Influences in Product Spaces: KKL and BKKKL Revisited

Transcription

1 Influences in Product Spaces: KKL and BKKKL Revisited Ehud Friedgut Abstract The notion of the influence of a variable on a Boolean function on a product space has drawn much attention in combinatorics, computer science and other fields. Two of the basic papers dealing with this notion are [KKL] and [BKKKL]. In this paper we survey the results in those papers and offer some simpler proofs, corrections, and extensions of the theorems presented there. We present several related open problems. 1 Introduction 1.1 Background The fundamental notion of the influence of a variable on a Boolean function on a product space arises naturally in game theory, economics, computer science, statistical physics, combinatorics and harmonic analysis. See [9],[3],[8],[2],[10], [11], for a (semi-random) collection of characteristic examples. In this paper we revisit two papers [KKL] and [BKKKL] that supplied basic insights about this important notion. We will expand and provide simpler proofs to some of the theorems proved in these two papers. We will present a conjecture as to what may replace a false statement presented in [BKKKL], and prove a related theorem that may be a step in this direction. Also we will present several conjectures and related open problems. Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem, Israel. ehudf@math.huji.ac.il 1

2 We begin with some definitions: Let X be a probability space. Let X n be the product space endowed with the product measure. For any x = (x 1,...,x n ) X n let s j (x) be the fiber containing x in the j th direction: s j (x) = {y : y i = x i for i j}. Let f : X n {0, 1}. The influence of the j th variable on f, denoted by I f (j) is the probability that, choosing x X n at random according to the probability measure, f is not fixed on s j (x): I f (j) = Pr({x : f is not constant on s j (x)}). One may think of f as some voting procedure, where n players each make some choice, and f decides between 0 and 1 according to the vector of their choices. In that case I f (j) is the probability that the j th player can decide the outcome of the vote if all other players make their choices randomly. It is quite natural to consider not only the influence of one player(coordinate) but also that of a coalition (a set of coordinates.) For any set J {1,...,n} and for any x X n let s J (x) = {y : y i = x i for i J}. Given a function f : X n {0, 1} as above the influence of a set of coordinates J, denoted by I f (J) is I f (J) = Pr({x : f is not constant on s J (x)}). We will consider only two different cases of the measure space X: the most basic case where X = {0, 1} with the uniform measure (This is the case covered in [KKL]) and the most general case X = [0, 1] with the uniform measure (covered in [BKKKL]). In this paper we will restrict ourselves to monotone functions, however, for our purposes, as observed in [KKL], this causes no loss of generality since every function f as above can be shifted into a function having the same expected value and for which all influences, both of individuals and of coalitions, are no greater than the corresponding influences for f. [KKL] was a breakthrough in the study of influences, due to the fact that it introduced the hammer of powerful techniques of discrete Fourier analysis, together with an appropriate chisel: a hypercontractive estimate due to Bonami ([4]) and, independently, Beckner ([1]). There are no known proofs of the results in this paper that do not use these two ingredients. 2

3 1.2 The Main Results in KKL and BKKKL The main theorem of [KKL] states that there always exists a variable whose influence is of order at least log(n)/n: Theorem 1.1 (KKL1) There exists a constant c > 0 such that the following holds: Consider {0, 1} n as a measure space with the uniform (product) measure. Let f : {0, 1} n {0, 1}. Let Pr({x : f(x) = 1}) = p. Then there exists a variable k such that I f (k) cp(1 p) log n n. The tribes example presented in [3] shows that, up to the value of c this theorem is tight (indeed, [3] was the paper in which Theorem 1.1 was first conjectured.) Since this example and related ones will inspire some of our conjectures let us recall it: Given n variables partition them into blocks (tribes) of size log(n) log log(n)+ c. Let the tribes be denoted by J 1,...,J k, with k n/ log(n). The tribes function takes on the value 1 if there exists a tribe that voted 1 unanimously: f(x) = 1 i x j = 1 j J i. A related function is the dual function g(x) = 1 f(x): g(x) = 1 i j J i such that x j = 1. As noted in [7] the original proof of Theorem 1.1 can be modified, and by using a convexity argument one can show the following: Theorem 1.2 There exists a constant c > 0 such that the following holds: Consider {0, 1} n as a measure space with the uniform measure. Let f : {0, 1} n {0, 1}. Let Pr({x : f(x) = 1}) = p. If for all k I f (k) δ then I f (k) > c log(1/δ)p(1 p). k Perhaps this is the place to mention a later result of Talagrand which is slightly stronger than Theorem 1.2. To start with, it is more general because it deals with a general product measure on {0, 1} n. It also gives a stronger result about the vector of influences. Let µ q be the measure on the 2-point space {0, 1} such that µ q (1) = q and let µ n q denote the n th-power product of this measure on the product space {0, 1} n. 3

