Mixing time for the Repeated Balls into Bins dynamics

We estimate the mixing time of the a nonreversible finite Markov chain called Repeated Balls-into-Bins (RBB) process. This process is a discrete time conservative interacting particle system with parallel updates. Place initially in $L$ bins $rL$ balls, where $r$ is a fixed positive constant. At each time step a ball is taken from each non-empty bin. Then \emph{all the balls} are uniformly reassigned into bins. We prove that the mixing time of the RBB process depends linearly on the maximum occupation number of balls of the initial state. Thus if the initial configuration is such that the maximum occupation number of balls is of order $L$ then the mixing time is of the same correct order. While if the initial configuration is more diluted then the equilibrium is reached in a time of order $(\log L)^c$.


Introduction
Consider L ∈ N bins where rL ∈ N balls are initially placed. At each discrete time step a ball is taken from each non-empty bin and all the balls are uniformly reassigned into bins. The occupation numbers of balls into bins is an ergodic discrete time finite Markov chain, called Repeated Balls-into-Bins process [1]. The RBB process arises naturally in queue theory and in parallel algorithmic contexts. In [3] and [4] we proved the propagation of chaos of the RBB process and studied some equilibrium properties of the limiting nonlinear process. This system is a conservative interacting particles system in discrete time with parallel updates we will thus call the balls particles and the bins sites. We can think to the RBB process as a zero-range process [11] on the complete graph with constant jump rates and parallel updates however, because of the parallel updating, it is not reversible. For this reason its invariant measure is difficult to compute and still unknown and the standard techniques to study the convergence rate to equilibrium cannot be used. The main result of this paper is Theorem 2.1 where we give the correct upper bound on the mixing time of the RBB process. This bound can be used to generate, by a Monte Carlo Markov chain, an approximate sample of the invariant measure. The mixing time of systems with parallel updating rules has been studied for Probabilistic Cellular Automata, see for example [10] and [7]. However PCA models are reversible, non conservative, and their invariant measure is usually known.
To prove Theorem 2.1 we use the path coupling technique of Bubley and Dyer [2]. This technique has been successfully applied in [6] to estimate the mixing time of the mean field zero range process, which is similar to the RBB process but has sequential updating and a known reversible stationary measure. A remarkable feature of the method used in [6] is that it does not require the reversibility and the explicit knowledge of the invariant measure of the model. We use the same approach of [6] although the parallel updating of the RBB process requires new ideas.
We briefly outline the strategy of the proof of Theorem 2.1. We show that, after a thermalization time depending linearly on the maximum occupation number of the initial state, the following happens. First the distribution of the site occupation number decays exponentially. Second there is a coupling such that the distributions of two copies of the RBB process started from two different configurations are close in total variation distance. In particular we show that this distance can be estimated in terms of the coalescing time of two RBB processes starting from configurations which differ only for one particle.
The paper is organized as follows. In Section 2 we state the notations and Theorem 2.1, in Section 3 we give its proof. Finally in the last two sections we prove the two lemmata on which the proof of the main result relies.

Notations and main result
We denote with Z + the set of the non-negative integers, N := Z + \ {0} and for L ∈ N the configuration space Ω := Z L + . For any denumerable set S its cardinality, finite or infinite, is denoted by |S| and for any n ∈ N we define [n] := {1, . . . , n}. Given η ∈ Ω let η ∞ := max x∈[L] η x . If µ and ν are two probability measures we denote with µ − ν their total variation distance.
To keep notation simple in the following the term constant means a number which may depend only on r, where rL is the fixed number of particles.
Theorem 2.1 Let (η(t)) t≥0 be the RBB process. Then there is a positive constant c such that for any ε ∈ (0, 1) and η ∈ Ω the mixing time This is the correct bound because for any η ∈ Ω the process needs at least η ∞ time steps to move all the particles thus the mixing time is at least of order η ∞ (see [8] §7.12 diameter bound ).

Proof of the main result
The proof of Theorem 2.1 is based on Lemmata 3.1 and 3.2. The first one states that, after a thermalization time depending linearly on the initial state, the distribution of the site occupation number of the RBB process decays exponentially.
Lemma 3.1 There are positive constants θ, κ, α such that for all η ∈ Ω, x ∈ [L] and t ≥ 0. In particular, for any a ≥ 0 The second lemma asserts that the distributions of two RBB processes started from two different configurations, again after a thermalization time depending linearly on them, are close in total variation distance.

Lemma 3.2 There exists a positive constant c such that
We can now prove Theorem 2.1.

