Mixing times for exclusion processes on hypergraphs

We introduce a natural extension of the exclusion process to hypergraphs and prove an upper bound for its mixing time. In particular we show the existence of a constant $C$ such that for any connected, regular hypergraph $G$ within some natural class, the $\varepsilon$-mixing time of the exclusion process on $G$ with any feasible number of particles can be upper-bounded by $CT_{\text{EX}(2,G)}\log(|V|/\varepsilon)$, where $|V|$ is the number of vertices in $G$ and $T_{\text{EX}(2,G)}$ is the 1/4-mixing time of the corresponding exclusion process with just two particles. Moreover we show this is optimal in the sense that there exist hypergraphs in the same class for which $T_{\mathrm{EX}(2,G)}$ and the mixing time of just one particle are not comparable. The proofs involve an adaptation of the chameleon process, a technical tool invented by Morris and developed by Oliveira for studying the exclusion process on a graph.


Introduction
Let G = (V, E) be a finite connected graph with vertex set V and edge set E. Fix k ∈ {1, . . . , |V |} and consider k indistinguishable particles moving on V using the following rules: 1. each vertex is occupied by at most one particle, 2. each edge e ∈ E rings at the times of a Poisson process of rate 1, independently of all other edges, 3. when an edge e = {u, v} rings, the occupancy states of vertices u and v are switched.
For each v ∈ V and t ≥ 0, let η t (v) = 1 if v is occupied at time t, and η t (v) = 0 if v is vacant at time t. The process (η t ) t≥0 is called the k-particle exclusion process on G: see Figure 1.
In this paper we are interested in a natural extension of the exclusion process to hypergraphs. Figure 1: Example transition of 3-particle exclusion process on K 5 . When the edge indicated rings, the single particle currently on that edge moves to the vertex at the other end of the edge.
Let G = (V, E) be a finite connected hypergraph, where E ⊆ P(V ), the power set of V . For each e ∈ E, denote by S e the symmetric group on the elements in e, and let f e : S e → [0, 1] be a probability measure on S e . We write f to denote {f e : e ∈ E}, the set of these measures. Consider k indistinguishable particles moving on V using rules 1. and 2. above and in addition: 3 . when an edge e rings, a permutation σ ∈ S e is chosen according to f e and every particle on a vertex in e moves simultaneously according to σ, i.e. a particle at vertex v moves to vertex σ(v). (Note that as σ is a permutation, rule 1. is preserved.) Setting η t (v) = 1 if v is occupied at time t and 0 otherwise, we obtain a process (η t ) t≥0 referred to as the k-particle exclusion process on (f, G), or simply EX(k, f, G): see Figure 2.
(Note that if each edge e ∈ E contains exactly two vertices, and f e puts all of its mass on the transposition belonging to S e , then EX(k, f, G) is just the k-particle exclusion process on the graph G, as above.) Our main aim in this paper is to study the total-variation mixing time of EX(k, f, G), and to establish an upper bound in terms of the mixing time of EX(2, f, G). Recall that for a continuous-time Markov process X on a finite set Ω with transition probabilities {q t (x, y)} and equilibrium distribution π, the total variation ε-mixing time is defined as T X (ε) := inf t ≥ 0 : max x∈Ω q t (x, ·) − π TV ≤ ε , (1.1) Figure 2: Example transition of 3-particle exclusion process on a hypergraph with 5 vertices and 3 edges (indicated by the different shaded regions, i.e. here there are two edges of size 3 and one of size 4). When the edge containing four vertices rings, the two particles currently belonging to that edge are permuted.
where · TV is the total-variation norm.
In several parts of the proof it will be useful to consider the associated process where the k particles are distinguishable. Suppose the particles are labelled 1, . . . , k and setη t (v) to be the label of the particle at vertex v at time t. If there is no particle at v at time t, setη t (v) = 0. The process (η t ) t≥0 is the k-particle interchange process on (f, G), or simply IP(k, f, G). Note that the exclusion process may be recovered from the interchange process simply by 'forgetting' the labels of the particles.
We will assume throughout that the hypergraph G is regular (every vertex has the same degree). We will also make the following assumptions about the set of measures f (with notation appearing below being formally defined in Section 2.1).

1.
For every e, f e is constant on the conjugacy classes of S e (i.e. in group-theoretic terms, f e is a class function). That is, if σ 1 and σ 2 are elements from S e with the same cycle structure, then f e (σ 1 ) = f e (σ 2 ).
2. For every e and each v ∈ e, σ∈Se f e (σ)1 {|σ(v)|=1} ≤ 1/5, where |σ(v)| denotes the size of the cycle containing v in σ. In other words, the probability (under f e ) of a vertex v ∈ e being a fixed point of σ is at most 1/5.
Remark 1.4. The exclusion process on a hypergraph G with the edge set E consisting only of edges of size 2 or 3 exhibits the negative correlation property (which we shall discuss further in the sequel). As a result, for this subset of hypergraphs we can actually extend the main theorem, replacing the T EX(2,f,G) (1/4) appearing in the right-hand side by T EX (1,f,G) (1/4).
Remark 1.5. A simple example suffices to show that Theorem 1.2 is optimal in the sense that we cannot replace T EX(2,f,G) (1/4) on the right-hand side with T EX (1,f,G) (1/4) (even under our standing assumptions). Let G = (V, E) with V = {1, 2, 3, 4} and E = {V }. Suppose f gives probability 1 − δ to permutations composed of two disjoint transpositions, and probability δ to 4-cycles. This hypergraph is clearly regular and f satisfies Assumption 1.1 for any δ ∈ (0, 1]. Notice that the 2-particle exclusion process cannot mix until a 4-cycle is chosen, whereas this event is not necessary for the 1-particle exclusion process to mix. Hence we have T EX(2,f,G) (1/4)/T EX (1,f,G) (1/4) → ∞ as δ → 0.

Motivation and connections with the literature
Our results contribute to the general question of when properties of a multi-particle system can be deduced from properties of a system with only a few particles. Arguably the most significant recent result in this area has come from Caputo, Liggett, and Richthammer (2010) who showed that the spectral gap of the interchange process on a graph is equal to the spectral gap of a random walker on the same graph, proving a conjecture of Aldous that had been open for 20 years. Proving results in this area is particularly important in applications since the large reduction in the size of the state space often makes it much easier to compute or estimate statistics. While interacting particle system models (e.g. exclusion process, interchange process, voter model, contact process, zero range process) on graphs have received considerable attention, there has so far been little study of such processes on hypergraphs. Studying these processes on hypergraphs is very natural though, as hypergraphs allow simultaneous interactions of multiple particles, rather than only pair-wise interactions. One model for which its analogue on hypergraphs has been recently studied is the voter model (Chung and Tsiatas (2014); Istrate, Bonchis, and Marin (2014)), for which various properties are considered, including the mixing time.
Any interchange process (with k = |V |) on a graph can be viewed as a card shuffle by transpositions, with notable examples being the top-to-random transposition shuffle (star graph), random-to-random transposition shuffle (complete graph) and nearest-neighbour transposition shuffle (the cycle). This connection is well-known, see e.g. Section 4A of (Diaconis and Saloff-Coste, 1993). The literature concerning mixing times of card shuffles by transpositions is extensive. In the following discussion we suppose that time is scaled so that shuffles are performed at unit rate (which amounts to setting the rate that each edge rings to 1/|E|).
The case of the top-to-random transposition shuffle has been dealt with by Flatto, Odlyzko, and Wales (1985) and Diaconis (1988) (among others) -this shuffle has mixing time cutoff at time |V | log |V |. The mixing time of the random-to-random transposition shuffle was first obtained by Diaconis and Shahshahani (1981), and is |V | 2 log |V | (with cutoff). More recently, Lacoin (2016) has studied the nearest-neighbour transposition shuffle and showed mixing (with cutoff) occurs at time π −2 |V | 3 log |V |. Jonasson (2012) has shown that the slowest a transposition shuffle can mix is of order |V | 4 log |V |, with the lollipop graph an example which achieves this (Erikshed, 2011). Achieving these results typically involves finding an argument tailored specifically to the model in question. If we care less about the specific constant multiple (at which mixing occurs) and instead focus on the order, a result of Oliveira (2013) can prove particularly useful as a general way of bounding mixing times of exclusion processes: Theorem 1.6 (Oliveira (2013)). There exists a constant C > 0 such that for every connected weighted graph G and every k ∈ {1, . . . , |V | − 1} and ε ∈ (0, 1/2), where T RW(G) (1/4) is the mixing time of the random walk on G.
Of course, transposition shuffles are just one class of shuffle, and there is significant interest in mixing times of more general shuffles in which multiple cards are moved simultaneously. A large class of time-homogeneous shuffles can be represented as interchange processes on hypergraphs. One of the simplest hypergraphs to consider is the complete uniform hypergraph (in which every edge contains the same number of vertices, and every possible such edge is in the hypergraph). If each edge has size d and if, when an edge rings, we move particles by application of a uniformly chosen cycle, this corresponds to shuffling cards by d-cycles. The mixing time (with cutoff) is known to be |V | d log |V |. This was first shown for the case of fixed d by Berestycki, Schramm, and Zeitouni (2011), and later extended to any d = o(|V |), see (Hough, 2015). A related model is the following: instead of choosing a cycle uniformly, we fix a conjugacy class Γ (so Γ is a set of permutations with the same cycle structure) with |Γ| = d (where |Γ| is the number of non-fixed points in any permutation from Γ), and every time an edge rings, we choose a permutation on the vertices in that edge uniformly from Γ. Sharp bounds (up to a multiplicative constant) on the mixing time of this model with d = o(|V |) were first obtained by Roichman (1996), with cutoff at time |V | d log |V | proven by Berestycki and Sengul (2014). Cutoff at the same time for the model with d ≥ |V |/2 was shown by Lulov and Pak (2002).
Our main result extends Theorem 1.6 to a class of hypergraphs. Furthermore, our results hold for a large class of measures acting on the symmetric group S |V | which goes beyond the standard framework studied by previous authors, in which a conjugacy class is fixed and then sampled from uniformly. Indeed, our measures f e can vary dramatically between edges e ∈ E, and furthermore we do not require each f e to be supported on a fixed conjugacy class.