4 Theorem 1.3 (Talagrand, [12]) Consider {0, 1} n as a measure space with the product measure µ q. There exists a constant c such that the following holds: Let f : {0, 1} n {0, 1}. Let Pr({x : f(x) = 1}) = p. Then If (k)/ log(1/i f (k)) cp(1 p)q(1 q). Moving to consider the influence of coalitions, by repeated applications of Theorem 1.1 one can deduce the corollary stated in [KKL]. It states that there always exists a relatively small coalition that has influence close to 1: Theorem 1.4 (KKL2) Let {0, 1} n be a measure space with the uniform measure. For any ε > 0 and for any balanced f : {0, 1} n {0, 1} (i.e. Pr({x : f(x) = 1}) = 1/2) there exists a set J with J = O(log(1/ε)n/ log(n)) such that I f (J) 1 ε. The theorem can be adjusted to the case where f is not balanced, but for simplicity we will stick to the balanced case. We will spell out the exact process of deriving this theorem later in this paper. Now let us consider the case where our base space is [0, 1]. The first theorem in [BKKKL] is the exact analogue of Theorem 1.1: Theorem 1.5 (BKKKL) There exists a constant c such that the following holds: Consider [0, 1] n as a measure space with the uniform measure. Let f : [0, 1] n {0, 1}. Let Pr({x : f(x) = 1}) = p. Then there exists a variable k such that I f (k) cp(1 p) log n n. As before, it is perhaps natural to assume that by repeated application of the above one may deduce a theorem about the influence of a coalition exactly as in the discrete case. Indeed such a theorem is stated in [BKKKL], but, as noted already in [7], this is false. A simple counterexample actually appears in [BKKKL] itself. 1.3 Outline Here is a brief outline of what follows in this paper: First, using Theorem 1.2 we will show how the reasoning presented in [BKKKL] actually can be used to get an extremely simple proof of Theorem 1.5, much simpler than the original one. We then will move on to study influences of coalitions. We will see why the 4

5 continuous case is more complicated than the discrete one. We will present a conjecture as to what should replace the erroneous statement in [BKKKL], present a series of conjectures concerning the discrete cube that would imply the continuous conjecture and prove a new theorem that is a step in that direction. We will conclude with several other related conjectures. 2 From Discrete to Continuous We begin by showing that the main result in [BKKKL], Theorem 1.5 may be derived rather simply from the discrete case. Claim 2.1 Theorem 1.2 implies Theorem 1.5. Proof: Let f : [0, 1] n {0, 1} be a monotone function. We start, as is done in [BKKKL], by discretizing the continuous cube. Let k = 3 log n (one can improve the constants by choosing k more carefully.) Subdivide [0, 1] n into 2 kn equal subcubes by subdividing every one of the base intervals into 2 k equal parts. This partition is fine enough that we may assume for our purposes that f is constant on each of the small subcubes, else replace f by an appropriate approximation. Note that this is valid due to the fact that f is monotone: the number of mixed subcubes is no more than the number of subcubes that touch the boundary of [0, 1] n. Now f corresponds in a natural way to a function on the discrete cube g : {0, 1} kn {0, 1}, by replacing the interval [r2 k, (r + 1)2 k ] with the binary expansion of r. Every variable i {1,...,n} is now replaced by k variables i j : 1 j k. The following simple key observation was made in [BKKKL]: i j I g (i j ) 2I f (i). (1) This together with Theorem 1.2 immediately yields the required result: Assume that for all i I f (i) c 1, and hence by (1) log n n i,j I g (i j ) 2c 1 log n n. This implies by Theorem 1.2 that I g (i j ) c 2 log(n)p(1 p), i,j 5

