Self-Diffusion Coefficient in the Kob-Andersen Model

The Kob-Andersen model is a fundamental example of a kinetically constrained lattice gas, that is, an interacting particle system with Kawasaki type dynamics and kinetic constraints. In this model, a particle is allowed to jump when sufficiently many neighboring sites are empty. We study the motion of a single tagged particle and in particular its convergence to a Brownian motion. Previous results showed that the path of this particle indeed converges in diffusive time-scale, and the purpose of this paper is to study the rate of decay of the self-diffusion coefficient for large densities. We find upper and lower bounds matching to leading behavior.


Introduction
Kinetically constrained lattice gases is a family of models divised by physicists in order to study glassy systems, that could be seen as the conservative version of kinetically constrained spin models, see e.g. [9,13,6]. In this paper we study one such model -the (k, d)-Kob-Andersen model. It is a Markov process living on the graph Z d , that depends on a parameter k ≥ 2. Each site of Z d may contain at most one particle, that can jump to an empty neighboring site if it has at least k empty neighbors both before and after the jump. When this constraint is satisfied, the particle jumps at rate 1. For any q ∈ (0, 1), this process is reversible with respect to the product Bernoulli measure of parameter 1−q.
When q is small, the constraint is difficult to satisfy, resulting in a significant lengthening of time scales related to this process. Indeed, for a particle to move around it must wait for sufficiently many vacancies to arrive at its vicinity. This fact gives rise to two important length scales of this model -for these vacancies to propagate and reach the particle they must form a droplet of some typical length scale ℓ. In the case k = d = 2, for example, for a droplet to advance it must find a close by vacancy, which is typically possible for ℓ ≈ 1/q. The second scale is the distance L within which such a droplet can be found, i.e., L ≈ q −ℓ , which in the case k = d = 2 is (to leading behavior) e −1/q . For higher values of k and d the mechanism which allows a droplet to move is based on the fact that a d − 1 dimensional layer parallel to the droplet could evolve like a (k − 1, d − 1)-Kob-Andersen model since one of the k required empty neighbors comes from the droplet. Particles are thus allowed to move in this layer if its size reaches the scale L of the (k − 1, d − 1) dynamics, which thus equals ℓ of the (k, d) dynamics. The details of this argument can be found in [17] (see also [14]).
At large times, the path of a marked particle converges to a Brownian motion with a coefficient called the self-diffusion, which is the subject of this paper. In [1] it has been proven that this coefficient is strictly positive for all q ∈ (0, 1), in contrast to the conjecture in the physics literature that below some non-zero critical q the path of tagged particles is no longer diffusive. In this work we find the dependence of this diffusion coefficient in q, showing that it decays very fast when q is small, in a similar way to the spectral gap [11].
We start by introducing the model and our result, and then prove a lower and an upper bound on the diffusion coefficient. The main tool we use is a variation formula of [15] for the diffusion coefficient. In order to bound it from below, as in [1], we compare the Kob-Andersen dynamics with a random walk on an infinite percolation cluster. The upper bound is obtained by identifying an appropriate test function related to the bootstrap percolation, a process which is closely related to the Kob-Andersen model.

Model and main result
The model we study here is defined on the lattice Z d . We denote by (e 1 , . . . , e d ) the standard orthonormal basis. The set of configurations is Ω = {0, 1} Z d , where 0 stands for an empty site and 1 for an occupied site. Given two sites x, y we denote x ∼ y if they are nearest neighbors. We also denote by Fix and integer k ∈ [2, d]. For η ∈ Ω and x ∼ y, we define the local constraint for the edge xy by is entirely occupied elsewhere. The generator of our Markov process describing the KA dynamics operating on a local function f is given by: where η xy is the configuration equal to η except that η xy (x) = η(y) and η xy (y) = η(x). In words, the only way a configuration can change is a particle (i.e. an occupied site) "jumping" to a neighboring empty site provided each of those site have at least (k − 1) other empty neighbors. We call this transition a legal KA-kf transition, or simply legal transition whenever the context allows it.
Observe that, from Formula 2.2, this process is reversible with respect to the product measure µ := ⊗ x∈Z d Ber(1 − q) for any q ∈ (0, 1).
