Lecture 7. The Cluster Expansion Lemma

Transcription

1 Stanford Universit Spring 206 Math 233: Non-constructive methods in combinatorics Instructor: Jan Vondrák Lecture date: Apr 3, 206 Scribe: Erik Bates Lecture 7. The Cluster Expansion Lemma We have seen the Lovász Local Lemma and its stronger variant, Shearer s Lemma, which is unfortunatel quite unwieldn applications. Quite recentl, researchers in mathematical phsics discovered an intermediate form of the lemma, which seems to give results close to Shearer s Lemma but it is much more easil applicable. First let us review a connection between Shearer s Lemma and statistical phsics which inspired this development. 7. The hard core model For a graph G on n vertices, the hard core repulsive gas model is a model where particles can appear on the vertices of G, in such a wa that neighboring vertices are not simultaneousl occupied. (G can be arbitraril but tpicall, regular lattice graphs are of interest in phsics.) The set of admissible configurations is thus identified with the independent sets of G. There is a fugacit parameter λ v associated with each vertex v, and the probabilit of a configuration I Ind(G) (that is, particles are placed exactl at the vertices in I) is where Z G is the partition function P(I) = λ v, Z G Z G (λ, λ 2,..., λ n ) = v I λ v. v I Observe that Z G is the multivariate generating function of independent sets, just like Shearer s polnomials (except for the alternating signs). An object of interest is the Talor expansion of log Z G around 0, also called the Maer expansion. Various sufficient conditions for the absolute convergence of this series have been identified over the ears. It eventuall became apparent that these conditions are related to the local lemmas we have discussed. In particular, a condition presented in (Dobrushin, 996) is: The Maer expansion is absolutel convergent for all λ i p i, if there are, 2,..., n > 0 such that p i i =, 2,..., n. ( + j ) If we substitute = x i x i, x i (0, ), we get the LLL condition: ( + j ) = x i x i = x i x j ( x j ). j Γ(i)

2 (Scott and Sokal, 2005) clarified the connection between the Maer expansion and the LLL: The Maer expansion is absolutel convergent for all λ i p i if and onlf Shearer s conditions are satisfied for p, p 2,..., p n and the graph G. Hence Dobrushin s result follows from the fact that LLL conditions impl Shearer s conditions. 7.2 Cluster expansion conditions The cluster expansion lemma (Bissacot et al., 20) is an intermediate form of the LLL, stronger than the LLL but weaker than Shearer s Lemma. Hence, we have the following ordering from weaker to stronger:. Smmetric LLL 2. Asmmetric LLL 3. Cluster expansion local lemma (CLL) 4. Shearer s Lemma Theorem 7. (CLL) Let E, E 2,..., E n dependenc graph G, and If there exist, 2,..., n > 0 such that be events on some probabilit space, with a (negative) P(E i ) p i. p i I Γ + (i) j I i =, 2,..., n, (7.) j then ( n ) P E i > 0. i= We can make a quick comparison to see that this result is, in fact, stronger than the LLL: I Γ + (i) j I j S Γ + (i) j S = j ( + j ), where the latter quantits the bound required in the LLL, as discussed above. Hence, the gain in CLL is that we sum up onl over independent subsets of the neighborhood, as opposed to all subsets. The cluster expansion lemma was originall proved in (Bissacot et al., 20) b analtic arguments involving the convergence of the Maer expansion. Here we present a combinatorial proof from (Harve and Vondrak, 205). We will show that Shearer s conditions are implied b the hpotheses. This wa we obtain a relationship between Shearer s coefficients and the coefficients in CLL. We note that if one wishes to avoid Shearer s Lemma, one can replace q S b P S = Pr[ i S E i]

3 and obtain a self-contained proof essentiall without an change (onln (7.3) we get an inequalit instead of an equalit). Proof of Theorem 7.. For S V = V (G) = [n], define Y S := I =. i I With this notation, the CLL assumption (7.) reads p i Y Γ + (i). (7.2) We make two observations: First, recall from Lectures 5 and 6 the identit An analogous identit for Y S also holds: Indeed, Y S = I = I (S a) I + q S = p a q S\Γ + (a). (7.3) Y S = Y S a + a Y S\Γ + (a). (7.4) a I I = Y S a + a \Γ + (a) I = Y S a + a Y S\Γ + (a). Second, observe that Y S is sub-multiplicative in the following sense: If S T =, then Y S T = I I J = I J. = Y S Y T T I T I T The inequalit holds because if I Ind(G), then I S and I T are also independent. We will show bnduction on S that q S Y V \S S V, a S. (7.5) Note that Y is indexed b the sets complementar to the indices of q. Thus the sets indexing Y shrink in the induction, while the sets indexing q grow. The reason for this can be traced back to (7.4) which contains a sign opposite to (7.3). For the base case of S = V, we use (7.4) and (7.2) to write Y V = Y V a + a Y V \Γ + (a) Y V a + p a Y Γ + (a)y V \Γ + (a) Y V a + p a Y V. The last inequalits a result of the sub-multiplicative propert shown above. Hence Y V a Y V p a = q a q.