6 and using (1) again i I f (i) c 2 2 log(n)p(1 p), and in particular there exists a variable i such that I f (i) c 2 2 log n p(1 p). n We now move on to the question of the influence of coalitions. Let us begin by sketching how repeated applications of Theorem 1.1 yield Theorem 1.4: Let f : {0, 1} n {0, 1} be a monotone function with E(f) = Pr(f = 1) = p 1. Let i 1 be a variable with large influence as guaranteed by Theorem 1.4, I f (i 1 ) = α > cp 1 (1 p 1 ) log(n)/n. From the definition of influence it follows that E(f(x) x i1 = 1) = p 1 + α/2 := p 2. Setting x i1 = 1 now gives a new function on n 1 variables with expected value p 2. Repeating the above argument we find a variable i 2 with large influence with respect to the new function, set it to be equal 1, etc. After O(log(1/ε)n/ log(n)) iterations we have a set J 1 of variables such that E(f(x) x j = 1 j J 1 ) > 1 ε/2. In the same manner we may find a set J 0 such that Putting these together shows that E(f(x) x j = 0 j J 0 ) < ε/2. I f (J 0 J 1 ) > 1 ε. Now why is this line of reasoning not valid in the continuous case? Why is it not true that for any function f : [0, 1] n : {0, 1} there exists a small coalition that dominates the function? The reason is that for a monotone function when one sets a variable x i to be equal to 1 the gain in the expectation of the function comes from the mixed fibers in the i th direction. In the discrete case every fiber consists of two points, hence if the measure of the union of the mixed fibers is α one gains α/2. Clearly this is not the case 6

7 in the continuous cube. The simplest counterexample is the following, The Subcube : f(x) = 1 1/n x i 1 i. (2) The expected value of f is (1 1/n) n 1/e. The influence of every variable is quite large - it is (1 1/n) n 1 1/e, however, since the mixed fibers have only a 1/n proportion where f = 0, setting one variable to be equal to 1 has almost no effect on the expectation of the function: E(f(x) x 1 = 1) = (1 1/n) n 1. Indeed if J is a coalition with J = o(n) then E(f(x) x j = 1 j J) = (1 1/n) n o(n) E(f). The astute reader must have noted that, on the other hand, setting even a single variable to be equal to 0 implies f = 0. (This may be compared to many real-life situations where the cooperation of a large coalition is needed to achieve a goal that can easily be foiled by one member defecting.) This suggests introducing the following notions and notations: for a variable j let E j=1 (f) = E(f(x) x j = 1) and define the influence of j towards 1 as Define similarly and I 1 f(j) = E j=1 (f) E(f). E j=0 (f) = E(f(x) x j = 0) I 0 f(j) = E(f) E j=0 (f). Note that I f (j) = If 1(j) + I0 f (j). As in the case of influence we make the analogous definitions for coalitions. If J {1,..., n} Define E J=1 (f) = E(f(x) x j = 1 j J) and similarly define If 1(J), E J=0(f) and If 0 (J). The previous example suggests that perhaps the following conjecture might be the correct continuous version of Theorem 1.4: 7

8 Conjecture 2.2 Let f : [0, 1] n {0, 1}. There exists a coalition J with J = o(n) such that either E J=1 (f) = 1 o(1) or E J=0 (f) = o(1). By following the approach of the proof relying on discretizing the continuous cube, the above conjecture would follow from the following one. Assume that, as in Israeli politics, one is given a set of parties and when building a coalition must either take a whole party or none of its members. Then perhaps it is still possible to build a small coalition, but perhaps with large influence only in one direction: Conjecture 2.3 Let m = nk. For i = 0,...,n 1, and let J i = {ik +1,ik + 2,...,ik + k}. Let f : {0, 1} m {0, 1}. Then there exists S {1,...,n} with S = o(n) such that for J = i S J i either E J=0 (f) = o(1) or E J=1 (f) = 1 o(1). The canonical tribes example (where the parties are the tribes) shows that one cannot strengthen this conjecture by replacing one-sided influence with the usual influence. In fact the tribes example is exactly the dual of what one gets when discretizing the subcube example (Although since we chose the tribes to be rather large by setting k = 3 log n we will get a function where only a minority in every tribe have the right to vote.) These examples are cases where one can build a small influential coalition but it uses representatives from all the parties (tribes). For our purposes it would suffice to prove a weaker conjecture that we state below - since we are interested in functions coming from monotone functions on the continuous cube one may assume that f in the above conjecture has a certain supermonotonicity: Let x,y {0, 1} m. We will say x dominates y if every one of the n blocks of k coordinates in x dominates the corresponding block of coordinates in y. We say that the block in x, dominates the block in y, if the first is the binary representation of an integer b x and the second of b y and b x b y. We will say f is supermonotone if whenever x dominates y, f(x) f(y). Conjecture 2.4 Conjecture 2.3 holds if we add the condition that f is supermonotone. 8

