2.1 The permutation model

For the permutation model, let $(U_{x})_{x\ge 1}$ be a sequence of i.i.d. $\mathrm {Uniform}([0,1])$ random variables. For each integer $n\ge 1$ , we denote by $V_{1:n}<V_{2:n}<\cdots <V_{n:n}$ the order statistics of $U_{1},U_{2},\cdots , U_{n}$ . Then it is easy to see that the random permutation $\Sigma ^{n}$ on $[n]$ such that $V_{i:n}=U_{\Sigma ^{n}(i)}$ for all $1\le i \le n$ is uniformly distributed among all permutations on $[n]$ . Define

(4) $$ \begin{align} \xi^{n}_{x} := \Sigma^{n}(x) \cdot \mathbf{1}(1\le x \le n). \end{align} $$

We now state our main result for the permutation model. We obtain a precise first-order asymptotic for the largest k rows and columns, as stated in the following theorem.

Theorem 2.1 (The permutation model).

Let $\xi ^{n}$ be the permutation model as above. For each $k\ge 1$ , denote $\rho _{k}(n)=\rho _{k}(\xi ^{n})$ and $\lambda _{k}(n)=\lambda _{k}(\xi ^{n})$ . Then for each fixed $k\ge 1$ , almost surely,

(5) $$ \begin{align} \lim_{n\rightarrow \infty} n^{-1}\rho_{k}(n) &= \frac{1}{k(k+1)}, \qquad \lim_{n\rightarrow\infty} n^{-1/2}\lambda_{k}(n) = \frac{2}{\sqrt{k-1}+\sqrt{k}}. \end{align} $$

Our proof of Theorem 2.1 proceeds as follows. We first establish a combinatorial lemma (Lemma 3.5) that associates the soliton lengths and numbers with a modified version of Greene-Kleitman invariants for BBS. We then utilize the tail bounds on longest increasing subsequences in uniformly random permutations in Baik, Deift and Johansson [Reference Baik, Deift and JohanssonBDJ99] for establishing the scaling limit for the lengths of the columns. For the row lengths, we use the characterization of soliton numbers as an additive functional of finite-capacity carrier processes [Reference Kuniba and LyuKL20]. Such a process becomes an exclusion process on the unit circle for the permutation model.

2.2 The independence model

To define the independence model, fix integers $n,\kappa \ge 1$ . Let $\mathbf {p}=(p_{0},p_{1},\cdots ,p_{\kappa })$ be a probability distribution on $\{ 0,1,\cdots ,\kappa \}$ . Let $\xi =\xi ^{\mathbf {p}}$ be the sequence $(\xi _{x})_{x\in \mathbb {N}}$ of i.i.d. random variables $\xi _{x}$ where

(6) $$ \begin{align} {\mathbb{P}}(\xi_{x}=i) = p_{i} \qquad \text{for } i=0,1,\dots,\kappa. \end{align} $$

For each integer $n\ge 1$ , define $\kappa $ -color BBS configuration $\xi ^{n,\mathbf {p}}$ of size n by

(7) $$ \begin{align} \xi^{n,\mathbf{p}}_{x} &= \xi^{\mathbf{p}}_{x} \cdot \mathbf{1}(1\le x \le n). \end{align} $$

We may further assume, without loss of generality, that $p_{i}>0$ for all $1\le i\le \kappa $ . Indeed, if $p_{i}=0$ for some i, then we can omit the color i entirely and consider the system as a $(\kappa -1)$ -color BBS by shifting the colors $\{i+1,\cdots , \kappa \}$ to $\{i,\cdots , \kappa -1 \}$ .

Through various combinatorial lemmas (see Section 3), we will establish that the soliton lengths $\lambda _{j}(n)$ of for the i.i.d. model are closely related to the extreme behavior of a Markov chain $(W_{x})_{x\in \mathbb {N}}$ defined on the nonnegative integer orthant $\mathbb {Z}_{\ge 0}^{\kappa }$ , which we call the ‘ $\kappa $ -color carrier process’. Denote $\mathbf {e}_{i}\in \mathbb {Z}^{\kappa }$ whose coordinates are all zero except the ith coordinate being 1.

Definition 2.2 ( $\kappa $ -color carrier process).

Let $\xi :=(\xi _{x})_{x\in \mathbb {N}}$ be $\kappa $ -color ball configuration. The ( $\kappa $ -color) carrier process over $\xi $ is a process $(W_{x})_{x\in \mathbb {N}}$ on the state space $\Omega :=\mathbb {Z}^{\kappa }_{\ge 0}$ defined by the following evolution rule: Denoting $i:=\xi _{x+1}$ if $\xi _{x+1}\in \{1,\dots ,\kappa \}$ and $i:=\kappa +1$ if $\xi _{x+1}=0$ ,

(8) $$ \begin{align} W_{x+1} - W_{x} = \begin{cases} \mathbf{e}_{i} - \mathbf{1}(i_{*}\ne 0) \, \mathbf{e}_{i_{*}} & \text{if } 1\le i \le \kappa \\ -\mathbf{1}(i_{*}\ne 0) \, \mathbf{e}_{i_{*}} & \text{if } i=\kappa+1, \end{cases} \end{align} $$

where $ i_{*}:=\sup \{ 1\le j< i \,:\, W_{x}(j)\ge 1 \}$ with the convention $\sup \emptyset = 0$ . Unless otherwise mentioned, we take $W_{0}=\mathbf {0}$ and $\xi =\xi ^{\mathbf {p}}$ with density $\mathbf {p}=(p_{0},\dots ,p_{\kappa })$ .

In words, at location x, the carrier holds $W_{x}(i)$ balls of color i for $i=1,\dots ,\kappa $ . When a new ball of color $1\le \xi _{x+1}\le \kappa $ is inserted into the carrier $W_{x}$ , then a ball of the largest available color that is smaller than $\xi _{x}$ is excluded from $W_{x}$ ; if there is no such ball in $W_{x}$ , then no ball is excluded. If $\xi _{x+1}=0$ , then no new ball is inserted, and a ball of the largest available color that is smaller than $\xi _{x}$ is excluded from $W_{x}$ . The resulting state of the carrier is $W_{x+1}$ . We call the transition rule (8) as the ‘circular exclusion’ (since a ball in the carrier’s possession is excluded from the carrier upon the insertion of a new ball according to the circular ordering). One can also view the carrier process as a multitype queuing system, where $W_{x}$ denotes the state of the queue and $W_{x}(i)$ is the number of jobs of ‘cyclic hierarchy’ i to be processed.

A large portion of this paper will be devoted to analyzing scaling limits of the carrier process $W_{x}$ over the i.i.d. configuration $\xi ^{\mathbf {p}}$ . In this case, $W_{x}$ is a Markov chain on the state space of the nonnegative integer orthant $\Omega $ . See Figure 1 for an illustration.

Figure 1 State space diagram for the carrier process $W_{x}$ for $\kappa =2$ . Red arrows illustrate the transition kernel at the ‘interior’ (gray) and ‘boundary’ (green) points in the state space. A single excursion (starting and ending at the origin) of ‘height’ $8$ is shown in a blue path with arrows.

Theorem 2.3 states the behavior of the carrier process in the subcritical regime $p_0>\max (p_1, \cdots , p_{\kappa })$ . Define a function $\pi :\Omega \rightarrow \mathbb {R}$ by

(9) $$ \begin{align} \pi(n_{1},n_{2},\cdots, n_{\kappa}) = \prod_{i=1}^{\kappa} \left(1-\frac{p_{i}}{p_{0}} \right) \left( \frac{p_{i}}{p_{0}} \right)^{n_{i}}. \end{align} $$

This is a valid probability distribution on $\Omega $ when $p_{0}>\max (p_{1},\cdots ,p_{\kappa })$ since

(10) $$ \begin{align} \sum_{n_{1}=0}^{\infty}\cdots \sum_{n_{\kappa}=0}^{\infty} \prod_{i=1}^{\kappa} \left( \frac{p_{i}}{p_{0}} \right)^{n_{i}} = \prod_{i=1}^{\kappa} \left(1-\frac{p_{i}}{p_{0}} \right)^{-1} \in (0,\infty). \end{align} $$

Note that $\pi $ is the the product of geometric distributions of means $p_{i}/(p_{0}-p_{i})>0$ for $i=1,\dots ,\kappa $ .

By using Theorem 2.3, we establish sharp scaling limit of soliton lengths for the independence model in the subcritical regime, which is stated in Theorem 2.4 below. (See Section 1.5 for a precise definition of Landau notations.)

Theorem 2.4 (The independence model – Subcritical regime).

