = E (f^{i_{1}} \dots f^{i_{p}}) E (f^{j_{1}} \dots f^{j_{q}}) \dots .

(3.2)

Then, the following definition is due to Sahi.⁵

Definition 3.1.

For functions f¹, …, fⁿ on a probability space X, we define

E_{n} (f^{1}, \dots, f^{n}) = \sum_{σ \in S_{n}} {(- 1)}^{C_{σ} - 1} E_{σ} (f^{1}, \dots, f^{n}) .

(3.3)

Using (3.3), one can easily verify that E₁, E₂, E₃ coincide with their earlier definitions. We note that the factor of 2 in the term 2E(f¹f²f³) in formula (2.1) comes from the two 3-cycles (123) and (213). More generally, E_n will have repeated terms because E_σ is unchanged if we rearrange the indices within a cycle. For example, for n = 4, we have

\begin{array}{l} \begin{aligned} E_{4} (f^{1}, f^{2}, f^{3}, f^{4}) & = 6 E (f^{1} f^{2} f^{3} f^{4}) - 2 [E (f^{1}) E (f^{2} f^{3} f^{4}) + E (f^{2}) E (f^{1} f^{3} f^{4}) + \dots] \\ + [E (f^{1}) E (f^{2}) E (f^{3} f^{4}) + E (f^{1}) E (f^{3}) E (f^{2} f^{4}) + \dots] \\ - [E (f^{1} f^{2}) E (f^{3} f^{4}) + E (f^{1} f^{3}) E (f^{2} f^{4}) + \dots] - E (f^{1}) E (f^{2}) E (f^{3}) E (f^{4}) . \end{aligned} \end{array}

We now give an explicit formula for E_n in a special case.

Lemma 3.2.

Let X = [0, 1] be the unit interval equipped with Lebesgue measure, and let fⁱ be the characteristic function

χ_{[0, a_{i}]}, 0 \leq a_{i} \leq 1

, with 0 ≤ a₁ ≤ ⋯ ≤ a_n ≤ 1. Then, we have

E_{n} (f^{1}, \dots, f^{n}) = a_{1} (1 - a_{2}) \dots (n - 1 - a_{n}) .

We note that the above formula implies that E_n is positive, i.e., Conjecture 1.1 holds for the Lebesgue measure on [0, 1]. While it is easy enough to give a direct proof the lemma, we prefer to postpone the proof to Subsection III B where we will derive it as a consequence of a more general result.

B. Algebraic properties of E_n

We first prove a recursive formula relating E_n to E_n−1.

Proposition 3.3.

We have E_n(f¹, …, fⁿ⁻¹, f) = e₁ + ⋯ + e_n−1 − e_n, where

e_{i} = \{\begin{cases} E_{n - 1} (f^{1}, \dots, f^{i} f, \dots, f^{n - 1}) & if 1 \leq i \leq n - 1, \\ E_{n - 1} (f^{1}, \dots, f^{n - 1}) E (f) & if i = n . \end{cases})

(3.4)

Proof.

We write f = fⁿ and consider expression () for E_n(f¹, …, fⁿ) as a sum over the symmetric group S_n. We decompose S_n as a disjoint union,

S_{n} = S^{(1)} \cup \dots \cup S^{(n)}, S^{(i)} = {σ \in S_{n} ∣ σ (i) = n} .

(3.5)

Then, S⁽ⁿ⁾ is a subgroup of S_n, naturally isomorphic to S_n−1. By (), we have

E_{n} (f^{1}, \dots, f^{n}) = Σ^{1} + \dots + Σ^{n}, Σ^{i} = \sum_{σ \in S^{(i)}} {(- 1)}^{C_{σ} - 1} E_{σ} (f^{1}, \dots, f^{n}) .

