Sharp Thresholds. Zachary Hamaker. March 15, 2010

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1 Sharp Threshols Zachary Hamaker March 15, 2010 Abstract The Kolmogorov Zero-One law states that for tail events on infinite-imensional probability spaces, the probability must be either zero or one. Behavior analogous in a natural sense to this exists on finite-imensional spaces as well. Events exhibiting this behavior are sai to have a sharp threshol. Let Q n = {0, 1} n be the iscrete hypercube with the probability measure P p efine by P p (ω 1, ω 2,..., ω n ) = p k (1 p) n k where k = ω 1 + ω ω n. An up event A Q n is symmetric if there exists a transitive permutation group on the inices uner which A is invariant. We present the following result, ue to Friegut an Kalai: Theorem 0.1. For every symmetric up-event A Q n, if P p (A) > ɛ, then P q (A) > 1 ɛ for q = p + c log(1/2ɛ)/ log n, where c is an absolute constant (it oes not epen on any of the other terms). 1 Introuction When the theory of ranom graphs was first evelope by Erös an Rényi, they notice that many graph properties exhibite a peculiar property. Their first such encounter was shown in [3], where they emonstrate that for the ranom graph G(n, p) (the graph of n vertices where each is connecte i.i.. with probability p), if p < (1 ɛ) log n/n the graph will almost surely contain an isolate vertex an if p > (1 + ɛ) log n/n, the entire graph will almost surely be connecte. Many other graph properties on ranom graphs were foun to exhibit such behavior, which we refer to as a sharp threshol. In Figure 1, we see an example of a sharp threshol. Here, the thick lines 0 an 1 correspon to the probability of a tail event with critical probability p. The otte lines show the continuous ensity function of the probability with respect to p for some given n. Here, δ relies on p, n an ɛ. In general, we can sen δ to zero as n goes off to infinity, regarless of the values of ɛ an p. 1

2 Figure 1. Since these first observations, many people have further stuie sharp threshols. They have been emonstrate for a multitue of events. Over the years, the bouns which etermine how sharp they are have also improve. In 1996, Friegut an Kalai emonstrate that every symmetric up-event has a sharp threshol [6] an provie optimal bouns up to a constant factor. Although not the final wor in the fiel, the repercussions of this result are strong. Applications of sharp threshols aboun. For many problems relating to ranom graphs, Boolean algebras an other fiels, sharp threshols serve a valuable purpose. If the problem can be shown to be easy when a relate event is almost always or almost never true, then we can confine our attention to the narrow ban aroun some particular value of p. Of course, for our purposes the most important application of sharp threshols relates to percolation. They are at the center of one of the easier approaches to proving Kesten s theorem, which states that P H 1/2. A proof using the result state below can be foun in [2]. We now start eveloping the tools necessary to prove the sharp threshol theorem of Friegut an Kalai. 2 Preliminaries Let A be an event in the hypercube Q n. For ω Q n, the ith variable ω i is pivotal if precisely one of ω = (ω 1, ω 2,..., ω n ) an ω = (ω 1, ω 2,..., 1 ω i,..., ω n ) is in A. Note that whether the ith coorinate is pivotal epens both on the point ω an the event A. The influence of the 2

3 ith variable on A is β i (A) = P p ({ω Q n : ω i is pivotal for A}). The following lemma, first prove by Margulis in 1974 an reiscovere in 1981 by Russo, states the erivative of P p (A) with respect to the influence. Recall A Q n is an up-event if for ω = (ω 1,..., ω n ) A an α i ω i for all i [n], we have α = (α 1,..., α n ) A as well. Lemma 2.1. Let A Q n be an up-event. Then p P p(a) = n β i (A). i=1 This result follows from the fact that as A is an up-event, we can show that for each inex P p is a constant unless the inex is pivotal, in which case it takes the value pβ i (A). A proof can be foun in [2] p The next result we nee is ue to Kahn, Kalai an Linial [7]. Theorem 2.2. Let A Q n p that with P p (A) = t. Then there exists an absolute constant c such max β i (A) ct(1 t) log n/n. i When applying Theorem 2.2, we will pull the larger of t an (1 t) insie of our constant as it will be greater than or equal to 1/2. In this way, we only nee to concern ourselves with the smaller term. No proof shall be presente, as it relies on finite-imensional Fourier analysis. While the techniques use are efinitional with the exception of Parseval s ientity an several inequalities ue to Beckner (cite in [7]), it is too involve to present here. A strictly combinatorial proof, also too intricate to be shown here, was presente by Falik an Samoronitsky in 2005 [4]. We note that both of the above results can be extene. The proof of Lemma 2.1 allows for each inex to have its own value p i. The extension of Theorem 2.2 in 1992 by Bourgain, Kahn, Kalai, Katznelson an Linial [1] is more substantive. Instea of being prove on Q n, BKKKL states that above hols for an arbitrary n-imensional probability space. A greatly simplifie proof was presente in 2004 by Friegut [5]. Let A Q n be an up-event. We say A is symmetric if there is a transitive permutation group Γ on [n] such that A is invariant uner Γ. In the case that the event A is symmetric, 3