Fix $\kappa \ge 1$ and let $\xi ^{n,\mathbf {p}}$ be as the i.i.d. model above. Denote $\lambda _{j}(n)=\lambda _{j}(\xi ^{n,\mathbf {p}})$ , $p^{*} := \max _{1\le i \le \kappa } p_{i}$ , and $r:=| \{ 1\le i \le \kappa \,\colon \, p_{i}=p^{*}\}|$ . Suppose $p_{0}>p^{*}$ and denote $\theta := p^{*}/p_{0}$ . Then for each fixed $j\ge 1$ ,

(13) $$ \begin{align} \lambda_{j}(n) = \log_{\theta} n + (r-1)\log_{\theta} \log n + \Theta(1). \end{align} $$

Furthermore, denote $\nu _{n}:=(1+\delta _{n}) \log _{\theta }\left ( \sigma n /(r-1)! \right )$ , where $\sigma := \prod _{i=1}^{\kappa }\left ( 1-\frac {p_{i}}{p_{0}}\right )$ and $\delta _{n} := \frac {(r-1) \log \log _{\theta }\left ( \sigma n /(r-1)! \right ) + \log (r-1)! }{\log \sigma n / (r-1)!}$ . Then for all $x\in \mathbb {R}$ ,

(14) $$ \begin{align} \exp(-\delta \theta^{-x}) &\leq {\mathop{\mathrm{lim\,inf}}\limits_{n\rightarrow \infty}}{\mathbb{P}}\left( \lambda_{j}(n) \leq x+\nu_{n} \right) \end{align} $$

(15) $$ \begin{align} &\leq {\mathop{\mathrm{lim\,sup}}\limits_{n\rightarrow \infty}}{\mathbb{P}}\left( \lambda_{j}(n) \leq x+\nu_{n} \right) \leq \exp\left(-\frac{C}{(r-1)!} \theta^{-(x-1)} \right)\sum_{k=0}^{j-1} \frac{\theta^{-k(x-1)}}{k! (r-1)! }, \end{align} $$

where $\delta>0$ , $C\ge 1$ are constants in Theorem 2.3.

Next, we turn our attention to the critical and the supercritical regime, where $p_{0} \le \max (p_{1},\cdots , p_{\kappa })$ . In this regime, the carrier process does not have a stationary distribution, and we are interested in identifying the limit of the carrier process in the linear and diffusive scales. A natural candidate for the diffusive scaling limit (if it exists) would be the semimartingale reflecting Brownian motion (SRBM) [Reference WilliamsWil95], whose definition we recall in Section 10. Roughly speaking, an SRBM on a domain $S\subseteq \mathbb {R}^{\kappa }$ is a stochastic process $\mathcal {W}$ that admits a Skorokhod-type decomposition

(16) $$ \begin{align} \mathcal{W} = X + RY, \end{align} $$

where X is a $\kappa $ -dimensional Brownian motion with drift $\theta $ , covariance matrix $\Sigma $ and initial distribution $\nu $ . The ‘interior process’ X gives the behavior of $\mathcal {W}$ in the interior of S. When it is at the boundary of S, it is pushed instantaneously toward the interior of S along the direction specified by the ‘reflection matrix’ R and an associated ‘pushing process’ Y. We say such $\mathcal {W}$ is a SRBM associated with $(S, \theta , \Sigma , R, \nu )$ . If $R=I-Q$ for some nonnegative matrix Q with spectral radius less than one, then such $\mathcal {W}$ is unique (pathwise) for possibly degenerate $\Sigma $ when $S=\mathbb {R}^{\kappa }_{\ge 0}$ [Reference Harrison and ReimanHR81]. If $\Sigma $ is nondegenerate and S is a polyhedron, a necessary and sufficient condition for the existence and uniqueness of such SRBM is that R is ‘completely- $\mathcal {S}$ ’ (see Definition 10.2) [Reference WilliamsWil95, Reference Kang and WilliamsKW07].

A crucial observation for analyzing the carrier process in the critical and supercritical regimes is the following. Of all the $\kappa $ coordinates of $W_{x}$ , some have a negative drift and some others do not. We call an integer $1\le i\le \kappa $ an unstable color if $p_{i}\ge \max (p_{i+1},\cdots ,p_{\kappa },p_{0})$ and a stable color otherwise. Since balls of color i can only be excluded by balls of colors in $\{i+1,\dots ,\kappa ,0\}$ , then the coordinate $W_{x}(i)$ is likely to diminish if the color i is stable but not if i is unstable. Denote the set of all unstable colors by $\mathcal {C}_{u}^{\mathbf {p}}=\{\alpha _{1},\cdots , \alpha _{r}\}$ with $\alpha _{1}<\cdots <\alpha _{r}$ and let $\mathcal {C}_{s}^{\mathbf {p}}:=\{0,1,\cdots ,\kappa \}\setminus \mathcal {C}_{u}^{\mathbf {p}}$ denote the set of stable colors. (See Figure 8 for illustration.) By definition, we have

(17) $$ \begin{align} p_{\alpha_{1}}\ge p_{\alpha_{2}} \ge \dots \ge p_{\alpha_{r}} \ge p_{\alpha_{r+1}}:=p_{0}. \end{align} $$

Now, we will construct a new process $X_{x}$ , which we call the ‘decoupled carrier process’ (see Section 6.1), that mimics the behavior of $W_{x}$ , but the values of $X_{x}$ on the unstable colors are unconstrained and thus can be negative. Since $W_{x}$ is confined in the nonnegative orthant $\mathbb {Z}^{\kappa }_{\ge 0}$ but $X_{x}$ is not, we need to add some correction process to $X_{x}$ that ‘pushes’ it toward the orthant $\mathbb {Z}^{\kappa }_{\ge 0}$ whenever $X_{x}$ has some of its coordinates going to negative. More precisely, in Lemma 6.3, we identify a ‘reflection matrix’ $R\in \mathbb {R}^{\kappa \times \kappa }$ and a ‘pushing process’ $Y_{x}$ on $\mathbb {Z}^{\kappa }$ such that

(18) $$ \begin{align} W_{x} = X_{x} + R Y_{x} \quad \text{for } x\ge 0, \end{align} $$

where $Y_{0}=\mathbf {0}$ , and for each $i\in \{1,\dots ,\kappa \}$ , the ith coordinate of $Y_{x}$ is nondecreasing in x and can only increase when $W_{x}(i)=0$ . We call the above as a Skorokhod decomposition of the carrier process (Our definition is motivated by the Skorokhod problem; see Definition 10.3.) This and the classical invariance principle for SRBM [Reference Reiman and WilliamsRW88] are the keys to establishing the following result on the scaling limit of the carrier process.

Theorem 2.5 (Linear and diffusive scaling limit of the carrier process).

Suppose $p_{0} \le \max (p_{1},\cdots , p_{\kappa })$ . Let $\alpha _{1}<\cdots <\alpha _{r}$ as before and define

(19) $$ \begin{align} \boldsymbol{\mu}=(\mu_{1},\dots,\mu_{\kappa}):=\sum_{j=1}^{r} \mathbf{e}_{\alpha_{j}} (p_{\alpha_{j}}- p_{\alpha_{j+1}}), \end{align} $$

where we let $p_{\alpha _{r+1}}=p_{0}$ .

1. (i) (Linear scaling) Almost surely,

   (20) $$ \begin{align} \lim_{x\rightarrow\infty} \, x^{-1} W_{x} =\lim_{x\rightarrow\infty} \, x^{-1} \left( \max_{0\le t \le x}W_{t}(i)\,;\, i=1,\dots,\kappa \right)= \boldsymbol{\mu}. \end{align} $$

2. (ii) (Diffusive scaling) Let $(\overline {W}_{t})_{t\in \mathbb {R}_{\ge 0}}$ denote the linear interpolation of $(W_{x}-x \boldsymbol {\mu })_{x\in \mathbb {N}}$ . Then as $n\rightarrow \infty $ ,

   (21) $$ \begin{align} (x^{-1/2}\overline{W}_{xt} \,;\, 0\leq t \leq 1 ) \Longrightarrow \mathcal{W} \, \text{ in }\, C([0,1]), \end{align} $$
   where $\mathcal {W}$ is an SRBM associated with data $(S, \mathbf {0}, \Sigma , R, \delta _{\mathbf {0}})$ (see Definition 10.1) with $S:=\{ (x_{1},\dots ,x_{\kappa })\in \mathbb {R}^{\kappa } \,:\, x_{i}\ge 0 \,\, \text {if } \mu _{i}=0 \}$ , $\Sigma $ the limiting covariance matrix (possibly degenerate) in (177), $R := \mathrm {tridiag}_{\kappa }(0,1,-1)$ , and $\delta _{\mathbf {0}}$ the point mass at $\mathbf {0}$ .

In Figures 2 and 3, we provide simulations of the carrier process $W_{x}=(W_{x}(1), W_{x}(2))$ for $\kappa =2$ in various regimes, numerically verifying Theorem 2.5. In Figure 2, we show the carrier process in diffusive scaling ( $n^{-1/2}$ ) at three different critical ball densities $\mathbf {p}$ . The carrier process in diffusive scaling converges weakly to an SRBM in $\mathbb {R}^{2}_{\ge 0}$ , whose covariance matrix depends on $\mathbf {p}$ and can be degenerate. For instance, at $\mathbf {p}=(4/11,4/11,3/11)$ , $W_{x}(2)$ is subcritical (since $p_{2}=3/11<4/11=p_{0}$ ), and $W_{x}(1)$ is critical, so the SRBM degenerates in the second axes.