9 Actually an even weaker conjecture would suffice: in attempting to prove Conjecture 2.4 we can follow the derivation of theorem 1.4 as long as there exists, at each stage of the iteration a variable with large influence towards 1. hence we can add to the last conjecture the hypothesis that there does not exist such a variable. This corresponds to the following: Conjecture 2.5 Let k,n,m,j i be as in Conjecture 2.3. Let f : {0, 1} m {0, 1} be supermonotone and balanced, i.e. Pr[f = 1] = 1/2. Assume further that for every i {1,...,n} I 1 f(j i ) = O(log(m)/m) = O(1/n) Then there exists S {1,...,n} with S = o(n) such that for J = i S J i E J=0 (f) = o(1). To understand the above conjecture better, note that its hypothesis is fulfilled by the discretization of the subcube, the dual of the tribes function. Of course, the assumption that f is balanced can be weakened to the asumption that the expectation of f is bounded away from 0 and 1. There is another rationale hiding behind the above conjecture - the idea that the only extremal examples where all variables simultaneously have influence O(log(n)/n) are tribe-type functions. The question of understanding the extremal functions seems to be quite challenging, yet such an understanding would be quite rewarding in the sense that the conjecture below, for example, would imply all the previous ones. Conjecture 2.6 For every c > 0 and ε > 0 there exists b > 0 such that if f : {0, 1} n {0, 1} and for all i I f (i) c log(n)/n. Then there exists a small tribe - i.e. a set J of variables with J b log(n) such that E J=0 (f) ε or E J=1 (f) 1 ε. It seems that the only hope of attacking this is via Fourier analysis. As we see, the simple subcube example inspires many conjectures - and we are not done yet. We will now present a theorem inspired by it and follow it by yet another conjecture concerning subsets of [0, 1] n with small projections. 9

10 Recall that in our function f(x) = 1 1/n x i 1 i no variable has large influence towards 1, but all variables have very large influence - a constant which is approximately 1/e. (This tradeoff is reminiscent of the discrete case where, as stated in Theorem 1.2 if all influences are small their sum must be large.) This tradeoff is capture by the following theorem which states that when all influences towards 1 are small (e.g. much smaller than log(n)/n) then there exists a variable with large influence (much larger than log(n)/n.) Theorem 2.7 Let c = c where c is the constant from Theorem 1.2. Then 6 for any balanced monotone function f : [0, 1] n {0, 1} if for all i then there exists i such that I f (i) α n I 1 f(i) α n ( n ) c /α. 2α The case where all influences towards 1 are o(1/n) is not interesting because this implies that f is very close to the function which is identically 1 (and in particular is not balanced.) On the other extreme α can always be assumed to be at most c log(n), else the assertion of the theorem is that there exists a variable with influence Ω(log(n)/n), which is always true. Now note that up to the value of c the theorem is tight at both ends of the spectrum: for the subcube when all influences towards 1 are of order 1/n and in the tribes example where the influences are of order log(n)/n. Proof: Our basic approach will be, as in the proof of Claim 2.1, to discretize the cube. Assume as usual that f is monotone and as before divide every base interval into k = 3 log(n) equal parts inducing a partition of the cube into 2 kn subcubes and assume f is constant on each small subcube. As before this defines in a natural manner a function g : {0, 1} kn {0, 1}. For i {1,...,n} and x [0, 1] n. Recall the definition of s i (x) the fiber in the i th direction through x : s i (x) = {y : y j = x j for j i}. 10