Heuristics and structure of the proof
The proof of Theorem 1.2 depends on the size of the vertex set V . If |V | is sufficiently small, the proof is fairly simple and we state the result as the following lemma: Lemma 1.7. There exists a constant C > 0 such that for every hypergraph G = (V, E) with |V | < 36, every f and every k ∈ {1, . . . , |V |/2} and ε > 0, On the other hand, the argument for |V | ≥ 36 is much more intricate and is split into two parts, the first being the following lemma which is of independent interest (and is stronger than needed for our main theorem, as it relates to the interchange process): Lemma 1.8. There exists a constant C > 0 such that for every hypergraph G = (V, E) with |V | ≥ 36, every f and every k ∈ {1, . . . , |V |/2} and ε > 0, Oliveira (2013) proves his main result (bounding the mixing time of the k-particle exclusion process by the mixing time of a random walker) by first relating the mixing time of a kparticle interchange process to that of a 2-particle interchange process. Roughly speaking, this is possible due to the fact that any time an edge of the graph under consideration rings, at most two particles move under interchange, and so it is pairwise interactions that determine the mixing rate. This contrasts with the exclusion process on hypergraphs considered here, in which many particles can move at the same time. Nevertheless, a suitable adaptation of the techniques appearing in (Oliveira, 2013) provides the proof of Lemma 1.8. Remark 1.9. Lemma 1.8 only holds when |V | is sufficiently large and k ≤ |V |/2. We cannot hope to remove these conditions and replace EX(4, f, G) with EX(2, f, G) in this statement, even for regular hypergraphs satisfying Assumption 1.1, as the following example illustrates. Let G = (V, E) with V = {1, 2, 3} and E = {V }, i.e. there is just a single edge which contains all three vertices in the hypergraph. Suppose that f gives probability 1 − δ to the conjugacy class of 3-cycles, and probability δ to the class of transpositions. For δ sufficiently small this satisfies Assumption 1.1. The 2-particle interchange process cannot mix until a transposition is chosen (as half of the states cannot be reached before this time), whereas this event is not necessary for the 2-particle exclusion process to mix, and hence it is straightforward to see that as The second part of the proof for |V | ≥ 36 requires showing that T EX(4,f,G) and T EX(2,f,G) are of the same order: Lemma 1.10. There exists a constant λ > 0 such that for any hypergraph G with |V | ≥ 36, and f , T EX(4,f,G) (1/4) ≤ λT EX(2,f,G) (1/4).
We now demonstrate that Theorem 1.2 follows simply from Lemmas 1.7, 1.8 and 1.10.
Proof of Theorem 1.2. The contraction principle (see Aldous and Fill (2002)) gives and so provided k ≤ |V |/2, we have the result for |V | ≥ 36 by Lemmas 1.8 and 1.10 and for |V | < 36 by Lemma 1.7. However, note that switching the roles of occupied and unoccupied vertices in EX(k, f, G) yields the process EX(|V | − k, f, G). It follows that and so the proof of Theorem 1.2 is complete.
We finish this section with a brief overview of the rest of the paper. In Section 2 we define formally the processes considered in this paper and present some preliminary results. In addition, we demonstrate that the negative correlation property, which is fundamental to the result in (Oliveira, 2013), fails to hold for the hypergraph setting. In Section 3 we prove Lemma 1.8 subject to the existence of a process with certain key properties that relate it to an interchange process: see Lemma 3.1 for the precise statement. This process is constructed in Section 6 and we show it has the desired properties in Section 7. Proving Lemma 3.1 is the most challenging (and technical) part of this paper.
In Section 4 we prove Lemma 1.10 by first characterizing every hypergraph as one of two types depending on how long it takes any two of four independent particles to meet.
We use some of the ideas developed in Section 4 to prove Lemma 1.7 in Section 5. A few of the more technical proofs required are included in two appendices.

Random walks, exclusion and interchange processes
We formally define the main processes studied in this paper, RW(f, G), RW(k, f, G), EX(k, f, G) and IP(k, f, G), by explicitly stating their generators. In the next section we shall present a graphical construction of these processes, similar to that of Liggett (1999) for the standard interchange and exclusion processes. This graphical construction will allow us to simultaneously define the processes on the same probability space, and thus directly compare them.
Recall S e as the group of permutations of elements in e. Our processes of interest evolve by the action of permutations from these groups. However, it will often be convenient to consider permutations as acting on V and we can easily do this by extending a permutation σ e ∈ S e to a permutation in S V by setting σ e (v) = v for all v / ∈ e. We can also consider such permutations as acting on a subset of V or on vectors with elements being distinct members of V . To do this we can define, for a set A ⊆ V , σ e (A) := {σ e (a) : a ∈ A}, and for a vector x of k distinct elements of V we define σ e (x) := (σ e (x(i))) k i=1 . Set notation: For k ∈ N we define and for a set A ⊆ V we write Generators: We now explicitly state the generators of the processes. For a hypergraph G and a suitable set of functions f , the simple random walk on G, RW(f, G), is the continuoustime Markov chain with state space V and generator for all u ∈ V and h : V → R.
We denote by RW(k, f, G) the product of k independent random walkers on G. This process is the continuous-time Markov chain with state space V k and generator The k-particle exclusion process EX(k, f, G), is the continuous-time Markov chain with state space V k and generator for all A ∈ V k and h : V k → R. The k-particle interchange process IP(k, f, G), is the continuous-time Markov chain with state space (V ) k and generator for all x ∈ (V ) k and h : (V ) k → R.

Graphical construction
We first construct an independent sequence of E-valued random variables {e n } n∈N such that each e n is identically distributed with P [e n = e] = 1/|E|. Given the sequence {e n } n∈N , let {σ n } n∈N be a sequence of permutations with σ n ∈ S en independently chosen and satisfying for each n ∈ N, P [σ n = σ] = f en (σ). Now that we have the sequence of edges that ring and the permutations to apply, it remains to determine the update times of the processes.
Let Λ be a Poisson process of rate |E| and for 0 < s < t denote by Λ[s, t] the number of points of Λ in [s, t]. For every 0 < s < t, we define a random permutation I [s,t] : V → V associated with the time interval [s, t] to be the composition of the permutations performed during this time; that is, We set I t := I [0,t] for each t > 0, and I (t,t] to be the identity. Note (cf Proposition 3.2 of Oliveira (2013)) that L[I (s,t] where we write L for the law of a process. We can lift the functions I [s,t] to functions on V k and (V ) k in the following way: for A ∈ V k , and for x ∈ (V ) k , The following proposition is fundamental: its proof follows by inspection.
Proposition 2.1. Fix s > 0. Then: 1. For each u ∈ V , the process {I [s,s+t] (u)} t≥0 is a realisation of RW(f, G) initialised at u at time s. We shall often write this process simply as (u RW t ) t≥s .