Figure 2 Simulation of the carrier process $W_{x}$ in diffusive scaling for $\kappa =2$ , $n=2\times 10^{5}$ , at three critical ball densities (left) $\mathbf {p}=(4/11,4/11,3/11)$ , (middle) $\mathbf {p}=(1/3,1/3,1/3)$ and (right) $\mathbf {p}=(4/11,3/11,4/11)$ . In all cases, the process converges weakly to a semimartingale reflecting Brownian motion on $\mathbb {R}^{2}_{\ge 0}$ whose covariance matrix is nondegenerate in the middle and degenerate in the other two cases.

Figure 3 Simulation of the carrier process $W_{x}$ in diffusive scaling for $\kappa =2$ , $n=2\times 10^{5}$ , at four supercritical ball densities $(a)$ $\mathbf {p}=(3/11,4/11,4/11)$ , $(b)$ $\mathbf {p}=(3/11,5/11,3/11)$ , $(c)$ $\mathbf {p}=(2/11,5/11,4/11)$ and $(d)$ $\mathbf {p}=(3/11,6/11,2/11)$ . The processes grow linearly at least in one dimension (the top row shows uncentered processes in diffusive scaling). As shown in the second row, after centering by the mean drift $\boldsymbol {\mu }$ , the processes converge weakly to semimartingale reflecting Brownian motion on domains $(a)$ $\mathbb {R}_{\ge 0}\times \mathbb {R}$ , $(b)$ $\mathbb {R}\times \mathbb {R}_{\ge 0}$ , $(c)$ $\mathbb {R}^{2}$ (no reflection) and $(d)$ $\mathbb {R}\times \mathbb {R}_{\ge 0}$ (with a degenerate covariance matrix).

In Figure 3, we show the carrier process in diffusive scaling at three different supercritical ball densities $\mathbf {p}$ . The carrier process has a nonzero drift $\boldsymbol {\mu }=(\mu _{1},\mu _{2})\in \mathbb {R}^{2}_{\ge 0}$ . If $\mu _{1},\mu _{2}>0$ , then the centered carrier process $W_{x}-x \boldsymbol {\mu }$ converges weakly to a 2-dimensional Brownian motion in diffusive scaling. If either $\mu _{1}$ or $\mu _{2}$ equals zero, then the diffusive scaling limit is an SRBM on $\mathbb {R}_{\ge 0} \times \mathbb {R}$ or $\mathbb {R}\times \mathbb {R}_{\ge 0}$ , which is the domain S in the statement of Theorem 2.5 (ii). For instance, for $\mathbf {p}=(3/11, 6/11, 2/11)$ as in Figure 3 (d), the SRBM is on domain $S=\mathbb {R}\times \mathbb {R}_{\ge 0}$ and has a degenerate covariance matrix, since $W_{x}(2)$ is subcritical and vanishes in the diffusive scale.

Using the linear and the diffusive scaling limit of the carrier process in Theorem 2.5, we obtain a sharp scaling limit of soliton lengths for the independence model in the critical and subcritical regimes. These results are stated in Theorems 2.6 and 2.7 below.

Theorem 2.6 (The independence model – Critical regime).

Suppose $p^{*}=p_{0}$ . Then for each fixed $j\ge 1$ , $\lambda _{j}(n)=\Theta (\sqrt {n})$ . Furthermore, let $\Sigma $ be a $\kappa \times \kappa $ covariance matrix defined explicitly in (177) and $R=\mathrm {tridiag}_{\kappa \times \kappa }(0,1,-1)$ . Let $\mathcal {W}$ be a semimartingale reflecting Brownian motion associated with data $(\mathbb {R}^{\kappa }_{\ge 0}, \mathbf {0}, \Sigma , R, \delta _{\mathbf {0}})$ (see Definition 10.1). Then as $n\rightarrow \infty $ ,

(22) $$ \begin{align} n^{-1/2} \lambda_{1}(n) \Longrightarrow \sup\, \lVert W \rVert_{1}, \end{align} $$

where $\Longrightarrow $ denotes weak convergence.

Multiple remarks on Theorems 2.4–2.7 are in order. These results extend the ‘double-jump’ phase transition on soliton lengths for the $\kappa =1$ case established by Levine, Lyu and Pike [Reference Levine, Lyu and PikeLLP20] to the multicolor case. As in the $\kappa =1$ case, we find that there exist three regimes – subcritical ( $\lambda _{1}(n)=\Theta (\log n)$ ), critical ( $\lambda _{1}(n)=\Theta (\sqrt {n})$ ) and supercritical ( $\lambda _{1}(n)=\Theta (n)$ ) – depending on whether the maximum ball density $p^{*}=\max (p_{1},\dots ,p_{\kappa })$ exceeds the empty box density $p_{0}$ . However, we find that the scaling behavior of the soliton lengths inside each regime is significantly more nuanced in the multicolor case than in the single-color case.

In the subcritical regime $p^{*}<p_{0}$ , we find all top soliton lengths $\lambda _{j}(n)$ for $j\ge 1$ are concentrated around $\log _{\theta } n + (r-1)\log _{\theta } \log n$ , where $\theta =p^{*}/p_{0}$ and r denotes the multiplicity of the maximum positive color $p^{*}$ , and the tail of $\lambda _{n}(n)$ has a Gumbel-type tail distribution. While this scaling coincides with that in the $\kappa =1$ case for $r=1$ , if $r\ge 2$ , then the top solitons are an asymptotically ‘a tad’ longer by $(r-1)\log _{\theta } \log n$ , which is caused by the competition between multiple maximal colors.

In the critical regime $p^{*}=p_{0}$ , we find that $\lambda _{1}(n)/\sqrt {n} \Rightarrow D$ , where the distribution of the nondegenerate random variable D depends on a SRBM on the orthant $\mathbb {R}^{\kappa }_{\ge 0}$ with zero drift and an explicit covariance matrix $\Sigma $ . This is the same SRBM to which the entire carrier process converges weakly in diffusive scaling as in Theorem 2.5. For instance, if $p^{*}$ is uniquely achieved, then the SRBM $\mathcal {W}$ is degenerate in all but one dimension. In particular, for $\kappa =1$ , our result recovers the corresponding result in [Reference Levine, Lyu and PikeLLP20]. In general, $\Sigma $ can depend on the entire $\mathbf {p}$ , capturing the intertwined interaction between balls of all colors in the multicolor case.

In the supercritical regime $p^{*}>p_{0}$ , Theorem 2.7 shows that $\lambda _{1}(n)/n \rightarrow p^{*}-p_{0}$ almost surely and the fluctuation of $\lambda _{1}(n)$ about its mean is of order $\sqrt {n}$ . While a central limit theorem (CLT) for $\lambda _{1}(n)$ in the supercritical regime was shown in [Reference Levine, Lyu and PikeLLP20] for the $\kappa =1$ case, we find in the multicolor case that the distribution of the fluctuation of $\lambda _{1}(n)$ does not always satisfy CLT. More precisely, the following corollary shows that CLT holds for $\lambda _{1}(n)$ if and only if the ball density is strictly decreasing on the unstable colors. (Recall (17).)

Indeed, suppose $p_{\alpha _{1}}>\dots >p_{\alpha _{r}}$ as in Corollary 2.8 (i). Then Theorem 2.5 states that $x^{-1/2}(W_{x}-\boldsymbol {\mu } x)$ converges weakly to the (nonreflecting) Brownian motion in $\mathbb {R}^{\kappa }$ with covariance matrix $\Sigma $ . Hence, in this case, Theorem 2.7 (i) immediately implies that

(29) $$ \begin{align} \frac{\lambda_{1}(n)-(p_{*}-p_{0})n}{\sqrt{n }} \Longrightarrow \sum_{i=1}^{\kappa} B^{i}(1), \end{align} $$

where $B=(B^{1},\dots ,B^{\kappa })$ is a Brownian motion in $\mathbb {R}^{\kappa }$ with zero drift and covariance matrix $\Sigma $ in Theorem 2.5. Since $B(1)$ is a standard normal vector with mean zero and covariance matrix $\Sigma $ , the result in Corollary 2.8 (i) follows.

If we are in the situation as in Corollary 2.8 (ii), then some of the consecutive unstable colors have the same ball density (i.e., $p_{\alpha _{j}}=p_{\alpha _{j+1}}$ ). For every such $\alpha _{j}$ , the corresponding coordinate has to remain nonnegative in the limiting SRBM. So in this case, the fluctuation of $\lambda _{1}$ about its mean in the diffusive scaling has a positive expectation. As an example, consider the case $\mathbf {p}=(p_{0},p_{1},p_{2})$ with $p_{1}>p_{2}=p_{0}$ (see Figure 3 (b)). In this case, the limiting SRBM $W=(W^{1},W^{2})$ is on the domain $\mathbb {R}\times \mathbb {R}_{\ge 0}$ , so the lower bound $W^{1}(1)+W^{2}(1)$ on the fluctuation in (24) has a positive expectation. This can be understood for the following reasons. Since $p_{1}>\max (p_{0},p_{2})$ , the number of color $1$ balls in the carrier grows linearly and makes the dominant contribution (of order n) to $\lambda _{1}(n)$ . However, the number of color $2$ balls in the carrier still contributes to $\lambda _{1}(n)$ by order $\sqrt {n}$ since $p_{2}=p_{0}$ . While the fluctuation of the number of color 1 balls around its mean $(p_{1}-p_{0})n$ has mean zero, the contribution of color 2 balls of order $\sqrt {n}$ is only visible in the diffusive scaling, and it is almost always of a positive amount.