11 let Z i (x) = 1 E(f(y) y s i (x)). If f is not constant on the fiber s i (x) then Z i (x) is the contribution of this fiber to I 1 f (i). We will make this precise shortly. Let i[0, 1] n be the union of the mixed fibers in the i th direction: Then i [0, 1] n = {x [0, 1] n : f is not constant on s i (x)}. If(i) 1 = Z i (x) i [0,1] n (3) where the integration is with respect to the uniform measure on [0, 1] n. We now want to connect Z i (x) to the contribution of s i (x) to the influences I g (i j ). This combines observations made in [BKKKL] and [7]. Claim 2.8 For all i and j I g (i j ) 2If(i) 1 = 2Z i (x) i [0,1] n (4) and for all i I g (i j ) 6Z i (x) log(1/z i (x)). (5) i [0,1] n j Proof of claim: The claim follows from the one dimensional case where i = 1 is the only variable and there is only one fiber in our space. In this case assume f is monotone so f(x) = 1 x z. Since changing the value of the j th variable in g corresponds to flipping the j th bit in the binary representation of x it is easy to see that for every j I g (1 j ) min(2z, 2 1 j ). This immediately implies (4). To get (5) we sum on j which gives I g (i j ) 2z log(1/z) + 4z 6z log(1/z). j 11

12 Returning to the proof of Theorem 2.7 we know that for all i i [0,1] n Z i (x) = I 1 f(i) α n (6) hence by (4) for all i and j I g (i j ) 2α n. Using Theorem 1.2 we deduce that I g (i j ) c log(n/2α), so there exists a variable i such that I g (i j ) c log(n/2α). n Using (5) we have i,j j 6Z i (x) log(1/z i (x)) c log(n/2α) i [0,1] n n, (7) but from (6) i [0,1] n Z i (x) α n so by concavity of the function z log(1/z) 6Z i (x) log(1/z i (x)) 6 α ( i [0,1] n log npr( i [0, 1] n ) α n Putting (7) and (8) together gives Pr( i [0, 1] n ) α n ( n ) c 6α 2α but Pr( i [0, 1] n ) is exactly the influence of i, so we are done. ). (8) Actually the same proof shows that under the assumptions of Theorem 2.7 the average influence is large: 12

13 Theorem 2.9 Let c = c where c is the constant from theorem 1.2. Let 6 f : [0, 1] n {0, 1} be such that for all i I 1 f(i) α n. Let g : {0, 1} kn {0, 1} be defined as in the proof of Theorem 2.7. Let p i = j Ig(i j) i j I g(i j ) (note that p i = 1.) Then for every i I f (i) α n ( n ) c np i α 2α and from the Arithmetic-Geometric Mean Inequality ( n ) c α If (i) α. 2α 2.1 Projections of Subsets of the Cube All theorems and conjectures in this paper can be interpreted as isoperimetric statements concerning subsets of either the discrete or the continuous cube, and their projections. Given A [0, 1] n let A 1 i be the projection of A onto the face of the cube that lies on the plane x i = 1. Let Vol(A) denote the volume of a set, taken in the appropriate dimension. Let Ḡ 1 i(a) = Vol(A 1 i) Vol(A) measure the growth of A when projected, and G 1 i(a) = Vol(A1 i) Vol(A) Vol(A)(1 Vol(A)) the normalized growth. Theorem 1.5 implies the following: Theorem 2.10 Let [0, 1] n be partitioned into two sets A and A c. Then there exists a direction i such that either G 1 i(a) or G 1 i(a c ) is of order Ω( log n n ). Once again, the example where A is the subcube gives rise to the idea that if G 1 i(a) is small for all i then A is close to being either a subcube or a union of subcubes. However, since the subcube occupies a large proportion 13

14 of every direction one cannot pack two such subcubes disjointly into [0, 1] n. So perhaps it is reasonable to expect that not only is G 1 i(a c ) large, but that this holds for any substantially large subset of A c. Here is a conjecture in this spirit, probably due to Nati Linial: Conjecture 2.11 Let [0, 1] n be partitioned into k subsets A 1,...,A k. Then there exists a direction i such that for all but at most one of the A j s ( ) log n G 1 i(a j ) = Ω. n 2.2 Continuous Juntas In [6] Boolean functions on {0, 1} n are studied in the case where the total sum of the influences is bounded. An example of such a function is the dictatorship: f(x) = x 1. For the dictatorship I f (1) = 1 and I f (j) = 0 for j 1. It turns out that whenever j I f(j) B then for every ε > 0 there exists a junta - a set J of size exp(cb/ε) and a function g that depends only on the coordinates in J such that f g 2 2 ε. In ongoing work of the author with Irit Dinur [5] the analogous case for functions f : {1, 2,...,r} n {0, 1} is studied. We have proven the analogous result in the case where f is monotone, however for junta-type theorems one cannot assume monotonicity, for, unlike influence, the control of the junta is not preserved under a shifting procedure that makes a function monotone. Since a few of the conjectures we have raised are very closely related to those in this paper we will end by quoting some results and conjectures. Consider {1,...,r} n endowed with the uniform measure. Theorem 2.12 There exists a constant c > 0 such that the following holds. Let ε > 0 and f : {1,...,r} n {0, 1} be a monotone function. Assume that I f (j) B. j Then there exists a set J {1,..., n} with J exp(cb/ε) and a function g : {1,...,r} n {0, 1} depending only on the coordinates in J such that f g 2 2 ε. Of course, as we have seen in this paper, this implies the same for monotone Boolean functions on the continuous cube. In the non-monotone case a 14