For each
initialised at A at time s. We shall often write this process simply as (A EX t ) t≥s .

For each
initialised at x at time s. We shall often write this process simply as (x IP t ) t≥s .

Total variation and mixing times
There are several equivalent definitions of total variation that we shall make use of in this paper. Suppose µ and ν are two probability measures on the same finite set Ω. Then the total variation distance between these measures is defined as We shall also make extensive use of the following equivalent definition, which relates the total variation distance to couplings of µ and ν: where the infimum is over all couplings (X, Y ) of random variables with X ∼ µ and Y ∼ ν.
We recall a simple result bounding the total variation of product chains (see e.g. pg 59 of Levin, Peres, and Wilmer (2008)): for n ∈ N and 1 ≤ i ≤ n, let µ i and ν i be measures on a finite space Ω i and define measures µ and ν on n i=1 Ω i by µ := n i=1 µ i and ν := n i−1 ν i . Then (2.5) Recall equation (1.1) as the definition of the mixing time of a continuous-time Markov process. We will require several general mixing-time bounds throughout this work, which we present here.
Proposition 2.4 (Aldous and Fill (2002)). Let X be a Markov process on a finite state space Ω with symmetric transition rates. Then the equilibrium distribution is uniform over Ω and for all 0 < ε < 1/2 and t ≥ 2T X (ε), for all ω 1 , ω 2 ∈ Ω.