(3.6)

To study Σⁱ, we consider the map

σ \mapsto \bar{σ}

defined by dropping n from the cycle decomposition of σ. Thus, for n = 5, we have (13)(245) ↦ (13)(24), (12)(34)(5) ↦ (12)(34), etc. Then,

σ \mapsto \bar{σ}

defines a bijection from each S⁽ⁱ⁾ to S_n−1. If σ is in S⁽ⁱ⁾ and i ≠ n, then i and n occur in the same cycle of σ, and dropping n does not change the cycle count. This gives

C_{σ} = C_{\bar{σ}}, E_{σ} (f^{1}, \dots, f^{n - 1}, f) = E_{\bar{σ}} (f^{1}, \dots, f^{i} f, \dots, f^{n - 1}),

which implies Σⁱ = e_i. If σ is in S⁽ⁿ⁾, then (n) occurs as a separate cycle in σ and we get

C_{σ} = C_{\bar{σ}} + 1, E_{σ} (f^{1}, \dots, f^{n - 1}, f) = E_{\bar{σ}} (f^{1}, \dots, f^{n - 1}) E (f),

which gives Σⁿ = −e_n. This proves the proposition.■

Lemma 3.2 is now an easy consequence.

Proof of Lemma 3.2.

Let

f_{i} = χ_{[0, a_{i}]}

⁠. Since a_i ≤ a_n for all i, we get

f_{i} f_{n} = χ_{[0, a_{i}]} χ_{[0, a_{n}]} = χ_{[0, a_{n}]} = f_{i} .

Now applying Proposition 3.3 with f = f_n, we deduce that

E_{n} (f^{1}, \dots, f^{n}) = ((n - 1) - E (f^{n})) E_{n - 1} (f^{1}, \dots, f^{n - 1}) = (n - 1 - a_{n}) E_{n - 1} (f^{1}, \dots, f^{n - 1}) .

The result follows by a straightforward induction on n.■

We next establish a useful formula for the partial sum P_c of E_n over the set of permutations containing a fixed cycle c.

Proposition 3.4.

Let S^c denote the set of permutations σ ∈ S_n that contain a fixed cycle c = (i₁, …, i_p) and let J_c = {j₁, j₂, …} = {1, …, n}\{i₁, …, i_p}, then we have

P_{c} ≔ \sum_{σ \in S^{c}} {(- 1)}^{C_{σ} - 1} E_{σ} (f^{1}, \dots, f^{n}) = \{\begin{cases} - E (f^{i_{1}} \dots f^{i_{p}}) E_{n - p} (f^{j_{1}}, f^{j_{2}}, \dots) & if p < n, \\ E (f^{1} \dots f^{n}) & if p = n . \end{cases})

(3.7)

Proof.

The set S^c consists of a single permutation if p = n. Otherwise, it consists of permutations of the form σ = c · τ, where τ is a permutation of J^c. Evidently, the number of cycles in σ and τ are related by C_τ = C_σ − 1. Thus, in this case, we have

{(- 1)}^{C_{σ} - 1} E_{σ} (f^{1} \dots f^{n}) = - E (f^{i_{1}} \dots f^{i_{p}}) {(- 1)}^{C_{τ} - 1} E_{τ} (f^{j_{1}}, f^{j_{2}}, \dots) .

(3.8)

Now, the result follows by summing () over τ.■

C. Proof of the n function inequality for rectangles in any dimension

By a rectangle in dimension k or a k-rectangle, we mean a subset of [0, 1]^k of the form

[0, r_{1}] \times \dots \times [0, r_{k}], 0 \leq r_{1}, \dots, r_{k} \leq 1 .

Theorem 3.5.

If fⁱ are characteristic functions of k-rectangles, then E_n(f¹, …, fⁿ) ≥ 0.

Proof.

We proceed by induction on k ≥ 1 and for a given k by induction on n ≥ 1. The base cases k = 1 and n = 1 are straightforward and the former by Lemma 3.2. Thus, we may assume k > 1 and n > 1, and we can write

f^{i} = g^{i} \times χ_{[0, a_{i}]},

where gⁱ is the characteristic function of a (k − 1)-rectangle. By symmetry of E_n, we may assume

0 \leq a_{1} \leq \dots \leq a_{n} \leq 1 .