4 Now assuming the inductive hpothesis, we check the general case: Hence = Y V \S + a Y (V \S)\Γ + (a) Y V \S + p a Y Γ + (a)y (V \S)\Γ + (a) Y V \S + p a Y (V \S) Γ + (a). Y V \S p a Y(V \S) Γ+ (a) p a q S\Γ+ (a) = q S, where the second inequalits the result of iterated applications of (7.5) (for lower cases of S ): If Γ + (a) = {a, a, a 2,..., a k }, then Y (V \S) Γ + (a) = Y (V \S) Γ+ (a) Y (V \S) Γ + (a) a k Y(V \S)+a+a Y (V \S) Γ + (a) a k Y (V \S) Γ + (a) a k a k q S\Γ + (a) q (S\Γ + (a))+a k = q S\Γ + (a). q (S\Γ + (a))+a k q (S\Γ + (a))+a k +a k a Now that (7.5) has been verified, we have (b another telescoping product) q S q Y Y S T S = T ( + ) > 0. Shearer s conditions are thus satisfied, and so we ma conclude P ( n i= E i) > Applications of cluster expansion In practice, cluster expansion is applicable similarl to the LLL, but it gives better constants Latin transversals Recall the problem of Latin transversals from Lecture 4. For a matrix A Z n n, we sampled a permutation π S n uniforml at random. The events to be avoided were i S so that E i,j,i 2,j 2 = {π S n : π(i ) = j, π(i 2 ) = j 2 }, i i 2, j j 2, A i j = A i2 j 2 p := P(E i,j,i 2,j 2 ) = n(n ). A negative dependenc graph G was defined b { } E i,j,i 2,j 2, E i,j E(G) {i,i 2,j 2, i 2 } {i, i 2} or {j, j 2 } {j, j 2}.

5 If evernteger appears as an entr of A at most k times, then it was argued that each event has degree in G of at most 4(n )k. In particular, for a 4-tuple a = (i, j, i 2, j 2 ), Γ(a) is a union of four cliques, each of size at most nk. This means the independent subsets of Γ(a) consist of either 0 or vertex from each clique. In order to appl Theorem 7., then, it suffices that n(n ) ( + nk) 4. To determine the optimal value of, we differentiate and solve That is, we can have ( + nk) 4 4nk( + nk) 3 = 0 p /(3nk) 33 ( + = 3 ) nk = 4nk = 3nk. nk = nk, meaning we need k 27 (n ). 256 This is an improvement of the requirement derived when using the LLL: k n 4e Multipartite Turán problem Here we can obtain a result almost as good as the one found using Shearer s Lemma. Recall from Lecture 6 that the dependenc graph was the line graph D = L(K r ). For an edge {i, j} in K r (i.e. a vertex in D), Γ({i, j}) is the union of two independent cliques, those edges of K r incident to i, and those edges incident to j. Hence I Γ + ({i,j}) I + ( + (r 2)) 2 = +, ( + (r 2))2 where we have considered the independent subsets formed b taking just {i, j} or b taking either 0 or element from each of the two disjoint cliques of size r 2 (there remain r 2 edges incident to i, and the same is true for j). Once again, we optimize b solving 2 ( + (r 2))2 + 2(r 2) ( + (r 2)) = 0 + (r 2) = 2(r 2) = r 2.

6 We can thus take p + ( + (r = 2))2 + 4(r 2) = 4r 7. Recall that the bound we obtained from the Heilmann-Lieb theorem was p 4r 8, and using more refined information about the roots of Hermite polnomials, it was enough to assume that p. So the bound from CLL is ver close to what we can get from Shearer s Lemma. 4r Θ(r /3 ) References Rodrigo Bissacot, Roberto Fernández, Aldo Procacci, and Benedetto Scoppola. An improvement of the lovász local lemma via cluster expansion. Combinatorics, Probabilit and Computing, 20 (05):709 79, 20. RL Dobrushin. Estimates of semi-invariants for the ising model at low temperatures. Translations of the American Mathematical Societ-Series 2, 77:59 82, 996. Alexander D Scott and Alan D Sokal. The repulsive lattice gas, the independent-set polnomial, and the lovász local lemma. Journal of Statistical Phsics, 8(5-6):5 26, Nicholas Harve and Jan Vondrak. An algorithmic proof of the Lovász Local Lemma via resampling oracles