Failure of negative correlation
We conclude this preliminary section with a quick example to demonstrate that the exclusion process on a hypergraph does not enjoy the negative correlation property satisfied by the exclusion process on a graph. We first recall the version of the negative correlation property of the exclusion process on a graph to which we refer, and whose proof may be found in (Liggett, 1985). Let B ⊂ V and let (A EX t ) t≥0 be a 2-particle exclusion process on a graph G = (V, E). Then for every t ≥ 0, Now suppose G = (V, E) is the hypergraph with V = E = {1, 2, 3, 4} (i.e. there is only one edge), and that f is concentrated uniformly on the six possible 4-cycles. Suppose (u RW t ) t≥0 and (v RW t ) t≥0 are two independent realisations of RW(f, G), and (A EX t ) t≥0 is a realisation of (2.6) Indeed, since the event {u RW t ∈ B} is less likely than seeing at least one incident in a unit-rate Poisson process by time t, we have On the other hand, the event {A EX t ∈ B} is at least as likely as the edge ringing exactly once by time t, with the chosen permutation satisfying σ({1, 2}) = B. That is, Inequality (2.6) is therefore satisfied for any t < 0.33. 3 From k-particle interchange to 4-particle exclusion: proof of Lemma 1.8 Given a hypergraph with vertex set V and a (k − 1)-tuple z ∈ (V ) k−1 , let (1), . . . , z(k − 1)} be the (unordered) set of coordinates of z and define a space As we shall see, most of the work required to prove Lemma 1.8 is to show the existence of a certain Markov process having some key properties, which we outline in the following Lemma.
Lemma 3.1. There exist constants c 1 , c 2 and κ 1 such that for every regular hypergraph G = (V, E) with |V | ≥ 36, every f , every k ∈ {1, . . . , |V |/2}, every x = (z, x) ∈ (V ) k , and every realisation (x IP t ) t≥0 of IP(k, f, G) started from state x, there exists a continuous-time Markov process (M t ) t≥0 := (z C t , R t , P t , W t ) t≥0 with state-space Ω k (V ) defined on the same probability space as (x IP t ) t≥0 satisfying: 3. for every t ≥ 0 and j ∈ N, 4. for every t ≥ 0 and c ∈ (V ) k−1 , The proof of Lemma 3.1 is deferred to Section 7 and is a proof by construction: in Section 6 we will explicitly define a process and then proceed to show that it has the desired properties.
We can now relate the total-variation distance between two realisations of IP(k, f, G) to a certain expectation involving the amount of ink in the chameleon process M in the statement of Lemma 3.1. The following result is similar to Lemma 6.1 of (Oliveira, 2013): we include a sketch of the proof to highlight the importance of constructing in Section 6 a chameleon process satisfying part 2 of Lemma 3.1.
and denote by x IP t an interchange process started from x. Letx be uniform from V \ O(z) and denote byx IP t an interchange process started fromx = (z,x). Then for any where the second and third equalities follow from parts 1 and 4 of Lemma 3.1, respectively. On the other hand, part 2 of Lemma 3.1 gives The result now follows by repeated application of the triangle inequality, as in the proof of Lemma 6.1 of (Oliveira, 2013).
Proof of Lemma 1.8. We combine part 3 of Lemma 3.1 with Lemma 3.2 to give for every t ≥ 0 and j ∈ N, for some universal positive constants c 1 , c 2 and κ 1 . We choose which gives the bound (using k ≤ |V |), for some positive c 3 . Therefore there exists a universal constant C such that for any ε ∈ (0, 1/2) and t > CT EX(4,f,G) (1/4) log(|V |/ε), 4 From 4-particle exclusion to 2-particle exclusion: proof of Lemma 1.10 We begin by characterizing every connected hypergraph in terms of how long it takes two independent random walkers on the hypergraph to arrive onto the same edge, which then rings for one of the walkers -a time we shall refer to as the meeting time of the two walkers (note that we do not require the two walkers to actually occupy the same vertex). It will be useful to consider such times, as we will be able to couple two independent walkers with a 2-particle interchange process, up until this meeting time (see Proposition 4.7 for this statement).
Formalising this, for y ∈ V 2 , let (y RW t ) t≥0 be a realisation of RW(2, f, G) with y RW 0 = y. Denote by Λ 1 and Λ 2 the Poisson processes used to generate the edge-ringing times for the two particles, and let {e 1 n } n∈N and {e 2 n } n∈N be the two sequences of edge-choices (all as in Section 2.2). Define M RW (y) to be the first time y RW t (1) and y RW t (2) are in the same edge which then rings in one of the processes: Remark 4.2. We note that this definition is similar to Definition 4.1 of (Oliveira, 2013), from where we borrow the dichotomy "easy/non-easy". However, for the case of hypergraphs, this characterisation does not reflect the associated difficulty of dealing with each case! One difference in the case of hypergraphs is that at the meeting time we cannot guarantee that the two independent walkers occupy the same site, and this results in the analysis being more challenging.
4.1 From 4-particle exclusion to 2-particle exclusion: easy hypergraphs Lemma 4.3. There exists κ > 0 such that for any easy hypergraph G, any f and 0 < ε < 1/2, In this section we will make use of this lemma only for the case |V | ≥ 36, but this result will later be used in its full form when dealing with the case of |V | < 36: see Section 5. The proof uses a coupling argument for two realisations of EX(k, f, G).
and (W EX t ) t≥0 be two realisations of EX(k, f, G) started from U and W respectively. We define the two processes on a common probability space, and will show how to couple them in such a way that we can lower-bound the probability that U EX κT = W EX κT for some κ > 0 to be determined, where T := T EX(k−1,f,G) (1/4). The result will then follow by applying (2.4).
We begin by allowing the two processes to evolve independently up to time 10T . Then, for any S ∈ V k and t ≥ 0, we have where the inequality follows from (2.2). Maximizing over S and again using (2.2) gives By the Markov property, for any A, B ∈ V k , for any coupling of (A EX t ) t≥0 and (B EX t ) t≥0 , by (2.4). Let a and b be two uniformly and independently chosen elements of A and B, respectively. Consider now the k-particle process A * which evolves in the same way as the exclusion process begun at A, but with the label of the particle started from position a being tracked (this process can be described formally via the graphical construction, see Section 2.2). Thus A * can be thought of as something 'between' an exclusion process (in which no labels are tracked) and an interchange process (in which all labels are tracked). It's clear that the k − 1 particles of (A \ {a}) * behave marginally as an exclusion process, while the particle started from a behaves (again marginally) as a random walk on G. Furthermore, the exclusion process A EX can be recovered from A * simply by 'forgetting' which position is occupied by the 'special' particle starting from a.
Up to time 10T we couple the processes A * and B * using a maximal coupling of the (k − 1)particle exclusion processes (A \ {a}) * and (B \ {b}) * . (Recall that a maximal coupling is one which achieves equality in the coupling inequality (2.4).) By Proposition 2.2 we have Given the choice of a and b, let F a,b denote the event that the other (k − 1) particles have Using this maximal coupling it follows from (4.4) that P [F a,b ] ≥ 499/500. Combining this with equations (4.2) and (4.3) we see that for any K ∈ N, From (4.5) we see that we now need to upper bound the probability that A EX and B EX do not agree by time (10 + K)T , on the event that the (k − 1) particles in (A \ {a}) * and (B \ {b}) * agree at time 10T . As pointed out above, this event is equivalent (on F a,b ) to the locations of the k particles in A * (10+K)T and B * (10+K)T not agreeing. We shall bound this probability by coupling the processes (A * 10T +t ) t≥0 and (B * 10T +t ) t≥0 in the following manner. Let Λ be a Poisson process of rate 2|E| (i.e. twice the usual rate), with associated edge-choices {e n } n∈N and permutations {σ n } n∈N as in Section 2.2. In addition, let {θ n } n∈N be an i.i.d. sequence of Bernoulli (1/2) random variables: these will be used to thin the events of Λ and ensure that all particles are moving at the correct rate. We make this modification as it allows us to more easily compare a certain time to the meeting time of two independent random walkers as defined in (4.1). We evolve (A * 10T +t ) t≥0 and (B * 10T +t ) t≥0 by applying permutation σ n to edge e n (in both processes) at the n th incident time of Λ if and only if θ n = 1, and continue to do this until the first time τ a,b that the 'special' particles a and b are in a common edge which then rings: We write e a,b for the edge that they meet on at time τ a,b , and σ a,b for the corresponding permutation. Note that, since we use a common set of innovations over the period [0, τ a,b ), the processes (A \ {a}) * and (B \ {b}) * still agree at time τ a,b −. Furthermore, the processes a * and b * when viewed marginally behave as independent random walks over the period [0, τ a,b ), and so τ a,b has the same distribution as the meeting time M RW (a, b) in (4.1).
We now partition the probability space according to the following four sets (for some K ∈ N which is yet to be determined): where the last inequality uses (2.5), (4.4) and the contraction principle.
Third, conditioned on the event E 3 a,b , by Lemma 4.4 we can couple the locations of all k particles in A * and B * at time τ a,b with probability at least 1/30, so that where the last inequality is obtained in the same way as (4.7).
Our fourth and final case to consider is E 4 a,b : on this event a simple case-by-case analysis shows that the positions of a * τ a,b and b * τ a,b can be made to agree with probability at least 1/2, as long as there are no other (already matched) particles on edge e a,b at time We use a union bound to control the probability of the complement. We have (4.10) We now upper bound this using (2.1) and (2.3) (using the same method as in (4.7)). This gives the following upper bound for (4.10): , since, on the event E 4 a,b , the size of edge e a,b is at most four and so for any choice of e a,b there are only two possibilities for the value of c (since c * We now combine the bounds in (4.5), (4.6), (4.8), (4.9) and (4.11) to see that By assumption, k ≤ |V |/2 if |V | < 36 and k ∈ {3, 4} if |V | ≥ 36, and so for all possible combinations of k and |V | being considered here. Combining this bound with that in Assumption 1.1, we obtain: But since τ a,b has the same distribution as M RW (a, b), and G is an easy hypergraph, Finally, by submultiplicativity of the function (see e.g. Lemma 4.12 of Levin et al. (2008)), we deduce that and so the statement of Lemma 4.3 is proved upon taking κ = 10 14 .
The proof of the following lemma can be found in Appendix A.
Lemma 4.4. Recall the definition of the event E 3 a,b in the proof of Lemma 4.3. Conditioned on this event, the processes A * and B * can be coupled such that with probability at least 1/30 the positions of all k particles agree at time τ a,b .
Proof of Lemma 1.10 for easy hypergraphs. We simply apply Lemma 4.3 for the case |V | ≥ 36 twice, first with k = 4 and then with k = 3 (and take ε = 1/4 both times). We deduce that and so it suffices to take λ = κ 2 (log 4) 2 .
4.2 From 4-particle exclusion to 2-particle exclusion: non-easy hypergraphs We begin with a result showing that for non-easy hypergraphs the average meeting time for two independent random walkers is unlikely to be quick. Intuitively, this follows from the following observations. We know there exists a pair of vertices such that random walkers started from these two states likely take a long time to meet. If we look at where these two walkers are at time of order T RW(f,G) (1/4), they will be close to uniform. Hence, starting random walkers from a uniform pair we see that they will likely still take a long time to meet. The proofs of Lemmas 4.5 and 4.6, and of Proposition 4.7 are (somewhat technical) extensions of corresponding results of Oliveira (2013), and can be found in Appendix A.
Lemma 4.5. For every non-easy hypergraph we have .
Poisson processes used to generate the edge-ringing times for the four random walkers, and let {e 1 n } n∈N be the four sequences of edge-choices (all as in Section 2.2).
We now defineM RW (O(x)) to be the first time any two of x RW t (4) first arrive onto the same edge which then rings for one of them. Formally, Lemma 4.6. Let x ∈ (V ) 4 . Then for any ε ∈ (0, 1), Next, we provide a bound which relates the total-variation distance between two 4-particle exclusion processes to the probability that any two of four independent walkers have 'met'.
Proposition 4.7. For any x ∈ (V ) 4 and s ≥ 0: Lemma 4.8. For every non-easy hypergraph G and any two realisations of EX(4, f, G), Proof. By Proposition 4.7 and the triangle inequality for total-variation, for any u, v ∈ (V ) 4 , (4.12) An identical argument to that used for equation (4.2) tells us that Applying the inequality in (4.12), with any u, v satisfying Using Proposition 2.3 and the contraction principle for the third term on the right-hand side gives the desired result.
We are now ready to prove the main result of this subsection.
Proof of Lemma 1.10 for non-easy hypergraphs. We in fact show that for any two realisations Combining Lemmas 4.6 and 4.8 we have that for every ε ∈ (0, 1), Now by Lemma 4.5, this becomes Setting ε = 10 −3 completes the proof. 5 From k-particle exclusion to 2-particle exclusion for small graphs: proof of Lemma 1.7 We begin by showing that any hypergraph G with |V | < 36 satisfies which certainly implies (5.1). Since G is easy, we can apply Lemma 4.3 multiple times to deduce that However, since |V | < 36 and k ≤ |V |/2 the statement of the proof is complete taking C = κ 15 (log(1/4)) 14 .