(3.9)

We note that the assumption () on a_i means that we have

E (f^{i_{1}} \dots f^{i_{p}}) = a_{l} E (g^{i_{1}} \dots g^{i_{p}}), l = \min {i_{1}, \dots, i_{p}} .

(3.10)

Moreover, it follows from () and () that if a₂ = ⋯ = a_n = 1, then we have

E_{n} (f^{1}, \dots, f^{n}) = a_{1} E_{n} (g^{1}, \dots, g^{n}) .

(3.11)

We now fix an index i > 1 and let C(i) denote all set of all cycles containing i, then we have

E_{n} (f^{1}, \dots, f^{n}) = \sum_{c \in C (i)} P_{c},

where P_c is as in Proposition 3.4. If i is not minimal in c, then P_c is independent of a_i by (). If i is minimal in c, then 1 ∉ c; hence, c has length p < n, and by () and (), we get

P_{c} = - a_{i} b_{c}, b_{c} = E (g^{i_{1}} \dots g^{i_{p}}) E_{n - p} (f^{j_{1}}, f^{j_{2}}, \dots) .

By induction on n, we have b_c ≥ 0 for such c. This means that E_n(f¹, …, fⁿ) decreases as we increase a₂, …, a_n subject, of course, to condition (). In particular, E_n decreases as we successively increase

a_{n} ↗ 1, a_{n - 1} ↗ 1, \dots, a_{2} ↗ 1 .

By (), we get E_n(f¹, …, fⁿ) ≥ a₁E_n(g₁, …, g_n), which is positive by induction on k.■

If f is the characteristic function of a rectangle, then any level set of f is either the same rectangle or empty. However, using the layer-cake principle² and multilinearity as in the Proof of Lemma 2.2, we obtain the following immediate extension of the previous result.

Corollary 3.6.

If f¹, …, fⁿ are positive, monotone functions whose level sets are (not necessarily homothetic) rectangles, then E_n(f¹, …, fⁿ) ≥ 0.■

D. Proof of the n function inequality in two dimensions

Our main result is as follows:

Theorem 3.6.

If f¹, …, fⁿ are positive and monotone on [0, 1]², then E_n(f¹, …, fⁿ) ≥ 0.

As before, we can deduce this from the special case of χ_a as in (2.5).

Lemma 3.7.

It suffices to prove Theorem 3.6 for $χ_{a^{1}}, \dots, χ_{a^{n}}$ , $a^{i} \in A (m)$ , for all m.

Proof.

This is proved along the same lines as Lemmas 2.2 and 2.3.■

In this section, we work with $A = A (m)$ and to simplify the notation for a¹, …, aⁿ in $A$ ⁠, we set

E_{σ} (a^{1}, \dots, a^{n}) = E (χ_{a^{i_{1}}}, \dots, χ_{a^{i_{p}}}) E (χ_{a^{j_{1}}}, \dots, χ_{a^{j_{q}}}) \dots,

(3.12)

E_{n} (a^{1}, \dots, a^{n}) = \sum_{σ \in S_{n}} {(- 1)}^{C_{σ} - 1} E_{σ} (a^{1}, \dots, a^{n}) .

(3.13)

Then, we have $E_{n} (χ_{a^{1}}, \dots, χ_{a^{n}}) = E_{n} (a^{1}, \dots, a^{n})$ ⁠.

To study the positivity of E_n, we first consider a special case.

Proposition 3.8.

If aⁱ ≡ mα_i are constant sequences with 1 ≤ α₁ ≤ ⋯ ≤ α_n ≤ 0, then

E_{n} (a^{1}, \dots, a^{n}) = α_{1} (1 - α_{2}) \dots (n - 1 - α_{n}) .

(3.14)

Proof.

Let L_n = E_n(a¹, …, aⁿ). Since α_i ≤ α_n, we have aⁱaⁿ = aⁱ for all i. Thus, we get

L_{n} = \sum_{i = 1}^{n - 1} L_{n - 1} - L_{n - 1} E (a^{n}) = (n - 1 - α_{n}) L_{n - 1}

by Proposition 3.3. Now, () follows by induction on n, the case n = 1 being obvious.■

We now prove the generalization of Proposition 2.6.