Another interesting behavior of the multicolor BBS is the order of subsequent soliton lengths, $\lambda _{j}(n)$ for $j\ge 2$ , in the supercritical regime, which depends drastically on the multiplicity r of the maximal ball density $p^{*}$ . That is, $\lambda _{j}(n)$ for all $j\ge 2$ is of order $\log n$ if $r=1$ , but they are of order $\sqrt {n}$ if $r\ge 2$ . The former case agrees with the results for the $\kappa =1$ case in [Reference Levine, Lyu and PikeLLP20]. There, it was shown that $\lambda _{2}(n)$ comes from the subexcursions of the carrier process below its running maximum. The height of such subexcursions has exponential tails, so we have order $\log (n)$ as the order of the maximum of n subexponential random variables. However, if $r\ge 2$ in the multicolor case, the discrepancy between the number of balls of two maximal colors is of order $\sqrt {n}$ and contributes to $\lambda _{2}(n)$ (see the proof of Theorem 2.7 (iii)). We remark that a duality between the subcritical and the supercritical regimes for $\kappa =1$ was established in [Reference Levine, Lyu and PikeLLP20], in the sense that $\lambda _{j+1}$ in the supercritical regime corresponds to $\lambda _{j}$ in the subcritical regime for $j\ge 1$ . Our results confirm a similar correspondence still holds asymptotically for the simple ( $r=1$ ) supercritical regime; but $\lambda _{j+1}$ in the nonsimple ( $r\ge 2$ ) supercritical regime corresponds to $\lambda _{j}$ in the critical regime.

3.1 Infinite capacity carrier process and soliton lengths

The definition of $\kappa $ -color BBS dynamics we gave in the introduction involves the nonlocal movement of balls. It can instead be defined using a ‘carrier’, which gives a localized characterization of the process and reveals a number of important invariants that fully determine the resulting solitons. For the simplest case $\kappa =1$ , imagine a carrier of infinite capacity sweeps through the time-t configuration $\xi ^{(t)}$ from the left, picking up each ball it encounters and depositing a ball into each empty box whenever it can. We will see that after we run this carrier over $\xi ^{(t)}$ , the resulting configuration is in fact $\xi ^{(t+1)}$ . Moreover, the maximum number of balls in the carrier during the sweep is in fact the first soliton length $\lambda _{1}$ .

Now we introduce the infinite-capacity carrier process and the carrier version of the $\kappa $ -color BBS dynamic. Denote

(30) $$ \begin{align} \mathcal{B}_{\infty} := \left\{ \mathbf{x}\in \{0,1,\cdots ,\kappa\}^{\mathbb{N}} \mid \mathbf{x} \text{ is nonincreasing and has finite support} \right\}, \end{align} $$

which is the set of ‘reversed’ semi-standard Young tableaux of shape $1\times \infty $ and letters from $\{0,\dots ,\kappa \}$ . Namely, an element in this set is an infinite string of letters consisting of finitely many nonincreasing nonzero letters followed by an infinite string of zeros. An element $\mathbf {x}$ in $ \mathcal {B}_{\infty }$ describes the state of the infinite-capacity carrier. If the carrier at state $\mathbf {x}$ encounters a new ball of color y, it produces a new carrier state $\mathbf {x}'$ and a new ball color $y'$ according to the ‘circular exclusion rule’: Inserting y into $\mathbf {x}$ , $y'$ is the largest letter in $\mathbf {x}$ with $y'<y$ , and $\mathbf {x}'$ is obtained by replacing the leftmost letter $y'$ in $\mathbf {x}$ with y. More precisely, define a map $\Psi :\mathcal {B}_{\infty }\times \{0,1,\cdots , \kappa \} \rightarrow \{0,1,\cdots , \kappa \} \times \mathcal {B}_{\infty }$ , $(\mathbf {x},y)\mapsto (y',\mathbf {x}')$ by

1. (i) Suppose $y\ge 1$ and denote $i^{*}=\min \{ i\ge 1 \mid \mathbf {x}(i)< y \}$ . Then $y'=\mathbf {x}(i^{*})$ and

   (31) $$ \begin{align} \mathbf{x}'(i) = \mathbf{x}(i)\mathbf{1}(i\ne i^{*}) + y\mathbf{1}(i=i^{*}) \qquad \forall i\ge 1. \end{align} $$

2. (ii) Suppose $y=0$ . Then $y'=\mathbf {x}(1)=\max (\mathbf {x})$ and

   (32) $$ \begin{align} \mathbf{x}'(i) = \mathbf{x}(i+1) \qquad \forall i\ge 1. \end{align} $$

Fix a $\kappa $ -color BBS configuration $\xi :\mathbb {N}\rightarrow \{0,1,\cdots , \kappa \}$ . Fix $\Gamma _{0} \in \mathcal {B}_{\infty }$ , and recursively define a new $\kappa $ -color BBS configuration $\xi '$ and a sequence $(\Gamma _{x})_{x\ge 0}$ of elements of $\mathcal {B}_{\infty }$ by

(33) $$ \begin{align} (\xi^{\prime}_{x+1},\Gamma_{x+1}) = \Psi(\Gamma_{x}, \xi_{x+1}) \qquad \forall x\in \mathbb{N}. \end{align} $$

We call the sequence $(\Gamma _{x})_{x\ge 0}$ the infinite capacity carrier process over $\xi $ . The carrier state $\Gamma _{x}$ is determined by the balls in the interval $[1,x]$ (see Figure 4 for an illustration). Unless otherwise mentioned, we will assume $\Gamma _{0} =\mathbf {0}=[0,0,0,\cdots ]\in \mathcal {B}_{\infty }$ . The induced update map $\xi \mapsto \xi '$ turns out to coincide with the $\kappa $ -color BBS evolution (1). See Remark 3.4 for more details.

Figure 4 Time evolution of the infinite capacity carrier process $(\Gamma _{x})_{x\ge 0}$ over the $7$ -color initial configuration $\xi $ , producing new configuration $\xi '$ consisting of existing ball colors. For instance, $\xi _{2}=2$ , $\Gamma _{2}=[2,0,0,\cdots ]$ , and $\xi ^{\prime }_{4}=5$ . Notice that $\xi '$ can also be obtained by the time evolution of the 7-color BBS applied to $\xi $ .

It is important to note that the carrier process $(W_{x})_{n\in \mathbb {N}}$ we introduced in (8) can be derived from the infinite-capacity carrier process $(\Gamma _{x})_{x\in \mathbb {N}}$ above by simply recording the number of balls of each color $i=1,\dots ,\kappa $ . That is,

(34) $$ \begin{align} W_{x} = (m_{1}(\Gamma_{x}),\dots,m_{\kappa}(\Gamma_{x})) \quad \text{for all } x\ge 0, \end{align} $$

where $m_{i}(\Gamma _{x})$ denotes the number of balls of color (letter) i in $\Gamma _{x}$ for $i=1,\dots ,\kappa $ .

Lemma 3.1 below states that the first soliton length $\lambda _{1}$ equals the maximum number of balls of positive colors in the associated carrier process.

Lemma 3.1. Suppose the initial $\kappa $ -color BBS configuration $\xi $ has finite support. Let $(W_{x})_{x\ge 0}$ and $(\Gamma _{x})_{x\ge 0}$ be as before. Then

(35) $$ \begin{align} \lambda_{1}(\xi)= \max_{x\ge 0} \,\, \lVert W_{x} \rVert_{1} = \max_{x\ge 0} \left( \# \text{ of positive letters in } \Gamma_{x} \right). \end{align} $$

For $\kappa =1$ , it is possible to precisely characterize all subsequent soliton lengths $\lambda _{2},\lambda _{3},\dots $ by applying the ‘excursion operator’ to the carrier process multiple times and taking maximum [Reference Levine, Lyu and PikeLLP20]. Roughly speaking, given the 1-dimensional carrier process $W=(W_{x})_{x\ge 0}$ for $\kappa =1$ , which starts at 0 and takes value 0 for all large x, let $\mathcal {E}(W)$ denote the new lattice path that describes the excursion heights above the record minimum of W away from the rightmost global maximizer of W. Then $\lambda _{2}=\max (\mathcal {E}(W))$ , and $\lambda _{3}=\max (\mathcal {E}^{2}(W))$ , and so on. We currently do not have a similar $\kappa $ -dimensional excursion operator for exactly describing the subsequent soliton lengths for the general multicolor case. However, we provide a lower bound on $\lambda _{j}$ in terms of the jth largest ‘excursion height’ of the carrier process, which is enough to obtain sharp asymptotics for $\lambda _{j}$ in the subcritical regime.