15 weaker result holds, where the size of the junta depends also on r: under the same assumptions, sans monotonicity, it can be shown that there exists a set J as above such that J exp(cb log r/ε). We actually conjecture that the monotonicity is not a necessary condition: Conjecture 2.13 Theorem 2.12 holds for any function f : {1,...,r} n {0, 1} regardless of monotonicity. This would follow from the following conjecture about functions on {0, 1} n which is reminiscent of Conjecture 2.3. We conjecture that in the case of the coordinates being partitioned into parties the junta can be composed from a bounded number of parties, regardless of their size: Conjecture 2.14 There exists a constant c such that the following holds. Let m = nk. For i = 0,...,n 1 Let J i = {ik + 1,ik + 2,...,ik + k}. Let f : {0, 1} m {0, 1} and assume I f (J i ) B. i Then for every ε > 0 there exists S {1,...,n} with S exp(cb/ε) and a function g : {0, 1} m {0, 1} depending only on the coordinates in i S J i such that f g 2 2 ε. 3 Acknowledgments It is a pleasure to thank the second K (Gil Kalai) and L (Nati Linial) for innumerable discussions of this topic throughout the years, from which all the ideas in this paper have stemmed. I would also like to thank Noga Alon, Irit Dinur, the first K (Jeff Kahn) and Avi Wigderson for useful conversations and remarks. Also, I wish to thank the anonymous referee for various remarks. References [1] W. Beckner, Inequalities in Fourier analysis. Annals of Mathematics, 102 (1975),

16 [2] I. Benjamini, G. Kalai, O. Schramm, Noise Sensitivity of Boolean Functions and Applications to Percolation, Inst. Hautes Etudes Sci. Publ. Math. No. 90, (1999), 5 43 (2001). [BKKKL] J. Bourgain, J. Kahn, G. Kalai, Y. Katznelson and N. Linial, The influence of variables in product spaces, Israel J. Math. 77(1992), [3] M. Ben-Or and N. Linial, Collective coin flipping, in Randomness and Computation (S. Micali, ed.), Academic Press, New York, 1990, pp Earlier version: Collective coin flipping, robust voting games, and minima of Banzhaf value, Proc. 26th IEEE Symp. on the Foundation of Computer Science, 1985, pp [4] Aline Bonami, Etude des coefficients Fourier des fonctiones de L p (G), Ann. Inst. Fourier (Grenoble) 20:2 (1970), [5] I. Dinur and E. Friedgut, Continuous Juntas, in Preparation. [6] E. Friedgut, Boolean functions with low average sensitivity depend on few coordinates, Combinatorica Vol 18 (1) 1998 pp [7] E. Friedgut, G. Kalai, Every monotone graph property has a sharp threshold, Proc. Amer. Math. Soc. 124 (1996), pp [KKL] J. Kahn, G. Kalai, and N. Linial, The influence of variables on Boolean functions, Proc. 29-th Ann. Symp. on Foundations of Comp. Sci., 68-80, Computer Society Press, [8] G. Kalai A Fourier-Theoretic Perspective for the Condorcet Paradox and Arrow s Theorem, to appear in Advances in Applied Mathematics. [9] Lehrer, E. An axiomatization of the Banzhaf value. Internat. J. Game Theory 17 (1988), no. 2, [10] M. Talagrand, How much are increasing sets correlated? Combinatorica 16, 1996, [11] M. Talagrand On boundaries and influences, Combinatorica 17, 1997, [12] M. Talagrand, On Russo s approximate 0-1 law, Annals of Probability, 1994 Vol 22, No