We now consider the trajectory of a tagged particle. Let µ 0 = µ(·|η(0) = 1) and, under the initial distribution µ 0 , X t the position at time t of the particle initially at 0. More precisely, (X t , η t ) t≥0 is the Markov process with generator: The following classic result gives a convergence for X t : Theorem 2.1. [15,8] For any q ∈ (0, 1), there exists a non-negative d × d matrix D(q) such that where B t is a d-dimensional Brownian motion process and the convergence holds in the sense of weak convergence of path measures on D(R + , R d ). Furthermore, D(q) is characterized by the following variational formula: where the infimum is taken over all local functions on Ω, and (τ y η) is the configuration defined by (τ y η)(z) = η(z − y) for all z ∈ Z d . Remark 2.2. A priori, the diffusion coefficient is a matrix. In our case, however, the model is invariant under permutation and inversion of the standard basis vectors. This forces the diffusion matrix to be scalar, equal to any arbitrary diagonal element.
In [1], it was first proved that D(q) > 0 for all q > 0. We will give in this paper the appropriate scale of D(q) when q → 0. The main result is the following: For q ∈ (0, 1) let D(q) the diffusion coefficient given by Theorem 2.1. Then for q sufficiently small: where exp (k−1) denotes the exponential function iterated (k − 1) times, and c, c ′ are constants only dependent on k and d.

Proof of the lower bound
The proof of the lower bound will closely follow the proof of [1], sections 4 and 5. However, we use more refined combinatorial properties of the KA model in order to obtain the correct scaling.
Throughout the proof, c and λ denote generic positive constants which only depends on d and k. We start by defining a coarse grained version of the lattice, depending on two scales: We also call external face of a box any connected component of the set of vertices at (graph) distance 1 from the box. See Figure 3.1.
We think of the blocks' corners (L + 1)i as vertices of a graph Z d ℓ = (L + 1)Z d . More precisely, the graph Z d ℓ is defined as follows : • Its vertices are sites of the form (L + 1)i, i ∈ Z d . • Its edges connect a vertex (L + 1)i with a vertex (L + 1)j for ||i − j|| 1 = 1.
From now on, sites on the original lattice will be denoted using the letters x, y, . . . , while vertices of Z d ℓ will be denoted with the letters i, j, . . . . Similarly to [1], the purpose of this coarse grained lattice is to define an auxiliary dynamics that has diffusive behavior on a larger scale.
Let us now recall a few definitions in relation with the coarse grained lattice, introduced in [11].
Definition 3.2. Let E be a subset of the standard basis with size |E| ≤ d − 1, V ⊂ Z d a set of sites, and fix a site x ∈ V . The |E|-dimensional slice of V passing through x in the directions of E is defined as V ∩ (x + spanE), where spanE is the linear span of E. Definition 3.3. Given the d-dimensional cube C n = [n] d and an integer k ≤ d we define the k th frame of C n as the union of all (k − 1)-dimensional slices passing through (1, . . . , 1). Next, we say that the box C n is frameable for the configuration η ∈ {0, 1} Cn if η is connected by legal KA-kf transitions to a configuration for which the k th frame of C n is empty.
-frameable for all configurations η ′ that differ from η in at most one site. We also require a good box to contain one extra empty site in addition to the ones required before.
Note that this definition slightly differs from [11] by requiring an additional empty site. The reason will be clarified in the proof of Lemma 3.14. Whenever the context allows it, we shall simply say that a box is good instead of (d, k)-good.
Example 3.5. A box is (2, 2)-good if it contains at least two empty sites in each row each column, and at least one additional empty site in the box. Proposition 3.6. Let d ≥ k ≥ 2 and ℓ be defined as in (3.1). Then : In particular, for an appropriate choice of constants in equation (3.2), it is much larger than 1/L.
Proof. The probability of being frameable is bounded by the probability that the frame is already empty. Since the size of the frame is less than dℓ k−1 , the first bound follows. The second bound is due to [17]. See also Proposition 3.26 in [11] and the explanation that follows.