Proposition 3.9.

If a has a descent at i, but a¹, …, aⁿ⁻¹ do not, then we have

2 E_{n} (a^{1}, \dots, a^{n - 1}, a) = E_{n} (a^{1}, \dots, a^{n - 1}, a^{+}) + E_{n} (a^{1}, \dots, a^{n - 1}, a^{-}) .

(3.15)

Proof.

This is proved for each term E_σ in (3.13) in exactly the same way as Proposition 2.6 by applying Lemma 2.5 to the unique factor of E_σ involving a = aⁿ in (3.12).■

We shall prove the next three theorems together by induction on n.

Theorem 3.10.

If a¹, …, aⁿ⁻², b are in

A

; S is a subset of Q₂; and χ_bχ_S = 0, then

E_{n} (χ_{a^{1}}, \dots, χ_{a^{n - 2}}, χ_{b}, χ_{S}) \leq 0 .

(3.16)

Theorem 3.11.

If a¹, …, aⁿ⁻², b, c are in

A

; b, c have a descent at i; and b_i+1 ≤ c_i+1, then

E_{n} (a^{1}, \dots, a^{n - 2}, b, c^{⋆}) \leq E_{n} (a^{1}, \dots, a^{n - 2}, b, c) .

(3.17)

Theorem 3.12.

For all a¹, …, aⁿ in

A

⁠, we have

E_{n} (a^{1}, \dots, a^{n}) \geq 0 .

(3.18)

Proof.

Let us write A(n), B(n), and C(n) for the assertions of Theorems 3.10, 3.11, and 3.12. Then, A(1), B(1) are vacuously true, while C(1) is evident. Therefore, it suffices to prove the implications A(n − 1) ∧ C(n − 1) ⟹ A(n) and A(n) ⟹ B(n) ⟹ C(n) for all n ≥ 2.

A(n − 1) ∧ C(n − 1) ⟹ A(n): By assumption, we have χ_bχ_S = 0, and we also have

χ_{a^{i}} χ_{S} = χ_{S^{i}}

⁠, where

S^{i} = S \cap S_{a^{i}}

⁠. Thus, by Proposition 3.3, we get

\begin{array}{l} E_{n} (χ_{a^{1}}, \dots, χ_{a^{n - 2}}, χ_{b}, χ_{S}) = e_{1} + \dots + e_{n - 2} + e_{n - 1} - e_{n}, \\ where e_{i} ≔ E_{n - 1} (χ_{a^{1}}, \dots, χ_{S^{i}}, \dots, χ_{a^{n - 2}}, χ_{b}), i \leq n - 2, \\ e_{n - 1} ≔ E_{n - 1} (χ_{a^{1}}, \dots, χ_{a^{n - 2}}, 0), \\ e_{n} ≔ E_{n - 1} (χ_{a^{1}}, \dots, χ_{a^{n - 2}}, χ_{b}) E (χ_{S}) . \end{array}

Now, e_n ≥ 0 by C(n − 1) and e_n−1 = 0 by () and (). In addition,

χ_{b} χ_{S^{i}} = (χ_{b} χ_{S}) χ_{a^{i}} = 0

⁠, and so by symmetry, we can apply A(n − 1) to conclude e_i ≤ 0 for i ≤ n − 2. This implies A(n), ().

A(n) ⟹ B(n): Define

S_{c}, S_{c^{⋆}}

as in () and put

S = S_{c^{⋆}} \ S_{c}

⁠, then by Lemma 2.7 we have

χ_{S} χ_{b} = (χ_{c^{⋆}} - χ_{c}) χ_{b} = χ_{c^{⋆} b} - χ_{c b} = 0 .

Thus, by A(n), (), we get

E_{n} (χ_{a^{1}}, \dots, χ_{a^{n - 2}}, χ_{b}, χ_{c^{⋆}} - χ_{c}) \leq 0

⁠, which implies B(n), ().

