Reconstruction Threshold for the Hardcore Model

The reconstruction problem on the tree was originally studied as a problem in statistical physics but has since found many applications including in computational phylogenetic reconstruction [8], the study of the geometry of the space of random constraint satisfaction problems [1,13] and the mixing time of Markov chains [5,16]. For a Markov model on an infinite tree the reconstruction problem asks when do the states at level n provide non-trivial information about the state at the root as n goes to infinity. In general the problem involves determining the existence of solutions of distribution valued equations and as such exact thresholds are known only in a small number of examples [4,10,5,23]. In this paper we analyze the reconstruction problem for the hardcore model on the k-regular tree, where each vertex of the tree has degree k. The hardcore model is a probability distribution over independent sets I weighted proportionally to λ |I| . Previously Brightwell and Winkler [7] showed that reconstruction is possible when λ > (e + o(1)) ln 2 k. Improving on their bound for the nonreconstruction regime, Martin [15] showed that non-reconstruction holds when λ < e -1 still leaving a wide gap between the two thresholds. Our main result establishes that the bound of Brightwell and Winkler is tight up to a ln ln k multiplicative factor. Theorem 1 The hardcore model on the k-regular tree has non-reconstruction when λ < (ln 2 -o(1)) ln 2 k 2 ln ln k . For a finite graph G the independent sets I(G) are subsets of the vertices containing no adjacent vertices. The hardcore model is a probability measure over σ ∈ I(G) ⊂ {0, 1} G such that where λ is the fugacity parameter and Z is a normalizing constant. The definition of the hardcore model can be extended to infinite graphs by way of the Dobrushin-Lanford-Ruelle condition which essentially says that for every finite set A the configuration on A is given by the Gibbs distribution given by a random boundary generated by the measure outside of A. Such a measure is called a Gibbs measure and there may be one or infinitely many such measures (see e.g. [11] for more details). For every λ, there exists a unique translation invariant Gibbs measure on the k-regular tree and it is this measure which we study. An alternative equivalent formulation of the hardcore model is as a Markov model on the tree. An independent set σ is generated by first choosing the root according to the distribution for some 0 < ω < 1. The states of the remaining vertices of the graph are generated from their parents' states by taking one step of the Markov transition matrix . It can easily be checked that π is reversible with respect to M and that this generates a translation invariant Gibbs measure on the tree with fugacity Restating Theorem 1 in terms of ω we have non-reconstruction when We will introduce some further notation which we will make use of in the proof. A particularly important role is played by θ, the second eigenvalue of M as is discussed in the following subsection. We denote by P 1 T , E 1 T (and resp. P 0 T , E 0 T and P T , E T ) the probability and expectations with respect to the measure obtained by conditioning on the root ρ of T to be 1 (resp. 0, and stationary). We let L = L(n) denote the vertices at depth n and σ(L) = σ(L(n)) denote the configuration on level n. We will write Pr T [•|σ(L) = A] to denote the measure conditioned on the leaves being in state A ∈ {0, 1} L(n) . The reconstruction problem on the tree essentially asks if we can recover information on the root from the spins deep inside the tree. In particular we say that the model has non-reconstruction if Pr in probability as n → ∞, otherwise the model has reconstruction. Equivalent formulations of non-reconstruction are that the Gibbs measure is extremal or that the tail σ-algebra of the Gibbs measure is trivial [21]. It follows from Proposition 12 of [20] that there exists a λ R such that reconstruction holds for λ > λ R and non-reconstruction holds for λ < λ R . The reconstruction problem is to determine the threshold λ R . A significant body of work has been devoted to the reconstruction problem on the tree by probabilists, computer scientists and physicists. The earliest such result is the Kesten-Stigum bound [14] which states that reconstruction holds whenever θ 2 (k -1) > 1. This bound was shown to be tight in the case of the Ising model [4,10] where it was shown that non-reconstruction holds when θ 2 (k -1) ≤ 1. Similar results were derived for the Ising model with small external field [2] and the 3-state Potts model [23] which constitute the only models for which exact thresholds are known. On the other hand, at least when k is large, the Kesten-Stigum bound is known not to be tight for the hardcore model [7]. As such, the most one can reasonably ask to show is the asymptotics of the reconstruction threshold λ R (k) for large k. The Kesten-Stigum bound is known to be the correct bound for robust reconstruction for all Markov models [12]. Robust reconstruction asks whether reconstruction is possible after adding a large amount of noise to the spins in level n. It was shown in [12] that when θ 2 (k -1) < 1 after adding enough noise to the spins at level n, the "information" provided by the modified spins at level n decays exponentially quickly. In both the colouring model and the hardcore model the reconstruction threshold is far from the Kesten-Stigum bound for large k. In the case of the hardcore model As such, given a noisy version of the spins at level n, the information on the root decays rapidly as n grows. In the colouring model close to optimal bounds [3,22] were obtained by first showing that, when n is small, the information on the root is sufficiently small. Then a quantitative version of [12] establishes that the information on the root converges to 0 exponentially quickly. The hardcore model behaves similarly. Indeed, the form of our bound in equation ( 2) is strikingly similar to the bound for the q-coloring model which states that reconstruction (resp. non-reconstruction) holds when the degree is at least (resp. at most) q[ln q + ln ln q + O(1)]. Our proof then proceeds as follows. We first establish that when ω satisfies (2) then even for a tree of depth 3 there is already significant loss of information of the spin at the root. In particular we show that if the state of the root is 1 then the typical posterior probability that the state of the root is 1 given the spins at level 3 will be less than 1 2 . The result is completed by linearizing the standard tree recursion as in [5,23]. In this part of the proof we closely follow the notation of [5] who analyzed the reconstruction problem for the Ising model with small external field. We do not require the full strength of their analysis as in our case we are far from the Kesten-Stigum bound. We show that a quantity which we refer to as the magnetization decays exponentially fast to 0. The magnetization provides a bound on the posterior probabilities and this completes the result. The reconstruction problem plays a deep role in the geometry of the space of solutions of random constraint satisfaction problems. While for problems with few constraints the space of solutions is connected and finding solutions is generally easy, as the number of constraints increases the space may break into exponentially many small clusters. Physicists, using powerful but nonrigorous "replica symmetry breaking" heuristics, predicted that the clustering phase transition exactly coincides with the reconstruction region on the associated tree model [18,13]. This picture was rigorously established (up to first order terms) for the colouring and satisfiability problems [1] and further extended to sparse random graphs by [19]. As solutions are far apart, local search algorithms will in general fail. Indeed for both the colouring and SAT models, no algorithm is known to find solutions in the clustered phase. It has been conjectured to be computationally intractable beyond this phase transition [1]. The associated CSP for the hardcore model corresponds to finding large independent sets in random k-regular graphs. The replica heuristics again predict that the space of large independent sets should be clustered in the reconstruction regime. Specifically this refers to independent sets of size sn where s > π 1 (R), the density of 1's in the hardcore model at the reconstruction threshold. It is known that the largest independent set is with high probability (2-o(1)) ln k k n [6]. On the other hand the best known algorithm finds independent sets only of size n which is equal to π 1 (R)n [25]. This is consistent with the physics predictions and it would be of interest to determine if the space of independent sets indeed exhibits the same clustering phenomena as colourings and SAT at the reconstruction threshold. Determining the reconstruction threshold more precisely thus has implications for the problem of finding large independent sets in random graphs. The reconstruction threshold plays a key role in the study of the rate of convergence of the Glauber dynamics markov chain for sampling spin systems on trees. This problem has received considerable attention (see e.g. [2,9,16,17,24]) and in the case of the Ising model, the mixing time is known to undergo a phase transition from θ(n ln n) in the non-reconstruction regime to n 1+θ (1) in the reconstruction regime [2]. In fact, the mixing time is n 1+θ(1) for any spin system above the reconstruction threshold. A similar transition was shown to take place for the colouring model [24]. Sharp bounds of this type are not known from the hardcore model, however, it is predicted that the Glauber dynamics should again be O(n log n) in the non-reconstruction regime. It is simple to show that non-reconstruction on the k-regular tree is equivalent to non-reconstruction on the (k -1)-regular tree. For ease of notation we establish our bounds for the k-ary tree noting that in equation ( 2) we have that ω(k + 1) -ω(k) = o(k) so the difference can be absorbed in the error term. Let T denote the infinite k-ary tree and let T n denote the restriction of T to its first n levels. Before reading further, it might help the reader to quickly recall the notation from the end of Section 1.1. As in [5] we analyse a random variable X which denotes weighted magnetization of the root which is