The following definitions describes blocks that contain a droplet which is able to propagate. See (2) At least one of the boxes in the sequence is (d, k)-frameable We say that i, j are (d, k)-block-connected for η if the following conditions hold : (1) There exist a (d, k)-super-good path connecting i and j whose length is at most 3L. We stress that being block-connected does not depend on the values of η (i) and η (j). This notion of being block-connected defines a percolation process on Z d ℓ . Let η be the configuration on the edges of Z d ℓ that gives the value 1 to an edge i ∼ j if i and j are block-connected and 0 otherwise. We denote by µ the measure on these configurations induced by µ. Lemma 3.9. µ is a stationary ergodic measure, that stochastically dominates a supercritical Bernoulli bond percolation, whose parameter tends to 1 as q tends to 0.
Proof. The probability that an edge is open depends only on the sites in the blocks adjacent to it. There are 2d such blocks, and the diameter of a block is d, hence the percolation process is 2d 2 -dependent. By [10], it suffices to prove that the probability to be block connected tends to 1 as q tends to 0. The probability for a (d − 1)−dimensional face of a box to be good tends to 1 as q goes to 0 (see Proposition 3.6). We now need to prove that the probability that a block satisfies the condition (1) is also large.
It will be convenient to restrict the super-good path that we seek to a two dimensional plane, as in [11]. We assume without loss of generality that j = i + (L + 1)e 1 .
By Proposition 3.6 and large deviations for oriented percolation [4] the following paths exist with high probability: (1) an up-right path of good boxes connecting i This provides a good path of boxes of length at most 3L.
It is thus left to show that one of these boxes is super-good. This is a consequence of Proposition 3.6 and the FKG inequality (since both being good and being frameable are increasing events).
Following [1], we compare the KA dynamics to a simple random walk on the infinite connected cluster of Z d ℓ , conditioned on the event that 0 is in this cluster. We will denote µ * (·) = µ (·|0 ↔ ∞). It is shown in [5,3] that this dynamics has a diffusive limit, given by a strictly positive diffusion matrix.
. Then D aux is bounded away from 0 uniformly in q.
Thus, for the rest of this section we will concentrate on proving an inequality of the form: for ∆(q) proportional to the bound to the diffusion coefficient in Theorem 2.3.
First, in order to compare the KA dynamics and the auxiliary one, we should put them on the same space.
Lemma 3.11. Fix u ∈ R d . Then where now the infimum is taken over local functions on Ω.
Proof. The proof follows the exact same steps as that of [1, Lemma 5.2].
The comparison of both dynamics will be via a path argument. We will use the construction of [11], by concatenating basic moves. The next definition describes a sequence of legal KA transitions which keeps track of the configuration η and the position of a marked particle z. (2) for any t ∈ [T ], the configurations M t−1 η and M t η are either identical or linked by a legal KA transition contained in V , (3) for any t ∈ [T ], z t is the new position of the particle that was at site z t−1 in M t η.
For t ∈ [T ] and η, whenever M t−1 η = M t η, a particle has jumped from a site to a neighbor. We denote by x t (η, z) (resp. y t (η, z)) the initial (resp. final) position of the particle during the jump. More precisely, x t and y t are such that The next lemma constructs a T -step move that exchanges a marked particle at the origin with a block-connected site. Proof. The proof is based on the construction of [11], explained in [14,Proposition 5.2.41]. In that proposition, the T -step move is propagating a site through a super-good path. Note that propagating simply means that the values of the configuration in the initial and final sites are being swapped. This does not mean that the marked particle initially at 0 reached the site (L + 1)e 1 , since in [14], the permutation move exchanges any two sites even if they are both occupied (see [14,Proposition 5.2.35]), which is not a legal KA transition.
We will show here briefly the idea of the proof when k = d = 2, repeating the construction of [11] with the appropriate adaptations. We will then explain how these adaptations apply to general k, d.