We introduce some notation. Let $\mathbf {0}=(0,0,\cdots ,0)\in (\mathbb {Z}_{\ge 0})^{\kappa }$ denote the origin, and write

(36) $$ \begin{align} M_{n} := \sum_{x=1}^{n}\mathbf{1}(W_{x}=\mathbf{0}) \end{align} $$

for the number of visits of $W_{x}$ to $\mathbf {0}$ during $[1,n]$ . For each $k\ge 1$ , let $T_{k}$ denote the time of the kth visit of $W_{x}$ to $\mathbf {0}$ and set $T_{0}=0$ . We say that the trajectories of $W_{x}$ restricted to the time intervals $[T_{k-1},T_{k}]$ between consecutive visits to $\mathbf {0}$ are its excursions. Also note that $M_{n}$ defined at (36) equals the number of complete excursions of the carrier process during $[1,n]$ . We will define the height of the carrier at site x by

(37) $$ \begin{align} \lVert W_{x} \rVert_{1} = W_{x}(1) + \dots + W_{x}(\kappa), \end{align} $$

which equals the number of balls of positive color that the carrier possesses at site x. Define the kth excursion height $h_{k}$ and height of the final meander $r_{n}$ by

(38) $$ \begin{align} h_{k} = \max_{T_{k-1}\le t \le T_{k}} \lVert W_{x} \rVert_{1}, \qquad r_{n} = \max_{T_{M_{n}}\le t \le n} \lVert W_{x} \rVert_{1}. \end{align} $$

Denote by $\mathbf {h}_{1}(n)\ge \mathbf {h}_{2}(n)\ge \cdots \ge \mathbf {h}_{M_{n}}(n)$ the order statistics of the excursion heights $h_{1},\cdots ,h_{M_{n}}$ . We then have the following lemma.

Proofs of Lemmas 3.1 and 3.2 are relagated to Section 12.

3.2 Finite capacity carrier processes and soliton numbers

In [Reference Kuniba and LyuKL20], it is shown that the row lengths of the invariant Young diagram of any $\kappa $ -BBS trajectory can be extracted by running carrier processes of finite capacities, as we will summarize in this subsection. This will provide one of the key combinatorial lemmas in the present paper.

First, fix an integer parameter $c\ge 1$ that we call capacity. Denote

(39) $$ \begin{align} \mathcal{B}_{c} = \{ [x_{1},\cdots,x_{c}]\in \{0,1,\cdots ,\kappa\}^{c}\mid x_{1}\ge \cdots \ge x_{c} \}, \end{align} $$

which can also be identified as the set of all $(1\times c)$ semistandard tableaux with letters from $\{0,1,\cdots , \kappa \}$ . Define a map $\Psi _{c}:\mathcal {B}_{c}\times \{0,1,\cdots , \kappa \} \rightarrow \{0,1,\cdots , \kappa \} \times \mathcal {B}_{c}$ , $([x_{1},\cdots ,x_{c}],y)\mapsto (y',[x_{1}',\cdots ,x_{c}'])$ by the following ‘circular exclusion rule’:

1. (i) Suppose $y>x_{c}$ and denote $i^{*}=\min \{ i\ge 1\mid x_{i}< y \}$ . Then $y'=x_{i^{*}}$ and

   (40) $$ \begin{align} [x_{1}',\cdots,x^{\prime}_{c}] = [x_{1},\cdots,x_{i^{*}-1},y,x_{i^{*}+1},\cdots,x_{c}]. \end{align} $$

2. (ii) Suppose $x_{c}\ge y$ . Then $y'=x_{1}$ and

   (41) $$ \begin{align} [x_{1}',x_{2}',\cdots,x_{c}'] = [x_{2},\cdots,x_{c},y]. \end{align} $$

Fix a $\kappa $ -color BBS configuration $\xi :\mathbb {N}\rightarrow \{0,1,\cdots , \kappa \}$ . Let $\Gamma _{0}=[0,\cdots ,0]\in \mathcal {B}_{c}$ , and recursively define a new $\kappa $ -color BBS configuration $\xi '$ and a sequence $(\Gamma _{x})_{x\ge 0}$ of elements of $\mathcal {B}_{c}$ by

(42) $$ \begin{align} (\xi^{\prime}_{x+1},\Gamma_{x+1}) = \Psi_{c}(\Gamma_{x}, \xi_{x+1}) \qquad \forall x\in \mathbb{N}. \end{align} $$

We call the sequence $(\Gamma _{x})_{x\ge 0}$ the capacity-c carrier process over $\xi $ . See Figure 5 for an illustration.

Figure 5 Time evolution of the capacity-3 carrier process $(\Gamma _{x})_{x\ge 0}$ over the $7$ -color initial configuration $\xi $ , with new configuration $\xi '$ consisting of existing ball colors. For instance, $\xi _{2}=2$ , $\Gamma _{2}=[2,0,0]$ , and $\xi ^{\prime }_{4}=5$ . Notice that while $\xi $ is the same as in the example in Figure 4, the new $7$ -color BBS configuration $\xi '$ is different. In this case, the map $\xi \mapsto \xi '$ does not agree with the $7$ -color BBS time evolution.

The following lemma, which is proven in [Reference Kuniba and LyuKL20], gives a closed-form expression of the row sums of the invariant Young diagram:

Lemma 3.3. Let $(\xi ^{(t)})_{t\ge 0}$ be a $\kappa $ -color BBS trajectory such that $\xi ^{(0)}$ has finite support. For each $c\ge 1$ , let $(\Gamma _{x;c})_{x\ge 0}$ denote the capacity-c carrier process over $\xi ^{(t)}$ . Then for all $k\ge 1$ and $t\ge 0$ , we have

(43) $$ \begin{align} \rho_{1}(\xi^{(0)}) + \cdots+ \rho_{k}(\xi^{(0)}) \equiv \sum_{x=1}^{\infty} \mathbf{1}(\xi^{(t)}_{x}>\min \Gamma_{x-1;k}), \end{align} $$

where $\min \Gamma _{x-1;k}$ denotes the smallest letter in $\Gamma _{x-1;k}$ .

3.3 Modified Greene-Kleitman invariants for BBS

One natural way to associate a Young diagram with a given permutation is to use the celebrated Robinson-Schensted correspondence (see [Reference SaganSag01, Ch. 3.1]), which gives a bijection between permutations and pairs of standard Young tableaux of the same shape. For each permutation $\sigma $ , record the common shape of the Young tableaux as $\Lambda _{\mathrm {RS}}(\sigma )$ . Let $\rho _{i}^{\mathrm {RS}}(\sigma )$ and $\lambda _{j}^{\mathrm {RS}}(\sigma )$ denote its ith row length and its jth column lengths, respectively. According to Greene’s theorem [Reference GreeneGre82], the sum of the lengths of the first k columns (resp. rows) of $\Lambda _{RS}(\sigma )$ is equal to the length of the longest subsequence in $\sigma $ that can be obtained by taking the union of k decreasing (resp. increasing) subsequences. That is, for each $k\ge 1$ ,

(44) $$ \begin{align} \rho_{1}^{\mathrm{RS}}(\sigma)) + \cdots+ \rho_{k}^{\mathrm{RS}}(\sigma)) &= \max\left(\left| \bigsqcup k \text{ increasing subsequences of } \sigma \right| \right), \end{align} $$

(45) $$ \begin{align} \lambda_{1}^{\mathrm{RS}}(\sigma)) + \cdots+ \lambda_{k}^{\mathrm{RS}}(\sigma)) &= \max\left(\left| \bigsqcup k \text{ decreasing subsequences of } \sigma \right| \right). \end{align} $$

The quantities on the right-hand sides are called the Greene-Kleitman invariants.

If we consider the $\kappa $ -color BBS trajectory started at $\xi ^{(0)}=\sigma \mathbf {1}([1,n])$ , then we obtain another Young diagram $\Lambda (\sigma ) := \Lambda (\xi ^{(0)})$ , whose $j^{\text {th}}$ column length equals the $j^{\text {th}}$ longest soliton length. Then a natural question arises: Do the sums of the first k rows and columns of $\Lambda (\sigma )$ relate to some type of Greene-Kleitman invariants? For the rows, we find that the correct modification is to localize the length of an increasing sequence into the number of ascents in a subsequence. However, for the columns, it turns out that we just need to impose that the k decreasing subsequences be noninterlacing. In fact, in Lemma 3.5, we establish these modified Greene-Kleitman invariants for BBS in the more general setting when $\sigma $ is an arbitrary $\kappa $ -color BBS configuration with finite support, where having 0’s and repetitions are both allowed.

Let $\xi :\mathbb {N}\rightarrow \{0,1,\cdots , \kappa \}$ be a $\kappa $ -color BBS configuration with finite support. For subsets $A,B\subseteq \mathbb {N}$ , denote $A\prec B$ if $\max (A)<\min (B)$ . We say $A,B$ are noninterlacing if $A\prec B$ or $B \prec A$ . We say $\xi $ is nonincreasing on $A\subseteq \mathbb {N}$ if $\xi _{a_{1}}\ge \xi _{a_{2}}$ for all $a_{1},a_{2}\in A$ such that $a_{1}\le a_{2}$ . Denoting the elements of A by $a_{1}<a_{2}<\cdots $ , define the number of ascents of $\xi $ in A by

(46) $$ \begin{align} \mathrm{NA}(A,\xi):= 1+\sum_{i=2}^{|A|} \mathbf{1}(\xi_{a_{i-1}}<\xi_{a_{i}}). \end{align} $$

Moreover, define the penalized length of A with respect to $\xi $ by

(47) $$ \begin{align} {\mathrm{L}}(A,\xi):= \left[ |A| - \sum_{i = \min A}^{\max A} \mathbf{1}(\xi_{i}=0) \right] \mathbf{1}(\xi \text{ is non-increasing on } A). \end{align} $$

Note that the summation in (47) is over the interval $[\min A, \max A]\cap \mathbb {Z}$ , which may contain A properly.