B(n) ⟹ C(n): This argument is similar to the Proof of Theorem 2.9. Let $M$ be the set of n-tuples a = (a¹, …, aⁿ) in $A$ for which E_n(a) achieves its minimum, and let $N$ be the subset of $M$ for which $λ (a) = E (a^{1}) + \dots + E (a^{n})$ achieves its maximum on $M$ ⁠. We claim that for a in $N$ each aⁱ is a constant sequence; by Proposition 3.8, this clearly implies C(n), E_n(a) ≥ 0.

If the claim is not true, then one of the sequences has a descent at some i. First suppose that only one sequence, by symmetry aⁿ = a, has a descent at i. By Proposition 3.9 and minimality of E_n(a), we deduce E_n(a) = E_n(a¹, …, aⁿ⁻¹, a^±). Thus, replacing a by a⁺ preserves E_n(a) but increases λ(a), which is a contradiction. If two sequences have a descent at i, then by symmetry we may assume these are aⁿ⁻¹ = b, aⁿ = c with b_i+1 ≤ c_i+1. Now, B(n), (3.17), implies that replacing c by c^⋆ does not increase E_n(a), but it does increase λ(a), which is a contradiction.■

This proves Theorem 3.6 for $χ_{a^{i}}$ and, thus, by Lemma 3.7, in general.

ACKNOWLEDGMENTS

This work was partially supported by the NSF (Grant Nos. DMS-1939600 and DMS-2001537) and the Simons Foundation (Grant No. 509766). The hospitality of the Institute for Advanced Study is gratefully acknowledged.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

APPENDIX: THE EQUIVALENCE OF CONJECTURES 1.1 AND 1.2

We start by recalling some basic facts about partitions and permutations. For more background and details involving these ideas, we refer the reader to Ref. 3.

A partition λ of n, of length l, is a weakly decreasing sequence of positive integers,

λ_{1} \geq λ_{2} \geq \dots \geq λ_{l} > 0 such that λ_{1} + \dots + λ_{l} = n,

we say that the λ_j are the parts of λ, and we write $l (λ) = l$ and $|λ| = n$ ⁠.

The conjugation action of S_n permutes the indices in the cycle decomposition (3.1) of an element σ. Thus, the class of σ is uniquely determined by its “cycle type,” i.e., the partition λ whose parts are the cycle lengths of σ, arranged in decreasing order. Moreover, if $m_{i} = m_{i} (λ)$ denotes the number of parts of size i, then the conjugacy class of cycle type λ contains n!/z_λ elements, where

z_{λ} = \prod_{i \geq 1} i^{m_{i}} (m_{i}) .

(A1)

For a function f on a probability space, we define its moments by the formula

p_{d} (f) = E (f^{d}) and p_{λ} (f) = p_{λ_{1}} (f) \dots p_{λ_{l}} (f) .

(A2)

Lemma A.1.

We have $E_{n} (f, \dots, f) = n! \sum_{| λ | = n} {(- 1)}^{l (λ) - 1} z_{λ}^{- 1} p_{λ} (f)$ .

Proof.

If σ is of class λ, then the number of disjoint cycles in σ is l(λ), and by (3.2), we have E_σ(f, …, f) = p_λ(f). Thus, the sum (3.3) for E_n(f, …, f) is constant over conjugacy classes, with class λ contributing n!/z_λ identical terms. This implies the result.■

If f is as above and u is a parameter, then we can define the formal logarithm

\log (1 - u f) = - \sum_{i \geq 1} u^{i} f^{i} / i .

(A3)

Proposition A.2

We have $\exp (E (\log (1 - u f))) = 1 - \sum_{n \geq 1} u^{n} E_{n} (f, \dots, f) / n!$

Proof.

Let

Z = E (\log (1 - u f))

⁠, then by () we have

Z = - \sum_{i \geq 1} u^{i} p_{i} (f) / i .