In order to construct the move M , we will construct a sequence of shorter, simpler, moves. The first of them is the column exchange move: The next move we will use is the framing move: Claim 3.16. Fix a box, and consider the configurations for which the box is good, and, in addition, its bottom row is empty. Then there exists a T-step move M whose domain consists of these configurations,  Note that if a is 1 and b is the marked particle, we are not allowed to exchange them directly, which is the reason we introduce the blue 0. When a box is framed, we are able to permute its sites: One last ingredient before constructing the large move M is the jump move, that will allow us to hop a particle over a row of empty sites: We are now ready to construct the move as shown in Figure 3.6, exchanging the marked particle initially at 0 with the particle/vacancy ⋆ initially at (L + 1)e. First we use Claim 3.15 in order to propagate the empty column, then we frame the box [1, ℓ] × [0, ℓ] using Claim 3.16. We can then frame the column {1} × [0, ℓ], and then use the permutation move (with the modification described above) in order to bring the marked particle to the position (ℓ − 1, ℓ − 1). We then use the jump move (Claim 3.18), and clean up the modifications to the box [0, ℓ] 2 . Using again Claim 3.15, we can move the marked particle to the bottom right box, and apply the same framing procedure as before in order to exchange it with ⋆ and move ⋆ to (0, 0). All that is left is to take the row of 0s back to its original position.
For general k, d the same proof as [11] will allow us to construct M , where the only modification is in the definition of the permutation move, which now takes into account the position of the marked particle. This is done in the exact same way as we have seen in Claim 3.17 for the case k = d = 2. • If k = 2, • If k ≥ 3, This concludes the proof of the inequality (3.3). Together with Proposition 3.10, we obtain the lower bound of Theorem 2.3.

Upper bound
In order to find an upper bound, we will look for a suitable test function to plug in equation (2.4). Without loss of generality we consider u = e 1 . Let for c > 0 that may depend on d and k but not on q.
We now define the k neighbor bootstrap percolation on the box [−ℓ, ℓ] d . It is usually seen as a deterministic process defined on Ω ℓ := {0, 1} [−ℓ,ℓ] d , but for our needs it is convenient to define the bootstrap percolation map BP : Ω ℓ → Ω ℓ : That is, empty sites remain empty and occupied sites become empty if they have at least k empty neighbors. Let BP ∞ (η) be the limiting configuration, that is: For more details about bootstrap percolation, see e.g. [12].
The following observation clarifies the relation between bootstrap percolation and the KA model. Proof. Since in a legal KA move the particle has at least k empty neighbors both before and after the exchange, it would be emptied for both states.
We say that a site x is in the bootstrap percolation cluster of the origin if there is a nearest neighbor path 0, x 1 , . . . , x n = x such that BP ∞ (η) (x i ) = 0 for i = 1, . . . , n. For any η ∈ Ω, let f (η) be the first coordinate of the rightmost site in the bootstrap percolation cluster of the origin for η [−ℓ,ℓ] d .  Proof. This is a direct consequence of [2, Lemma 5.1] and the choice of ℓ in (4.1), as we set c = β − (d, k) following the notations of [2].  Proof. If both x and y are outside [−ℓ, ℓ] d , xy is not pivotal. If both are inside [−ℓ + 1, ℓ − 1] d , then c xy (η) = c xy (η [−ℓ,ℓ] d ). By Observation 4.1, xy can not be pivotal.
Assume xy is pivotal. Then either x or y is in the bootstrap percolation cluster of the origin for either η or η xy , hence B must occur for either η or η xy .
We can now estimate the right hand side of equation (2.4) for our choice of f .
Summing both contributions now yields the expected bounds.

Further questions
Theorem 2.3 shows how the self-diffusion constant decays as q → 0 up to a constant for k ≥ 3 and logarithmic correction for k = 2. In [16, equation (6.26)], it is conjectured for the case k = d = 2 that the true behavior is D(q) ≈ exp(−γq −1 ) for γ = π 2 9 . In view of recent works related to the Fredrickson-Andersen model [7], where similar scaling is observed and the exact constant γ could be identified, it seems reasonable that such a result could also be obtained for the Kob-Andersen model.
The methods used here could also be applied in other models for which the combinatorial structure allows a construction of a T -step move as in Lemma 3.14. In particular, it is natural to consider other kinetically constrained lattice gases, or even look for universality results on the self-diffusion coefficient.