The chameleon process
Our aim in this section is to construct a continuous-time Markov process which satisfies the properties of (M t ) t≥0 outlined in Lemma 3.1. We will call this process the chameleon process. In Section 7 we will prove Lemma 3.1 by demonstrating that the chameleon process does indeed have the desired properties.
The chameleon process was originally constructed (in a different form but to serve a similar purpose) by Morris (2006), and then adapted by Oliveira (2013) to analyse the mixing time of the k-particle interchange process on a graph (as opposed to on a hypergraph, as we consider here). It is built on top of an underlying interchange process, with the aim of helping to describe the distribution of the location of the kth particle in this process, conditional on the locations of the k − 1 other particles.
Unlike in a k-particle interchange process which always has k particles, the chameleon process has |V | particles (one at each vertex), although not all particles are distinguishable from each other. In addition, each particle has an associated colour : one of black, red, pink and white (which correspond to the processes z C t , R t , P t , W t respectively, appearing in the statement of Lemma 3.1). The movement of particles in the chameleon process follows that of the underlying interchange process in the sense that the locations of particles in both processes are updated using the same functions I as described in the graphical construction of Section 2.2. At some of the updates of the underlying interchange process we will colour some of the red and white particles pink (precisely when this happens is rather involved and is the subject of Section 6.2). To provide some insight into when these pinkening events occur, consider the chameleon process of (Oliveira, 2013): Here, if the vertices at the endpoints of a ringing edge are occupied by a red and a white particle then both of these particles are recoloured pink. In the lazy version of the interchange process on a graph (in which nothing happens with probability 1/2 when an edge rings), when an edge rings with endpoints occupied by a red and a white particle, with probability 1/2 they switch places and with probability 1/2 they do not move. Colouring both particles pink encodes the fact that at either vertex just after the edge rings we may have a white particle or a red particle, and these are equally likely.
We wish to use this notion of pinkening to encode similar events in the interchange process on hypergraphs, but the situation here is quite different since more than two particles are moved when an edge rings, and the way in which they move depends on the permutation chosen. As a result, describing precisely when these pinkenings occur for our version of the chameleon process is rather complicated, but the underlying motivation can be explained relatively simply. Suppose that an edge e rings and a permutation σ is chosen to move the particles on that edge. To decide which particles to pinken, we make use of the fact that f e is constant on conjugacy classes to construct another permutationσ with the same cycle structure as σ. Figure 3 gives an indication of howσ will be produced from knowledge of σ and the particle colours in the case of σ being a single cycle: by modifying the trajectories of four particular particles we are able to ensure that not only doesσ have the same cycle structure as σ, but that the trajectories of all black particles in the edge are identical under both permutations (a required property -see part 1 of Lemma 3.1). We then look for vertices v such that under σ a red particle is moved to v and underσ a white particle is moved to v; a certain subset of these particles will be pinkened. It is for this reason (i.e. needing to know the colours of four particular particles) that we are able to relate the mixing time of k particles to that of just four particles in Lemma 1.8. The chameleon process also updates at additional times (compared to its corresponding interchange). We refer to these additional updates as depinkings, as at these times we get the opportunity to collectively recolour all pink particles in the system either red or white. As in (Oliveira, 2013), we will only perform a depinking once there are a large number of pink particles (compared to the number of red and white) in the system.

New permutations from old
The first step towards constructing the chameleon process is describing how to generate the permutationσ (the new permutation) from σ (the old permutation), which is the subject of this section. We begin with the case of σ being a single cycle (of size at least 3), then consider when σ is a product of disjoint transpositions, and finally by combining these two methods we describe how to constructσ for general permutations σ. We conclude the section by describing an algorithm to generate a certain set A which plays a crucial role in the construction of the chameleon process. otherwise. (6.1) For d = 3 we similarly define the function β 0 : For a cycle σ ∈ C d we may write where for m ∈ N we write σ m for the composition of m copies of σ (and where we may sometimes write σ d ≡ σ 0 for the identity permutation).
Lemma 6.2. For d ≥ 3 and any i, j ∈ [d ]: 1. The function β i is self-inverse; 2. functions β i and β j commute for i = j.
Proof. Part 1 follows directly from the definition of β i . Part 2 only applies when d ≥ 4, and follows from the observation that the transpositions in Remark 6.1 corresponding to β i and β j commute for i = j.
Now consider what happens when we apply the function β to a uniformly chosen cycle σ ∈ C d . Clearly, for any set A chosen independently of σ, the resulting cycle β A (σ) will also be uniformly distributed on C d . Most importantly, this remains true even when A is allowed to depend upon σ, as long as a certain condition is met, as explained in the following Lemma. We denote by P Ω the power set of a set Ω.
Lemma 6.5. Let d ≥ 3 and suppose A :

6)
and that σ is chosen uniformly from C d . Then β A(σ) (σ) is also uniform on C d . Moreover, if we average over the input permutation σ, then the output β A(σ) (σ) is independent of the choice of A.
Although Lemma 6.5 is relatively simple, its importance should be emphasised at this point. We shall make use of the function β A to generate the random permutationsσ used in the construction of the chameleon process, and in doing so the input A will depend on the state of the chameleon process. The second part of Lemma 6.5 will be used to guarantee that the permutationσ = β A(σ) (σ) is independent of A. (The permutationsσ will be used to generate an interchange processx IP , and so it will be crucial that these do not depend on the state of the process.)

Composition of transpositions
Fix d ∈ 2N and let T d be the set of products of disjoint transpositions: For σ ∈ T d , we define an ordering, denoted ≺, of the transpositions in σ as follows: (a i a j ) ≺ (a k a ) if and only if (a i ∧ a j ) < (a k ∧ a ). Without loss of generality we shall always suppose that any σ ∈ T d is written such that (a 1 a 2 ) ≺ (a 3 a 4 ) ≺ · · · ≺ (a d−1 a d ), and a 2i−1 < a 2i for all 1 ≤ i ≤ d/2.

Given a set A ⊆ [d ]
and a permutation σ ∈ T d , we can produce a new permutation β A (σ) by multiplying σ (on the right) by a particular set of disjoint transpositions, as follows: This new permutation satisfies analogous properties to those already observed to hold (in Lemmas 6.4 and 6.5) for β A (σ) when σ ∈ C d . The proofs are straightforward, but it is worth emphasising that part 2 of Lemma 6.6 (that β A is an involution) holds precisely because we only multiply by the transpositions for which a 4i−1 < a 4i−2 .
Lemma 6.6. For any σ ∈ T d and set A ⊆ [d ]: Lemma 6.7. Suppose A : T d → P [d ] satisfies for all σ ∈ T d ,

A(β A(σ) (σ)) = A(σ)
and σ is uniform from T d . Then β A(σ) (σ) is also uniform on T d . Moreover, if we average over the input permutation σ, then the output β A(σ) (σ) is independent of the choice of A.

General permutations
By combining the ideas from the previous two sections we can now describe the algorithm for the construction ofσ (which will be given by β A(σ) (σ) for some function β A(·) (·) to be defined) when σ ∈ S n is a general permutation. The first step is to decompose the input permutation σ into its canonical cyclic decomposition form. Indeed, except for transpositions, the function β A(·) (·) will act independently on each cycle in a given permutation's decomposition.
Suppose σ has canonical cyclic decomposition form (where we omit fixed points): where K denotes the number of cycles in σ of size at least 3, and ρ 0 is a (possibly empty) product of disjoint transpositions.
For i = 0, 1, . . . , K we write m i for the minimal element of ρ i , and write d i for the size of the non-trivial orbit of ρ i . (For example, if σ = (1 4)(2 9)(3 7 6 8 5) ∈ S 9 then K = 1, m 0 = 1, m 1 = 3, d 0 = 4 and d 1 = 5.) Given the elements of the non-trivial orbit of ρ i , there is an obvious natural bijection between permutations of those elements and permutations of the set [d i ] = {1, . . . , d i }, in which the minimal element m i is mapped to 1. Rather than writing out this correspondence in detail, in order to ease notation in what follows we shall simply consider ρ i to be a member of the set C d i etc, even though the set of elements belonging to ρ i will not in general be {1, . . . , d i }.

With this understanding in mind, suppose that A(σ) is a vector of the form
Then we can easily extend the idea of our functions β A to apply to general permutations.
Definition 6.8. Let σ ∈ S n be a permutation with cyclic decomposition (6.7), and assume that A is a function on S n satisfying (6.8). Then we defineβ A(σ) (σ) to be the composition of the permutations obtained by applying the functions β A i (ρ i ) separately to each ρ i : where β A i (ρ i ) (ρ i ) are as defined in Sections 6.1.1 and 6.1.2 (but with m i replacing the element 1, as already explained).
Definition 6.8 says thatβ A(σ) (σ) is obtained from σ by modifying each of its cycles of size at least 3, and the set of disjoint transpositions, independently using functions β A i (·) (·) with which we are already familiar. We therefore have the following corollary to Lemmas 6.4 and 6.6: Corollary 6.9. For any σ ∈ S n and function A on S n satisfying (6.8): 1.β A(σ) (σ) belongs to the same conjugacy class as σ; Furthermore, note that if we choose a random permutation σ ∈ S n according to a law f which is constant on conjugacy classes, then given the sizes of the cycles in the decomposition of σ, the elements of [n] belonging to each cycle are (marginally) uniform. We can therefore also obtain a corollary to Lemmas 6.5 and 6.7: Corollary 6.10. Suppose that A is a function on S n satisfying (6.8), and that for all σ ∈ S n with cyclic decomposition (6.7) and each i = 0, 1, . . . , K, If σ is chosen according to law f on S n which is constant on conjugacy classes thenβ A(σ) (σ) also has law f on S n . Moreover, if we average over the input permutation σ, then the output β A(σ) (σ) is independent of the choice of A.