(A4)

Writing p_k = p_k(f) and p_λ = p_λ(f) for simplicity, we get

\exp (Z) = \prod_{i \geq 1} \sum_{m_{i} \geq 0} {(- 1)}^{m_{i}} {(u^{i} p_{i})}^{m_{i}} / i^{m_{i}} m_{i}! = \sum_{λ} {(- 1)}^{l (λ)} z_{λ}^{- 1} p_{λ} u^{| λ |} .

(A5)

Now, the result follows from Lemma A.1.■

Proposition A.3.

If f₁, f₂, … are functions on a probability space, then we have

1 - \exp (E (\log (1 - \sum_{i} f_{i} t^{i}))) = \sum_{n \geq 1} \sum_{i_{1}, \dots, i_{n}} E_{n} (f_{i_{1}}, \dots, f_{i_{n}}) t^{i_{1} + \dots + i_{n}} / n! .

Proof.

Let us write A = f₁t + f₂t² + ⋯, then by Proposition A.2, we get

1 - \exp (E (\log (1 - A))) = \sum_{n \geq 1} E_{n} (A, \dots, A) / n!,

and by multilinearity of E_n, we have

E_{n} (A, \dots, A) = \sum_{i_{1}, \dots, i_{n}} E_{n} (f_{i_{1}}, \dots, f_{i_{n}}) t^{i_{1} + \dots i_{n}}

⁠.■

Theorem A.4.

For a set of functions $I$ on a probability space, the following are equivalent:

For all n, we have E_n(f₁, …, f_n) ≥ 0 if $f_{1}, \dots, f_{n} \in I$ .
The power series $1 - \exp (E (\log (1 - \sum_{i} f_{i} t^{i})))$ has positive coefficients if $f_{1}, f_{2}, \dots \in I$ .

Proof.

The first statement implies the second by Proposition A.3. The converse was proved in Ref. 5, but we recall it here for completeness. Let p₁, p₂, …, p_n be the first n primes; define

k = p_{1} p_{2} \dots p_{n}, k_{j} = k / p_{j}, N = k_{1} + \dots + k_{n},

and consider possible solutions of the equation s₁k₁ + ⋯ + s_nk_n = N, where s₁, …, s_n are integers

\geq 0

⁠. If some s_j were 0, then p_j would divide the left side but not the right; thus, we must have all s_j > 0 and, hence, that s₁ = ⋯ = s_n = 1. Now, it follows from Proposition A.3 that the coefficient of t^N in the power series

1 - \exp (E (\log (1 - \sum_{j = 1}^{n} f_{j} t^{k_{j}})))

is precisely E_n(f₁, …, f_n). Thus, the second statement implies the first.■

The previous theorem proves the equivalence of Conjectures 1.1 and 1.2. In particular, our Theorem 3.6 implies Conjecture 1.2 for the Lebesgue measure on the unit square in $R^{2}$ ⁠.

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2022

Author(s)

On the extension of the FKG inequality to n functions

I. INTRODUCTION

II. THE INEQUALITY FOR THREE FUNCTIONS

A. Proof of the three function inequality in two dimensions

III. THE INEQUALITY FOR n FUNCTIONS

A. The definition of E_n

B. Algebraic properties of E_n

C. Proof of the n function inequality for rectangles in any dimension

D. Proof of the n function inequality in two dimensions

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

APPENDIX: THE EQUIVALENCE OF CONJECTURES 1.1 AND 1.2

REFERENCES

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On the extension of the FKG inequality to n functions

I. INTRODUCTION

II. THE INEQUALITY FOR THREE FUNCTIONS

A. Proof of the three function inequality in two dimensions

III. THE INEQUALITY FOR n FUNCTIONS

A. The definition of En

B. Algebraic properties of En

C. Proof of the n function inequality for rectangles in any dimension

D. Proof of the n function inequality in two dimensions

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

APPENDIX: THE EQUIVALENCE OF CONJECTURES 1.1 AND 1.2

REFERENCES

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A. The definition of E_n

B. Algebraic properties of E_n