a function of the leaf states of the tree. We define X = X(n) on T n by Since E T [P[σ ρ = 1|A]] = P[σ ρ = 1] = π 1 , from the above expression, we have that E[X] = 0. Also, X ≤ 1 since P[σ ρ = 1|A] ≤ 1. We will make extensive use of the following second moments of the magnetization. With these definitions in hand, by the definition in (3) we can characterize non-reconstruction as follows. Proposition 2.1 Non-reconstruction for the model (T , M ) is equivalent to where In the remainder of the proof we derive bounds for X. We begin by showing that already for a 3 level tree, X becomes small. Then we establish a recurrence along the lines of [5] that shows that once X is sufficiently small, it must converge to 0. As this part of the derivation follows the calculation in [5] we will adopt their notation in places. Non-reconstruction is then a consequence of Proposition 2.1. In the next lemma we determine some basic properties of X. The following relations hold: Proof: Note that for any random variable which depends only on the states at the leaves, f = f (A), we have Parts a) and b) therefore follow since X is a random variable that is a function of the states at the leaves. For part c) we proceed as follows. The first and last equalities below follow from ( 4). The second part of c) follows by combining this with a). The following proposition estimates typical posterior probabilities which we will use to bound X. For a finite tree T let T i be the subtrees rooted at the children of the root u i . Proposition 2.3 For a finite tree T we have that a) For any configuration at the leaves b) Let A be the set of leaf configurations . Then c) Let β > ln 2 -ln ln 2 and ω = 1 k ln k + ln ln k -ln ln ln k -β . Then in the 3 level k-ary tree T 3 we have that Proof: Part a) is a consequence of standard tree recursions for Markov models established using Bayes rule. For part b) first note that Now, where the first and third equations follow by definition of conditional probabilities and the second follows from (5) which establishes b). For part c), we start by calculating the probability of certain posterior probabilities for trees of small depth. Note that with our assumption on ω we have that λ = ω(1 + ω) k = e -β ln 2 k ln ln k By part a), since σ(L) ≡ 1 under P 1 we have that Also, Using the two equations above, we have that w.p. k Applying part a) to a tree of depth 2, we have By part b) with A as defined, and ( 6) we have that after substituting the expressions for λ and ω, We can now calculate the values of P 1 T3 [σ ρ = 0|σ(L)] as follows. By part a) By Chernoff bounds, and the bound on p from (7), Finally, by the definition of A, and hence, By taking k large enough above, we conclude that for β and large enough k, Proof: By part c) of Lemma 2.2, and part c) of Proposition 2.3, Next, we present a recursion for X and complete the proof of the main result. The developement of the recursion follows the steps in [5] closely so we follow their notation and omit some of the calculations in this short version. With T and x as defined previously, let y be a child of x and let T be the subtree of T rooted at y (see Figure 1). Let A be the restriction of A to the leaves of T . Let Y = Y (A ) denote the magnetization of y. Note: In the above construction, the vertex y is a vertex "at the same level" as x, and not a child of x as it was in Lemma 2.5. Lemma 2.6 With the notation above, Ŷ = θZ. The proof follows by applying Bayes rule, the Markov property and Lemma 2.2. These facts also imply that Lemma 2.7 For any tree T , With these lemmas in hand we can use the following relation to derive a recursive upper bound on the second moments. We will use the expansion Taking r = π 01 Y Ŷ , by Lemma 2.7 we have where the last inequality follows since X ≤ 1 with probability 1. Let ρ = Y 1 /Y and ρ = Z 1 /Z. Below, the moments Y etc. are defined according to the appropriate measures over the tree rooted at y (i.e. T ) etc. By applying Lemmas 2.2, 2.5 and 2.6, we have the following relations. Applying (π 01 ) -1 E 1 T [•] to both sides of (8), we obtain the following. where and If A-∆B ≥ 0, this would already give a sufficiently good recursion to show that X(n) goes to 0, so we will assume is negative and try to get a good (negative) lower bound. First note that by their definition ρ , ρ ≥ 0. Further since Similarly, ρ ≤ (π 1 ) -1 = 1 + 2ω ω . Since E 1 T [ Ŷ 2 ] and Z ≥ 0, it follows from ( 9) that (1 -θ) + θρ ≥ 0. Together with the fact that ρ ≥ 0, this implies that B ≤ 1. Since A is multi-linear in (ρ , ρ ), to minimize it, its sufficient to consider the extreme cases. When ρ = 0, A is minimized at the upper bound of ρ and hence A ≥ 1 -π 01 ω 1 + ω = 0. When ρ = (π 1 ) -1 , Hence, we have Applying this recursively to the tree, we obtain the following recursion for the moments. We bound the (1 + x) k -1 term as, When ω = 1 k ln k + ln ln k -ln ln ln k -β and β > ln 2 -ln ln 2, by Lemma 2.4, for k large enough, X(3) ≤ ω 2 . Hence by equation (10) we have that X(n) → 0 and so by Proposition 2.1 we have non-reconstruction. Since reconstruction is monotone in λ and hence in ω it follows that we have non-reconstruction for ω ≤ ω for large k. This completes the proof of Theorem 1.