Choosing the set A
We have described in Section 6.1.3 how to generate a new permutationσ =β A(σ) (σ) from a permutation σ such that it has the same law as σ. We now detail our method for choosing the vector A(σ) appearing in the definition ofβ A , in such a way that the conditions of Corollary 6.10 are satisfied; an illustrative example can be found in Figures 4 and 5. Our choice of A will depend not only on σ but also on particular subsets of vertices in the edge under consideration. (Later on these subsets will be specified in the chameleon process, but for now we keep them as general subsets.) Indeed, given an edge e ∈ E and a permutation σ ∈ S e , the function A is of the form A(R, W, σ), where R and W are two disjoint subsets of V .
Lemma 6.11. Fix an edge e ∈ E. For any disjoint subsets R, W of V and permutation σ ∈ S e , the functions A i (R, W, ρ i ) defined in (6.9) and (6.10) satisfy Proof. The idea here is that, as in part 3 of Corollary 6.9, β A i (R,W,ρ i ) (ρ i )(x) = ρ i (x) unless x belongs to a special set of three or four elements (whose exact definition depends upon the conjugacy class of ρ i ). Furthermore, β A i (R,W,ρ i ) permutes all elements of such a special set amongst themselves, and so the numbers of red and white vertices within the set are unchanged by the action of β A i (R,W,ρ i ) . (See Figure 5 again for a pictorial example.) We provide some details here for the case when ρ i is a cycle of length d i ≥ 4: the arguments for 3-cycles and ρ 0 are similar. Suppose j ∈ A i (R, W, ρ i ). Without loss of generality, suppose that Since j ∈ A i (R, W, ρ i ), from equation (6.4) we deduce that Therefore k ∈ A i (R, W, β A i (R,W,ρ i ) (ρ i )). The other cases follow similarly. This shows that , but an identical argument shows the reverse implication and we deduce the result.

Construction of the chameleon process
In this section we detail the construction of the chameleon process. The connection to the algorithm described in the previous section to generateσ from σ will be made clear in Lemma 6.13. In order to deal with edges of size 2, it will be convenient to modify the graphical construction of IP(k, f, G) introduced in Section 2.2, by doubling the rate at which edges ring, and compensating for this by making the process lazy.
More formally, consider the following ingredients: We now define σ θn n to equal σ n if θ n = 1 and to be the identity if θ n = 0. We modify the definition of the maps I [s,t] from Section 2.2 as follows: The thinning property of Poisson processes implies that the joint distribution of the maps I [s,t] , 0 ≤ s ≤ t < ∞, is the same as in Section 2.2. The chameleon process will be built on top of this modified interchange process. (1), . . . , z(k − 1)} denotes the (unordered) set of coordinates of z. The chameleon process is a continuous-time Markov process with state-space