Lemma 3.5. Let $(\xi ^{(t)})_{t\ge 0}$ be a $\kappa $ -color BBS trajectory such that $\xi ^{(0)}$ has finite support. Then for each $k,t\ge 0$ , we have

(48) $$ \begin{align} \rho_{1}(\xi^{(0)}) + \cdots+ \rho_{k}(\xi^{(0)}) &\equiv \max_{A_{1}\sqcup \cdots \sqcup A_{k} = \mathbb{N} } \sum_{i=1}^{k} \mathrm{NA}(A_{i}, \xi^{(t)}), \end{align} $$

(49) $$ \begin{align} \lambda_{1}(\xi^{(0)}) + \cdots+ \lambda_{k}(\xi^{(0)}) &\equiv \max_{A_{1}\prec\cdots \prec A_{k} \subseteq \mathbb{N} } \sum_{i=1}^{k} {\mathrm{L}}(A_{i},\xi^{(t)}). \end{align} $$

The proof of Lemma 3.5 may be found in Section 12.2.

4.1 Proof of Theorem 2.1 for the columns

Our proof of Theorem 2.1 for the columns relies on Lemma 3.5 and the sharp asymptotic of longest decreasing subsequence of a uniform random permutation due to Baik, Deift and Johansson [Reference Baik, Deift and JohanssonBDJ99].

Proof of Theorem 2.1 for the columns.

Fix an integer $k\ge 1$ . It suffices to show that, almost surely,

(50) $$ \begin{align} \lim_{n\rightarrow\infty} n^{-1/2}\sum_{i=1}^{k} \lambda_{i}(n) = 2\sqrt{k}. \end{align} $$

For each integer $k\ge 1$ , let $L(k)$ denote the length of the longest increasing subsequence in a uniformly random permutation of k letters. By Lemma 3.5, recall that

(51) $$ \begin{align} \lambda_{1}(n) + \cdots + \lambda_{k}(n) = \max \left\{ \sum_{i=1}^{k} L(A_{i}, \xi^{n}) \,|\, A_{1}\prec \dots \prec A_{k} \subseteq [1,n] \right\}. \end{align} $$

We view a random permutation as a ranking among n i.i.d. $\mathrm {Uniform}([0,1])$ random variables $U_{1},\cdots , U_{n}$ . If $A\subseteq \{1,\cdots , n\}$ , then the ranking of $U_{i}$ for $i\in A$ gives a uniform random permutation of A, which we call a random permutation of $[n]$ restricted on A. Moreover, one can also see that if we restrict a random permutation on multiple disjoint subsets, then these smaller permutations are independent. Hence, if $A_{1}\prec \cdots \prec A_{k}$ are noninterlacing subsets of $[0,n]$ , then the permutations restricted on these subsets are independent. Moreover, since the random permutation model $\xi ^{n}$ does not assign color $0$ on any site in $[0,n]$ , for any increasing subsequence $A\subseteq [0,n]$ and its supporting interval $I=[\min A, \max A]$ ,

(52) $$ \begin{align} {\mathrm{L}}(A,\xi^{n}) = |A| \le |I| = {\mathrm{L}}(I,\xi^{n}) \overset{d}{=} L(|I|). \end{align} $$

It follows that

(53) $$ \begin{align} \sum_{i=1}^{k} \lambda_{i}(n) \overset{d}{=} \max \left\{ \sum_{i=1}^{k} L(n_{i}) \bigg|\, \sum_{i=1}^{k} n_{i}=n,\, L(n_{1}),\dots, L(n_{k}) \text{ are indepenent} \right\}. \end{align} $$

Baik, Deift and Johansson [Reference Baik, Deift and JohanssonBDJ99] proved the following tail bounds for $L(n)$ (see also equations (1.7) and (1.8) in [Reference Baik, Deift and JohanssonBDJ99] or p. 149 in [Reference RomikRom15]): There exist positive constants $M, c, C$ such that for all $m \ge 1$ ,

(54) $$ \begin{align} &\text{(Lower tail):} \,\, {\mathbb{P}}\left( m^{-1/6} (L(m) - 2 \sqrt{m} ) \le - t\right) \le C \exp(-c t^3) \,\, \text{ for all } t \in [M, 2m^{1/3}]; \end{align} $$

(55) $$ \begin{align} &\text{(Upper tail):} \,\, {\mathbb{P}}\left( m^{-1/6} (L(m) - 2 \sqrt{m} ) \ge t\right) \le C \exp(-c t^{3/5}) \,\, \text{for all } t \in [M, m^{5/6} -2m^{1/3} ]. \end{align} $$

Taking $t = (\log m)^2$ , we obtain

(56) $$ \begin{align} {\mathbb{P}}\left( |L(m) - 2 \sqrt{m} | \ge (\log m)^{2} m^{1/6} \right) \le 2C \exp(-c(\log m)^{6/5}). \end{align} $$

Fix $\varepsilon>0$ . Note that if $m \ge \varepsilon \sqrt {n}$ , then for any fixed $d>0$ ,

(57) $$ \begin{align} {\mathbb{P}}\left( |L(m) - 2 \sqrt{m} | \ge (\log m)^{2} m^{1/6} \right) = O(n^{-d}). \end{align} $$

Now, denote the random variable in the right-hand side of (53) by X. We write $X=\max (Y,Z)$ , where

(58) $$ \begin{align} Y &= \max \{ L(n_1) + \cdots + L(n_k) : n_1 + \cdots + n_k =n, n_i \ge \varepsilon \sqrt{n} \text{ for all } i \} , \end{align} $$

(59) $$ \begin{align} Z &= \max \{ L(n_1) + \cdots + L(n_k) : n_1 + \cdots + n_k =n, n_i < \varepsilon \sqrt{n} \text{ for at least one } i \}. \end{align} $$

Denote $\mathcal {A}:=\{ (n_{1},\dots ,n_{k})\,:\, n_{1}+\dots +n_{k}=n,\, n_{i}\ge \varepsilon \sqrt {n} \text { for all } i \}$ . For each $\eta =(n_{1},\dots ,n_{k})\in \mathcal {A}$ , denote $Y_{\eta }:=L(n_{1})+\dots +L(n_{k})$ and $M_{\eta }:=2(\sqrt {n_{1}}+\dots +\sqrt {n_{k}})$ . Then by a union bound and (57),

(60) $$ \begin{align} {\mathbb{P}}(| Y_{\eta} - M_{\eta}| \ge k (\log m)^{2} m^{1/6} ) = O(n^{-d}). \end{align} $$

Note that $Y=\max _{\eta \in \mathcal {A}} Y_{\eta }$ , and since there are at most $n^k$ partitions of $[n]$ into k intervals, $|\mathcal {A}|\le n^{k}$ . So by a union bound, we have

(61) $$ \begin{align} {\mathbb{P}}\left( \big| Y - \max_{\eta\in \mathcal{A}} M_{\eta} \big|\right) \le \sum_{\eta\in \mathcal{A}} {\mathbb{P}}\left(| Y_{\eta} - M_{\eta}| \ge k (\log m)^{2} m^{1/6} \right) = O(n^{-d}) \end{align} $$

for any fixed $d>0$ . The deterministic optimization problem

(62) $$ \begin{align} \max_{\eta\in \mathcal{A}} M_{\eta} = \max \{ 2 \sqrt{n_1} + \cdots + 2 \sqrt{n_k} : n_1 + \cdots + n_k =n, n_i \ge \varepsilon \sqrt{n} \forall \, \text{ i} \} \end{align} $$

achieves its maximum when $\sum _{i=1}^{k} |n_{i}-(n/k)|$ is minimized, in which case we have $|n_{i}-(n/k)|\le 1$ for all $1\le i \le k$ . Denoting the maximizer as $n_{1},\cdots ,n_{k}$ , it follows that, for all $1\le i \le k$ ,

(63) $$ \begin{align} |\sqrt{n_{i}} - \sqrt{n/k}| \le \frac{1}{\sqrt{n_{i}} + \sqrt{n/k}} \le \frac{1}{2\sqrt{(n/k)-1}}. \end{align} $$

So this yields, for all sufficiently large $n\ge 1$ ,

(64) $$ \begin{align} &{\mathbb{P}}\left( | Y - 2\sqrt{kn}|> 2k (\log n)^{2} n^{1/6} \right) \\ \nonumber &\qquad \le {\mathbb{P}}\left( | Y - 2\sqrt{kn}| > k (\log n)^{2} n^{1/6} + \frac{k}{\sqrt{(n/k)-1}} \right) = O(n^{-d}) \end{align} $$

for any fixed $d>0$ .