By Lemma 3.2 and equation (3.3) we have
for any t ≥ c η ∞ ∨ (log L) c . Using sub-additivity, ergodicity of the RBB process and Lemma 3.1, the last therm of equation (3.4) can be bounded by which, taking c ≥ 2, is smaller than a positive constant times L −2 and the result follows.
The proofs of Lemmata 3.1 and 3.2 will be discussed in the next two sections.

Exponential decay of the distribution of the site occupation number
To prove Lemma 3.1 we need some preliminary results. The first one states that the RBB process at time t is stochastically dominated by a Maxwell-Boltzmann distribution with tL particles and L sites. This gives a bound on the number of particles per site of the RBB process and it will be crucial because the Maxwell-Boltzmann distribution is negatively associated (see e.g. [5]).

Lemma 4.1
The RBB process is monotone, furthermore there exists a random vectorB(t) with Maxwell-Boltzmann distribution with tL particles and L sites such that: for any x ∈ [L].
Proof. The graphical construction of (η(t)) t≥0 leading to equation (2.1) is a monotone coupling. In fact if we start two copies (η(t)) t≥0 and (η ′ (t)) t≥0 of the RBB process which use the same U(1), U(2), . . . and such that for some t ≥ 0 and η . This implies that the RBB process is monotone (see [9] Definition 2.3).
The next result states that if we start the RBB process from any configuration, after a fixed thermalization time, with high probability there are order L empty sites. Lemma 4.2 There exists a constant ε 0 ∈ (0, 1) such that for any ε ∈ (0, ε 0 ] for any t ≥ 2r and L ≥ 2. Proof. Fix L ≥ 2, it is enough to show that (4.3) holds for t = ⌊2r⌋. In fact, assuming that it holds for t k := ⌊2r⌋ + k where k ∈ Z + , then by Markov property and (4.3) follows for any t ≥ t 0 = ⌊2r⌋.

Coalescing time
To prove Lemma 3.2 we use the path coupling technique.
We reduce the problem of bounding the total variation distance of the distributions of two copies of the RBB process starting from different initial configurations to the problem of bounding the coalescing time of two tagged particles coupled to the RBB process.

Theorem 5.1
There is a positive constant c such that As the proof of Theorem 5.1 needs some extra work we first use it to prove Lemma 3.2. Proof of Lemma 3.2 Recall that P t η = P η (η(t) ∈ ·) is the distribution of η(t) when η(0) = η. We say that two configurations η, ξ ∈ Ω are adjacent if there are x, y ∈ [L] such that ξ x = η x − 1, ξ y = η y + 1 and ξ z = η z for any z ∈ [L] \ {x, y}. We observe that for any η, ξ ∈ Ω there is a sequence of adjacent configurations η := ζ 0 , ζ 1 , . . . , ζ k := ξ, with k ≤ rL and max j∈[k] ζ k ∞ ≤ η ∞ ∨ ξ ∞ . By triangle inequality Thus we have to bound P t ζ j−1 − P t ζ j for two adjacent configurations. We can consider a process (χ(t)) t≥0 , defined at the beginning of this section, starting from the initial condition χ j−1 such that η X (0) = ζ j−1 and η Y (0) = ζ j . Then, if τ is the coalescing time defined in (5.1), by Theorem 5.4 of [8] and Theorem 5.1, we have Proof. For any t ≥ 0 we have thus for any a > 0 we can write Because η(t) and U(t) are independent P χ (η z (t) > a | U(t)) = P χ (η z (t) > a) where in the last inequality we used (3.2). The result follows if one can choose a and t such that the last term in the above inequality is less than 1/2. Taking c 1 := ⌊1/α⌋ + 1, t =t := c 1 η ∞ and c 2 such that this happens.
Next lemma links the coalescing time with the starting site occupation numbers of the tagged particles.
From the last two lemmata the next result follows There is a positive constant β such that for any χ = (η, Proof. Let t := c 1 η ∞ , a := c 2 log(1 + η ∞ ) and h := η x 0 ∨ η y 0 + ⌊2r⌋ + 1, with c 1 and c 2 as in Lemma 5.2. Then by Markov property By Lemma 5.3 we have that From this the result follows.
We finally are in the position to prove Theorem 5.1 Proof of Theorem 5.1 Let β be as in Proposition 5.4 and let t and a be two positive parameters we will adjust later. Consider the decreasing sequence of events E 1 , E 2 , . . . defined by By Markov property As E h implies η(t h ) ∞ ≤ a, by Proposition 5.4, we have P χ(t h ) (τ > aβ) ≤ 1 − (1 + a) −β and P χ (E h+1 ) ≤ P χ (E h ) 1 − (1 + a) −β ≤ P χ (E h ) exp − (1 + a) −β .