Formal description of chameleon process
. Note that, due to the nature of the state-space, we can distinguish between the various black particles, whereas any two red/white/pink particles are indistinguishable from each other.
We denote the state at time t of the chameleon process started from M 0 = (z, R, P, W ) as M t = (z t , R t , P t , W t ). We say that a particle at vertex v at time t is red at time t if v ∈ R t (and similarly for the other colours). We define now a notion of ink, which represents the amount of redness either at a vertex or in the whole system. A vertex v has 1 unit of ink at time t if v ∈ R t and half a unit if v ∈ P t . Formally then, we define for each v ∈ V and t ≥ 0, We are now able to complete our formal definition of the chameleon process corresponding to an interchange process on a hypergraph. We set T = 20T EX(4,f,G) , and call T the phase length. As stated previously, the chameleon process is time-inhomogeneous, and behaves differently depending on which phase we are in. There will be just two different kinds of phase: those in which no colour-changing is permitted and particles are just moved around the graph according to the underlying interchange process; and those in which colour-changing (pinkening of red and white particles) can occur. Furthermore, there will be (deterministic) times at which depinking can occur. To be more precise, the chameleon process is updated at the incident times {τ n } of the Poisson process Λ and also at deterministic times 2iT , i ∈ N.
To describe which particles are pinkened during a colour-changing phase, let σ (= σ n ) be the permutation applied to some edge e (= e n ) at time t = τ n and once again recall the cyclic decomposition from (6.7): Given that t is in a colour-changing phase, we define subsets of V in the following way.
We note that, by construction, if j = 0 then L i,j t contains two vertices, with one in R t− and the other in W t− .
For cycles ρ i with d i ≥ 4, we define a set L i,j t for each j ∈ A i (R t− , W t− , ρ i ) as follows: i (m i ) contains a red particle, and the other three contain white particles), then set i (m i ) contains a white particle, and the other three contain red particles), then set Once again, this ensures that L i,j t contains two vertices, one in R t− and the other in W t− . For ρ 0 (the product of disjoint transpositions), we proceed similarly. For each j ∈ A 0 (R t− , W t− , ρ 0 ) satisfying a 4j−1 < a 4j−2 , we define L 0,j t as follows: (If a 4j−1 > a 4j−2 then set L 0,j t = ∅.) We then let The particles at the pairs of vertices selected in this way are those that we wish to pinken at time t. However, it turns out to be useful to limit the number of pinkenings that can occur (during a single colour-changing phase) so that the total number of pinks cannot exceed either the number of reds or the number of whites (this will be crucial to be able to appeal directly to a result of (Oliveira, 2013) in the proof of Lemma 7.2). In order to achieve this, we pick (arbitrarily) a subset L * t of with the property that |L * t | is as large as possible while still satisfying Note that L * t is a set of pairs of vertices, with each pair containing one red and one white particle. Finally, we letL t be the union of the elements of L * t . It is precisely the particles at vertices inL t that we will pinken at time t.
Box 6.12. Formal description of chameleon process updates There are three kinds of updates to the chameleon process -which update is performed at time t depends on the value of t.
In this case, we pinken all particles in the setL t (half of which belong to R t− , the others to W t− , by design). Regardless of which particles are pinkened, we then update as the interchange process at this time (using σ n ). Formally then, we update as ).
• e n = {w, r} for some w ∈ W t− , r ∈ R t− , and σ n = id.
In this case, we pinken both particles on the edge. Formally update as Depinking: For t = 2iT with i ∈ N, if |P t− | ≥ min{|R t− |, |W t− |} then we generate a Bernoulli(1/2) random variable Y i : if Y i = 1, we colour all pink particles red, otherwise we colour all pink particles white. Hence the update is Recall again the example from Figure 4, and suppose this represents the state at time t− of a chameleon process. Then Figure 6 represents the state at time t (assuming that t belongs to a colour-changing phase, and that the associated random variable θ equals 1). The result of updating the chameleon process from the state pictured in Figure 4.
Particles belonging to the set L t = {1, 2, 3, 4, 5, 8, 12, 24} have been pinkened, and then all particles have been moved according to the permutation σ = (5 21) (8 10)(16 20)(3 12 22)(1 6 17 18 19 2 13 25 24 9 7 15 14 4 11 23). (The number of particles that we are allowed to pinken depends upon the values of |R t− | and |W t− | of course, but here we have assumed for simplicity thatL t = L t .) The connection between the permutationsβ A (σ n ), which we spent time developing in Section 6.1, and the chameleon process is made explicit in the following lemma, which shall later be employed in the proof of part 2 of Lemma 3.1.
Lemma 6.13. Suppose t = τ n is in a colour-changing phase and |e n | > 2. Then there exists a permutation g : V → V such that both of the following statements hold: 1. for any vertex u containing (at time t−) a particle which is pinkened at time t in the chameleon process, 2. for any vertex u containing a particle which is not pinkened at time t, the particle at vertex g(u) at time t− has the same colour at time t as the particle at vertex u at time t−.
Proof. This follows simply by comparing the construction ofβ A(R t− ,W t− ,σn) (σ n ) with the construction of the chameleon process.
7 Proof of Lemma 3.1 In this section we show that the chameleon process satisfies the properties outlined in Lemma 3.1. Part 1 follows immediately from the construction of the chameleon process, since each black particle moves identically in the chameleon process and the underlying interchange process.
In order to prove the other three parts, we will need to understand the evolution of the total amount of ink in the chameleon process. We first of all note that the number of pink particles accumulates over time until we have a large number of them; at the next depinking time all pink particles are recoloured (either red or white) and the process of accumulation starts again. The process will continue in this manner until either we have no white particles or we have no red particles (which will occur immediately after some depinking). At this point, no more pink particles can be made and so there is no more recolouring of particles. In order to bound the mixing time of the interchange process we need a good understanding of how quickly the chameleon process reaches the state where no more recolouring can occur. There are two factors which affect this: the time we must wait between depinking events and how the process behaves at depinking times.
Writing x = (z, x), for each j ∈ N let D j (x) denote the jth depinking time of a chameleon process started from state (z, {x}, ∅, V \ (O(z) ∪ {x})) ∈ Ω k (V ). Let ink x t denote the total amount of ink in the process at time t; note that 0 ≤ ink x t ≤ |V | − k + 1. Motivated by Oliveira (2013), recall that in part 2 of Lemma 3.1 we defined the event This is the event that all initially-white particles are eventually coloured red. We shall make use of the following result concerning ink x t , whose proof may be found in (Oliveira, 2013). It is applicable in this setting because the event Fill x is independent of z C t (as it depends only on the outcomes of coin-flips at depinking times and these do not affect z C t ) and because ink x t is a martingale (clear from the construction). We note also that this result is identical to Lemma 3.1 part 4, and thus serves as its proof.
Consider now the expectation on the right-hand side of the statement of Lemma 3.2: an identical argument to that in Section 6 of (Oliveira, 2013) shows that this can be bounded in terms of the tail probability of the time of the jth depinking.
Lemma 7.2. There exist positive constants c 1 and c 2 such that for every j ∈ N, We therefore see that we need good control on the probability that there have only been a few depinkings by time t. Here we cannot simply rely on results from (Oliveira, 2013), since our chameleon process constructed in Section 6 clearly obeys very different dynamics. We shall need the following fundamental result -a lower bound on the number of red particles that are lost (due to pinkening) during a colour-changing phase of the chameleon process (where we start the phase with more white particles than red). The proof is deferred to Section 7.1.
Lemma 7.3. Suppose |V | ≥ 36 and consider a chameleon process with initial configuration (z, R, P, W ) satisfying |P | < |R| ≤ |W |. Then This result allows us to bound the probability appearing in the statement of Lemma 7.2.
Lemma 7.4. There exists a universal constant κ 1 > 0 such that for every interchange process on a regular hypergraph G = (V, E), every j ∈ N and x ∈ (V ) k , if |V | ≥ 36 then Proof. Thanks to Lemma 7.3, the proofs of Lemmas 6.2 and 9.2 of (Oliveira, 2013) can be emulated to show that there exists a positive constant κ such that E[e D j (x)/κT | Fill x ] ≤ e j for all j ∈ N. Thus by Markov's inequality, Writing κ 1 = 20κ completes the proof.
Combining Lemmas 7.2 and 7.4 completes the proof of part 3 of Lemma 3.1.
It therefore only remains to show that the chameleon process also satisfies part 2 of Lemma 3.1.
Let {τ n } n∈N denote the update times of the chameleon process {M t } t≥0 ; thus eachτ n is either an incident time of the Poisson process Λ from Section 6.2, or a depinking time (of the form 2iT with i ∈ N, as in Box 6.12). For each j ∈ N, consider a process {M j t } t≥0 which is identical to {M t } t≥0 for all t <τ j but evolves as the interchange process (i.e. with no further recolourings) for all t ≥τ j . More formally, for all t ≥τ j , where I is the map used in the modified graphical construction of the interchange process {x IP t } (see Section 6.2).
Notice that the almost-sure limit of {M j t } t≥0 as j → ∞ is the chameleon process {M t } t≥0 . As a result, by the dominated convergence theorem, it suffices to prove that for each j ∈ N and b ∈ V , where ink j t (b) is the amount of ink at vertex b in the process M j t . We prove this by induction on j. The case j = 1 is trivial since the particle initially at x is the only red particle (and there are no pink particles). For the inductive step we wish to show that almost surely For t <τ j , these are equal since the two processes evolve identically for such times. The update at timeτ j of process {M j+1 t } is a chameleon step and could be of two types: also an update of the interchange process (i.e.τ j is an incident time of the Poisson process Λ), or not (i.e. it is a depinking time). Suppose we are in the first case. We condition on the common state of M j and M j+1 at timeτ j−1 . We want to show that almost surely By the strong Markov property at timeτ j−1 we can construct a chameleon process {M j t } (with associated interchange processx IP ) which is identical to {M t } for all t <τ j , but for all t ≥τ j evolves as an interchange process (i.e. with no further recolourings) and uses permutation choices: • σ n if t = τ n is in a constant-colour phase, •β A(σn) (σ n ) if t = τ n is in a colour-changing phase.
We claim that for all b ∈ V , almost surely (where ink is the ink process underM j ). Ifτ j is in a constantcolour phase, then the statement is immediate (since all three processes update in exactly the same way). Ifτ j is in a colour-changing phase and the particle which is at b at timeτ j has just been pinkened in the chameleon process then ink j+1 τ j (b) = 1/2 and by Lemma 6.13, , and so the statement is true. Finally, ifτ j is in a colour-changing phase but the particle at b at timeτ j has not just been pinkened, then the three expectations are all equal since σ j andβ A(σ j ) (σ j ) have the same distribution, by Corollary 6.10 and Lemma 6.11 (and black particles move identically under each by Corollary 6.9). We thus have We are left to deal with the second case, whenτ j is not an update of the interchange process.
In this case there must be a depinking at timeτ j . We wish to show (7.1) holds, so again use the strong Markov property at timeτ j−1 to obtain where the second equality follows from the fact that an independent Bernoulli(1/2) random variable is used to determine the outcome of a depinking. This completes the induction, and with it the proof of part 2 of Lemma 3.1.