Next, if $n_i < \varepsilon \sqrt {n}$ , then we use the trivial upper bound $L(n_i) \le n_i \le \varepsilon \sqrt {n}$ ; otherwise, if $n_i> \varepsilon \sqrt {n}$ , we continue to use the tail bound for $|L(n_{i})-2\sqrt {n_{i}}|$ in (57). Hence,

(65) $$ \begin{align} {\mathbb{P}}\left( Z> 2\sqrt{(k-1)n} + 2 k (\log n)^2 n^{1/6} + k \varepsilon \sqrt{n} \right) = O(n^{-d}), \end{align} $$

where the first term bounds the contribution from at most $k-1$ intervals of size $\ge \varepsilon \sqrt {n}$ , the second term is given by the BDJ tail bound in (57) and the last term gives a trivial bound for intervals of size $<\varepsilon \sqrt {n}$ . Hence, if we choose $\varepsilon <2/k(\sqrt {k-1}+\sqrt {k})$ , then (64) and (65) give us

(66) $$ \begin{align} {\mathbb{P}}\left( Z>Y \right) &\le {\mathbb{P}}\left( Y< 2\sqrt{kn} + 2k (\log n)^{2} n^{1/6} \right) \\ \nonumber &\qquad + {\mathbb{P}}\left( Z> 2\sqrt{(k-1)n}+ 2k (\log n)^{2} n^{1/6} + \frac{2\sqrt{n}}{\sqrt{k-1}+\sqrt{k}} \right) \\ &= O(n^{-d}) \nonumber \end{align} $$

for each fixed $d>0$ . Now note that, for each $t>0$ ,

(67) $$ \begin{align} {\mathbb{P}}\left( \left| \left(\frac{1}{\sqrt{n}}\sum_{i=1}^{k} \lambda_{i}(n) \right) - 2\sqrt{k} \right|> t \right) &= {\mathbb{P}}\left( |\max(Y,Z)-2\sqrt{kn}|>t\sqrt{n} \right) \\ &\le {\mathbb{P}}\left( |Y-2\sqrt{kn}|>t\sqrt{n} \right) + {\mathbb{P}}\left( Z> Y \right). \nonumber \end{align} $$

Hence, by choosing $t=1/\log n$ , for any fixed $d>0$ , (64) and (66) yield

(68) $$ \begin{align} {\mathbb{P}}\left( \left| \left(\frac{1}{\sqrt{n}}\sum_{i=1}^{k} \lambda_{i}(n) \right) - 2\sqrt{k} \right|> \frac{1}{\log n} \right) = O(n^{-d}). \end{align} $$

Then the assertion follows from the Borel-Cantelli lemma.

4.2 Circular exclusion process and the row lengths

In this subsection, we prove Theorem 2.1 for the rows. By Lemma 3.3, this can be done by analyzing the carrier process over the uniform random permutation $\xi ^{n}$ . Let $\mathbf {X}:=(U_{x})_{x\ge 1}$ be a sequence of i.i.d. $\mathrm {Uniform}([0,1])$ random variables. For each capacity $k\ge 1$ , we may define the carrier process $(\boldsymbol {\Gamma }_{x})_{x\ge 0}$ over $\mathbf {X}$ using the same ‘circular exclusion rule’ we used to define the map $\Psi $ in Section 3.2. More precisely, denote $\mathcal {C}_{k}=\{ (x_{1},\cdots ,x_{k})\in [0,1]^{k}\mid x_{1}\ge \cdots \ge x_{k} \}$ . Define a map $\phi :\mathcal {C}_{k}\times [0,1]\rightarrow \mathcal {C}_{k}$ , $[x_{1},\cdots ,x_{k},y]\mapsto [x_{1}',\cdots ,x_{k}']$ by

1. (i) If $y>x_{k}$ , then denote $i^{*}=\min \{ i\ge 1\mid x_{i}< y \}$ and let

   (69) $$ \begin{align} [x_{1}',\cdots,x^{\prime}_{k}] = [x_{1},\cdots,x_{i^{*}-1},y,x_{i^{*}+1},\cdots,x_{k}]. \end{align} $$

2. (ii) If $x_{k}\ge y$ , then $[x_{1}',\cdots ,x_{k}'] = [x_{2},\cdots ,x_{k},y]$ .

Then the k-point circular exclusion process $(\boldsymbol {\Gamma }_{x})_{x\ge 0}$ over $\mathbf {X}$ is defined recursively by

(70) $$ \begin{align} \boldsymbol{\Gamma}_{x+1} = \phi(\boldsymbol{\Gamma}_{x}, U_{x+1}). \end{align} $$

See Figure 6 for an illustration. Note that $(\boldsymbol {\Gamma }_{x})_{x\ge 0}$ forms a Markov chain on state space $\mathcal {C}_{k}$ . When $\boldsymbol {\Gamma }_{0}=[0,0,\cdots ,0]$ , we call $(\boldsymbol {\Gamma }_{x})_{x\ge 0}$ the carrier process over $\mathbf {X}$ with capacity k.

Figure 6 Evolution of a 4-point circular exclusion process. The states in the unit circle are ordered clockwise. Each newly inserted point (black dot) annihilates the closest preexisting point in the counterclockwise direction (light blue dot).

In the following lemma, which will be proved in Section 4.3, we show that the k-point circular exclusion process converges to its unique stationary measure $\pi $ , which is the distribution of the order statistics from k i.i.d. $\mathrm {Uniform}([0,1])$ variables.

Now we derive Theorem 2.1 for the row asymptotics.

Proof of Theorem 2.1 for the rows.

Let $\mathbf {X}=(U_{x})_{x\ge 1}$ denote an infinite sequence of i.i.d. $\mathrm {Uniform}([0,1])$ random variables, $\Sigma ^{n}$ be the random permutation on $[n]$ induced by $U_{1},\cdots ,U_{n}$ , and $\xi ^{n}=\Sigma ^{n}\mathbf {1}([1,n])$ be the random n-color BBS configuration as defined at (4). Fix an integer $k\ge 1$ and let $(\boldsymbol {\Gamma }_{x})_{x\ge 0}$ be the k-point circulr exclusion process over $\mathbf {X}$ . Also, let $(\Gamma _{x})_{x\ge 0}$ be the capacity-k carrier process over $\xi ^{n}$ as defined in Section 3.2. By construction, for each $1\le x \le n$ , we have

(72) $$ \begin{align} \mathbf{1}(\xi^{n}(x)>\min \Gamma_{x-1}) = \mathbf{1}(U_{x}>\min \boldsymbol{\Gamma}_{x-1}). \end{align} $$

Thus, according to Lemma 3.3, almost surely,

(73) $$ \begin{align} n^{-1}\left( \rho_{1}(\xi^{n}) + \cdots+ \rho_{k}(\xi^{n}) \right) = n^{-1}\sum_{x=1}^{n} \mathbf{1}(U_{x}> \min \boldsymbol{\Gamma}_{x-1}). \end{align} $$

By Lemma 4.1 and Markov chain ergodic theorem, almost surely,

(74) $$ \begin{align} \lim_{n\rightarrow \infty} n^{-1}\left( \rho_{1}(\xi^{n}) + \cdots+ \rho_{k}(\xi^{n}) \right) &= {\mathbb{P}}\left(U_{k+1}> \min(U_{1},\cdots,U_{k}) \right) = \frac{k}{k+1}. \end{align} $$

Then the assertion follows.

4.3 Stationarity and convergence of the circular exclusion process

We prove Lemma 4.1 in this subsection. We will assume the stationarity of the circular exclusion process as asserted in the following proposition, which will be proved at the end of this section.

Proof of Lemma 4.1 .

For convergence, we use a standard coupling argument. Namely, fix arbitrary distributions $\pi _{0}$ and $\bar {\pi }_{0}$ on $\mathcal {C}_{k}$ and let X = (U _x ) _x≥1 denote a sequence of i.i.d. $\mathrm {Uniform}([0,1])$ variables. Let $(\boldsymbol {\Gamma }_{x})_{x\ge 0}$ be k-point circular exclusion processes over $\mathbf {X}$ with initial distribution π ₀ and let $(\bar {\boldsymbol {\Gamma }}_{x})_{x\ge 0}$ be k-point circular exclusion processes over $\mathbf {X}$ with initial distribution π-₀. These two processes are naturally coupled since they evolve simultaneously over the same environment $\mathbf {X}$ . Let $\tau =\inf \{ x\ge 0\mid \boldsymbol {\Gamma }_{x}=\bar {\boldsymbol {\Gamma }}_{x} \}$ denote the first meeting time of the two chains (see Figure 7). By the coupling, $\boldsymbol {\Gamma }_{s}=\bar {\boldsymbol {\Gamma }}_{s}$ and $s\le x$ imply $\boldsymbol {\Gamma }_{x}=\bar {\boldsymbol {\Gamma }}_{x}$ . A standard argument shows

(75) $$ \begin{align} d_{TV}(\pi_{x},\bar{\pi}_{x}) \le {\mathbb{P}}(\boldsymbol{\Gamma}_{x} \ne \bar{\boldsymbol{\Gamma}}_{x}) = {\mathbb{P}}(\tau> x), \end{align} $$

where $\pi _{x}$ and $\bar {\pi }_{x}$ denote the distributions of $\boldsymbol {\Gamma }_{x}$ and $\bar {\boldsymbol {\Gamma }}_{x}$ . We claim that

(76) $$ \begin{align} {\mathbb{P}}(\tau> t) \le {\mathbb{P}}(\boldsymbol{\Gamma}_{0}\ne \bar{\boldsymbol{\Gamma}}_{0}) \left(1 - \frac{1}{(2k)^{k-1}k!} \right)^{\lfloor t/k \rfloor}. \end{align} $$

According to Proposition 4.2, this will imply Lemma 4.1 by choosing $\bar {\pi }_{0}=\pi $ .