4 the influence of each inex will be ientical. We then enote the influence β(a). We are now prepare to prove the sharp threshol theorem of Friegut an Kalai for symmetric up-events. 3 Result The following proof is as presente in [6]. Theorem 3.1. For every symmetric up-event A Q n with 0 < ɛ < 1/2 an P p (A) < ɛ, we have P q (A) > 1 ɛ for q p + c log(1/(2ɛ)) log n where c is an absolute constant. Proof. Let A be a symmetric up-event. Then the influence of any two inices is the same, with each inex having influence β(a). Using Lemma 2.1 an Theorem 2.2 with c 1 as in the latter, we compute the following lower boun Therefore, r P r(a) = nβ(a) nc 1 P r (A) log n n = c 1P r (A) log n. r log P r(a) = P r r(a) P r (A) c 1P r (A) log n P r (A) For p such that P p (A) ɛ, efine q = p + log(1/(2ɛ)). Then c 1 log n log(p q (A)) log(p p (A)) + q p = c 1 log n. (1) c 1 log nr log(ɛ) + log(1/(2ɛ)) = log(1/2). We are now half of the way there. The rest of the way uses the same approach, with one slight variation. Then For 1/2 P r (A) < 1 ɛ, our lower boun changes as follows: r P r(a) = nβ(a) nc 1 (1 P r (A)) log n n = c 1(1 P r (A)) log n. r log(1 P r(a)) = (1 P r r(a)) (1 P r (A)) c 1(1 P r (A)) log n (1 P r (A)) = c log n. Note the irection of the inequality changes compare to Equation 1 as we are now taking the erivative of P r (A). Then by efining q = q + log(1/(2ɛ)) c 1 log n log(1 P q (A)) log(1 P q )(A)) q q, we see Thus for c = 2c 1 an q = p + log(1/(2ɛ)) c log n we have P q (A) > 1 ɛ. c 1 log n r log(1/2) log(1/(2ɛ)) = log(ɛ). 4

5 The above result can be generalize in several ways. When p oes not rely n, the above boun is sharp except for improvements upon the constant c. If p ecreases with n, then the boun can be improve by using the value q = p + cp log(1/p) log(1/(2ɛ)). It can also be aapte log n to probabilities with a finite number of possible values, so long as all but one of these values has appropriately small probability. Further generalizations can be foun in [6]. References [1] J. Bourgain, J. Kahn, G. Kalai, Y. Katznelson an N. Linial (1992), The influence of variables in prouct spaces, Israel Jour. Math. 77, [2] B. Bollobás an O. Rioran (2006), Percolation, Cambrige U. Press, Cambrige UK. [3] P. Erös an A. Rényi, On the evolution of ranom graphs, (1960) Publ. Math. Inst. Hungar. Aca. Sci. 5, [4] D. Falik an A. Samorognitsky (2007), Ege-isoperimetric inequalities an influences, Comb. Prob. Comput. 16, [5] E. Friegut (2004), Influences in prouct spaces: KKL an BKKKL revisite, Comb. Prob. Comput. 13, [6] E. Friegut an G. Kalai (1996), Every monotone graph property has a sharp threshol, Proc. Amer. Math. Soc. 124, [7] J. Kahn, G. Kalai an N. Linial (1988), The influence of variables on Boolean functions, Proc. 29-th Ann. Symp. on Founations of Comp. Sci., 68 80, Computer Society Press 5