Proof of Lemma 7.3
In order to prove Lemma 7.3, we need to show that during a colour-changing phase (started with more white particles than red particles) the number of pink particles we create is in expectation at least a constant times the number of red particles at the start of that phase. We prove this result in this section.
Suppose we wish to lower-bound the number of white particles that are pinkened (which we shall refer to as the number of pinkenings) during the first colour-changing phase [T, 2T ].
Since we start with 1 red particle, there will be more white particles at time T than red. We will wish to apply the following analysis for a general colour-changing phase (and not just the first) but the calculations will carry through since we are assuming the number of white particles is at least the number of red.
We make a change to the chameleon process in this section in order to ease our analysis -we remove the condition that we only pinken if we have fewer pink particles than either red or white particles and replace the setL t in the formal description (Box 6.12) with the potentially larger set L t . Although this means we can end up with more pinkening events, this will only happen if a certain number of pinkening events have already happened (since pink particles are only created at times of pinkening events), and in that case we will be happy regardless.
We shall refer to this new process as the modified chameleon process.
Let a ∈ V . We define t a to be the smallest integer n such that T < τ n ≤ 2T and a ∈ e n . If no such n exists we set t a = ∞. Also, we set φ a = τ ta with notation τ ∞ = ∞; hence φ a is the first time (after time T ) that vertex a is in a ringing edge of the underlying Poisson process. We define a third variable, F a , set to be equal to * in the case φ a = ∞. If, on the other hand, φ a < ∞, there are six possible cases. Let c ta denote the cycle of σ ta containing vertex a and |c ta | denote the number of elements in c ta . For ease of notation we write d for |ct a | 4 and m for the smallest element in c ta .
Remark 7.5. From this construction it is easy to see that (for any b, c, d ∈ V ) 1. In case 1 above, we have 2. In case 3 above, we have 3. In all other cases, we have We now present a method to count the number of pinkenings during a colour-changing phase of the modified chameleon process. For ease of notation we shall write I for the map I [0,T ] . The proofs of the first three results below are fairly simple extensions of equivalent results in Oliveira (2013) and can be found in Appendix B.
Lemma 7.6. Consider a modified chameleon process with starting configuration (z, R, P, W ) satisfying |P | < |R| ≤ |W |. Then the number of pinkenings during (T, 2T ) is at least the number of b ∈ I(W ) such that one of the following holds: • F a = (b, c) for some a ∈ I(R) and c ∈ I(W ) with φ a = φ b = φ c , and θ ta = 1, • F a = (b, c, d) for some a ∈ I(R) and c, d ∈ I(W ) with φ a = φ b = φ c = φ d , and θ ta = 1.
In bounding the expected number of pinkenings during a colour-changing phase, it turns out to be useful to have a lower bound on the probability that F a = * given φ a = ∞. This is because even if vertex a is in a ringing edge during time interval [T, 2T ], in order for the particle initially at a to be pinkened in the modified chameleon process at this time, it is necessary (but not sufficient) that F a = * . The proof of this lemma makes use of part 2 of Assumption 1.1.
Proposition 7.8. Consider a modified chameleon process with initial configuration (z, R, P, W ).
Then for any vertices a, b, c, d, .
We now present the main result of this section -a version of Lemma 7.3 but proved for the modified chameleon process. As explained earlier in this section, this implies the corresponding result for our original chameleon process.
Lemma 7.9. Suppose |V | ≥ 36 and consider a modified chameleon process with initial configuration (z, R, P, W ) satisfying |P | < |R| ≤ |W |. Then Proof. Write N (b) for the set of vertices that share at least one edge of the hypergraph with b. By Lemma 7.6, we have Note now that the event {F a = (b, c, d), φ a = φ b = φ c = φ d } is determined entirely by the process after time T , and in particular is independent of the process between times 0 and T , and hence of the map I = I [0,T ] . This is also true for the event Recalling Remark 7.5 we see that the expectation of the above is equal to Using Proposition 7.8 and P [θ ta = 1] = 1/2 we obtain the bound .
Consider the final probability in the above equation. We can write it as where we have made use of the fact that the choice of the permutations (which determines |F a |) is independent of the choice of the edges that ring. We next make use of the regularity of the hypergraph (and that all edges ring at the same rate) to obtain that (7.5) Combining (7.2), (7.3), (7.4) and (7.5) gives a∈V e: e a P [F a = * ] .
Using Lemma 7.7, where {a RW t } is a realisation of RW(f, G) started from a. Since From (7.6), we deduce that Finally, the assumptions in Lemma 7.3 on the sizes of the sets P , R and W imply that 3|W | ≥ |W | + |R| + |P | = |V | − k + 1 ≥ |V |/2 , and since |V | ≥ 36 we arrive at our stated result:

A Technical proofs for Section 4
Here we include some of the more technical proofs required to compare the mixing time of EX(4, f, G) with that of EX(2, f, G).
Proof of Lemma 4.4. Recall from the proof of Lemma 4.3, that where Λ is a Poisson process of rate 2|E|. We continue to write e a,b for the edge that the special particles a * and b * meet on at time τ a,b , and σ a,b for the permutation corresponding to time τ a,b . In addition, let θ a,b denote the Bernoulli (1/2) random variable attached to the meeting time (used to thin the events of Λ). We condition on the event E 3 a,b occurring, where we recall that We now define two events which will be used to determine the coupling strategy at time τ a,b : is the event that the permutation σ a,b moves the set of two special particles to a new set of positions; event J 2 (σ a,b ) further specifies that the two positions to which σ a,b moves the special particles should either both contain another particle of A * and B * (i.e. one of the already-matched k − 1 particles) or both be empty.
With this notation in place, we can describe our coupling of A * and B * at time τ a,b : (i) if θ a,b = 0 then we move nothing, and set A * (ii) if θ a,b = 1 but event J 2 (σ a,b ) fails to hold, then we apply permutation σ a,b to both A * and B * ; (iii) if θ a,b = 1 and event J 2 (σ a,b ) holds, we apply σ a,b to A * and permutationσ a,b to B * , whereσ . (Here and throughout we use the convention that composition of permutations corresponds to multiplication on the right: σ • ρ = ρσ.) To show that this is a valid coupling of A * and B * , it suffices to show that in case (iii) the permutationσ a,b belongs to the same conjugacy class as σ a,b , and that there is a bijection between the two permutations. By inspection, the cyclic decomposition ofσ a,b is obtained from that of σ a,b just by exchanging the elements σ a,b (a * τ a,b − ) and σ a,b (b * τ a,b − ), and so both permutations belong to the same conjugacy class. Moreover, there is a bijection between them since Furthermore, it follows from the above analysis that our coupling strategy in case (iii) leads to the positions of a * τ a,b and b * τ a,b agreeing, without breaking any matches between the alreadycoupled non-special (k − 1)-particles. (Recall that the non-special particles are moving according to an exclusion process, and so we only care about their positions, not their labels, in the above coupling.) Thus in order to complete this proof, we need to show that P θ a,b = 1, J 2 (σ a,b ) | E 3 a,b ≥ 1/30. Note first of all that (using both parts of Assumption 1.1) and so a simple union bound gives P J 1 (σ a,b ) | E 3 a,b ≥ 1/5. Therefore, But conditioned on J 1 (σ a,b ) and E 3 a,b both holding, J 2 (σ a,b ) is the event that two of the positions in e a,b not containing a * τ a,b − or b * τ a,b − either both contain a matched particle or are both empty; since |e a,b | ≥ 5 this probability is at least 1/3, thanks to part 1 of Assumption 1.1, and so our proof is complete.

We have
We decompose the sum over u above into u ∈ Good i,j and u ∈ Bad i,j , where For the Good terms, we have to the sum. However, the left-hand side in the above equation is at most 2 −20 by the choice of T . We deduce that and thus However, for each {u 1 , u 2 } ∈ Good i,j we know that We deduce that P {x(i), x(j)} EX 20T ∈ Bad i,j } ≤ ε + ε −1 2 −19 . Plugging this into (A.3) gives 4 i=2 i−1 j=1 {u 1 ,u 2 }∈Bad i,j P M RW ((u 1 , u 2 )) ≤ 20T P {x(i), x(j)} EX 20T = {u 1 , u 2 } ≤ 12(ε + ε −1 2 −19 ).
Combining this with (A.2) and (A.1) gives Proof of Proposition 4.7. This proof is similar to the proof of Proposition 4.6 in (Oliveira, 2013 We present a coupling of {x IP t } t≥0 and {x RW t } t≥0 such that the two processes agree up to timē M RW (O(x)). The coupling has state-space S := (V ) 2 × V 2 which we split into two parts: ∆ := {(z, z) : z ∈ (V ) 2 } and ∆ . Denote by q(·, ·) the transition rates. We construct the coupling as follows: In the third situation with F a = (b, c, d) for some a ∈ I(R), c, d ∈ I(W ) with φ a = φ b = φ c = φ d and θ ta = 1 we have two cases. The first case is if |c ta | = 2. We denote by (a 1 a 2 ) ≺ · · · ≺ (a l−1 a l ) the ordered transpositions in the cyclic decomposition of σ ta and denote by ρ 0 the composition of these transpositions. There are four sub-cases which are all similar, and we just prove the result for one of them. So suppose there exists j ∈ {1, . . . , l/4 } with a = a 4j−3 . Then we have I [T,φa) (b, c, d) = (a 4j−1 , a 4j−2 , a 4j ). Since φ a = φ b = φ c = φ d , we have (a 4j−1 , a 4j−2 , a 4j ) = (b, c, d).
Since a ∈ I(R), b, c, d ∈ I(W ) and φ a = φ b = φ c = φ d , we have that a ∈ I [0,φa) (R) and b, c, d ∈ I [0,φa) (W ) and hence j ∈ A 0 (R φa− , W φa− , ρ 0 ) with L 0,j φa = {a, b}. Since θ ta = 1 we deduce that the particle at b at time T is pinkened at time φ a . The other three sub-cases follow similarly. Since a ∈ I(R), b, c, d ∈ I(W ) and φ a = φ b = φ c = φ d , we have that a ∈ I [0,φa) (R) and b, c, d ∈ I [0,φa) (W ) and hence j ∈ A i (R φa− , W φa− , σ ta ) for some 1 ≤ i ≤ K with L i,j φa = {a, b}. Since θ ta = 1 we deduce that the particle at b at time T is pinkened at time φ a .