Figure 7 Joint evolution of two 3-point circular exclusion processes. The states in the unit circle are ordered clockwise. A newly inserted point annihilates one of the closest preexisting points in the counterclockwise direction. Blue (resp., red) dots represent points that are shared (resp., not shared) in both processes. The two chains meet after the fifth transition.

To bound the tail probability of meeting time $\tau $ , we will show that two circular exclusion processes ‘synchronize’ after k steps with probability at least $1/k!$ , in the sense that

(77) $$ \begin{align} {\mathbb{P}}(\boldsymbol{\Gamma}_{x+k}= \bar{\boldsymbol{\Gamma}}_{x+k}\mid \boldsymbol{\Gamma}_{x}\ne \bar{\boldsymbol{\Gamma}}_{x}) \ge \frac{1}{(2k)^{k-1} k!} \qquad \text{for all } x\ge 0. \end{align} $$

Then the claim (76) follows since

(78) $$ \begin{align} {\mathbb{P}}(\tau>Nk) &= {\mathbb{P}}(\boldsymbol{\Gamma}_{Nk}\ne \bar{\boldsymbol{\Gamma}}_{Nk}\mid \boldsymbol{\Gamma}_{0}\ne \bar{\boldsymbol{\Gamma}}_{0}) {\mathbb{P}}(\boldsymbol{\Gamma}_{0}\ne \bar{\boldsymbol{\Gamma}}_{0}) \end{align} $$

(79) $$ \begin{align} &\le {\mathbb{P}}(\boldsymbol{\Gamma}_{0}\ne \bar{\boldsymbol{\Gamma}}_{0})\prod_{i=1}^{N} {\mathbb{P}}(\boldsymbol{\Gamma}_{ik}\ne \bar{\boldsymbol{\Gamma}}_{ik}\mid \boldsymbol{\Gamma}_{(i-1)k}\ne \bar{\boldsymbol{\Gamma}}_{(i-1)k}) \end{align} $$

(80) $$ \begin{align} &\le {\mathbb{P}}(\boldsymbol{\Gamma}_{0}\ne \bar{\boldsymbol{\Gamma}}_{0}) \left(1-\frac{1}{(2k)^{k-1}k!} \right)^{N}. \end{align} $$

We begin with the following simple observation for a sufficient condition of meeting. Let $\mathbf {X}=(U_{t})_{t\ge 1}$ be a sequence of i.i.d. $\mathrm {Uniform}([0,1])$ variables. Fix $t\ge 1$ and let $\boldsymbol {\Gamma }_{x}=[x_{1},\cdots ,x_{k}]$ and $\bar {\boldsymbol {\Gamma }}_{x}=[\bar {x}_{1},\cdots ,\bar {x}_{k}]$ be arbitrary elements of $\mathcal {C}_{k}$ . Superpose the two k-point configurations into a one $2k$ -point configuration $0\le y_{1}\le y_{2} \le \cdots \le y_{2k}\le 1$ . For a special case, suppose $y_{2k}<1$ . Observe that on the event $\{y_{2k} < U_{t+k}<\cdots <U_{t+1}\le 1 \}$ , we have

(81) $$ \begin{align} \boldsymbol{\Gamma}_{x+k} = [U_{t+1},U_{t+2},\cdots, U_{t+k}] = \bar{\boldsymbol{\Gamma}}_{x+k}, \end{align} $$

as all of the k points in $\boldsymbol {\Gamma }_{x}$ and $\boldsymbol {\Gamma }_{x}$ will be successively annihilated from the largest to the smallest by inserting $U_{t+1},\cdots , U_{t+k}$ .

For the general case, regard each $U_{s}$ as a uniformly chosen point from the unit circle $S^{1}$ . Then the $2k$ points $y_{1},\cdots ,y_{2k}$ will divide $S^{1}$ into disjoint arcs of lengths, say, $\ell _{1},\cdots ,\ell _{m}$ , for some $2\le m \le 2k$ . If the points $U_{t+1},\cdots , U_{t+k}$ are strictly decreasing in the counterclockwise order within one of the m arcs, then by circular symmetry and a similar observation, we will have $\boldsymbol {\Gamma }_{x+k} = \bar {\boldsymbol {\Gamma }}_{x+k}$ . Noting that

(82) $$ \begin{align} {\mathbb{P}}\left( \begin{matrix} U_{t+1},\cdots, U_{t+k} \text{ are strictly decreasing in the} \\ \text{ counterclockwise order within an arc of length } \ell \end{matrix} \right) = \frac{\ell^{k}}{k!} \end{align} $$

and $\ell _{1}+\cdots +\ell _{m}=1$ , Hölder’s inequality yields

(83) $$ \begin{align} &{\mathbb{P}}\left(\boldsymbol{\Gamma}_{x+k}= \bar{\boldsymbol{\Gamma}}_{x+k}\mid \boldsymbol{\Gamma}_{x}=[x_{1},\cdots,x_{k}],\, \bar{\boldsymbol{\Gamma}}_{x}=[\bar{x}_{1},\cdots,\bar{x}_{k}]\right) \\&\quad\qquad \ge \sum_{i=1}^{m} \frac{\ell_{i}^{k}}{k!} \ge \frac{1}{k!} \frac{\left(\ell_{1}+\cdots+\ell_{m} \right)^{k}}{m^{k-1}} = \frac{1}{m^{k-1} k!} \ge \frac{1}{(2k)^{k-1} k!}. \nonumber \end{align} $$

This shows the assertion.

Lastly in this section, we prove Proposition 4.2.

Proof of Proposition 4.2 .

We show $\pi $ is a stationary distribution for the Markov chain $(\boldsymbol {\Gamma }_{s})_{s\ge 0}$ . Let $X_{(1)} < X_{(2)}< \cdots < X_{(k)}$ be the order statistics from k i.i.d. uniform RVs on $[0,1]$ . Let Y be an independent $\mathrm {Uniform}([0,1])$ random variable. After a new point Y is inserted to the preexisting list of k points $X_{(1)} < X_{(2)} < \cdots < X_{(k)}$ , the updated list of points will be

(84) $$ \begin{align} X_{(1)} < \cdots < X_{(I-1)} < Y < X_{(I+1)} < \cdots < X_{(k)}, \end{align} $$

where $I \in \{1, 2, \cdots , k\}$ is the random index such that $Y \in ( X_{(I)}, X_{(I+1)} )$ . For $I = k$ , the interval $( X_{(k)}, X_{(k+1)} )$ denotes the union of $(0, X_{(1)})$ and $(X_{(k)}, 1)$ . In this case, the point $X_{(k)}$ is deleted and Y is added as the smallest or largest point depending on which sub-intervals it falls.

We claim that (84) is still the order statistics from k i.i.d. uniforms on $[0,1]$ , which would prove that the distribution of k i.i.d. uniform points remains invariant under the transition rule. To show this, take a bounded test function $f:[0,1]^k \to \mathbb {R}$ . First, we write

(85) $$ \begin{align} &\mathbb{E} \left[ f( X_{(1)},\cdots , X_{(I-1)}, Y, X_{(I+1)}, \cdots , X_{(k)})\right] \end{align} $$

(86) $$ \begin{align} &\quad = \sum_{ i=1}^k \mathbb{E} [ f( X_{(1)}, \cdots , X_{(i-1)}, Y, X_{(i+1)}, \cdots, X_{(k)}) \mathbf{1}_{ Y \in ( X_{(i)}, X_{(i+1)} ) } ] \end{align} $$

(87) $$ \begin{align} &\quad = \sum_{ i=1}^{k-1} \frac{1}{k!} \int_{ z_{1} < \cdots < z_{i} < y < z_{i+1} < \cdots < z_{k} } f( z_{1}, \cdots, z_{i-1}, y, z_{i+1}, \cdots, z_{k}) \, dz_{1} \cdots dz_{k} dy \end{align} $$

(88) $$ \begin{align} &\hspace{1cm} + \frac{1}{k!} \int_{z_{1}< \cdots < z_{k}<y } f(z_{1}, \cdots, z_{k-1}, y) \, dz_{1} \cdots dz_{k} dy \end{align} $$

(89) $$ \begin{align} &\hspace{1.3cm} + \frac{1}{k!} \int_{y<z_{1}< \cdots < z_{k} } f(y,z_{1}, \cdots, z_{k-1}) \, dz_{1} \cdots dz_{k} dy. \end{align} $$

Integrating out $z_{i}$ and denoting $z_{0}:=0$ ,

(90) $$ \begin{align} &\quad = \sum_{ i=1}^{k-1} \frac{1}{k!} \int_{ z_{1} < \cdots < z_{i-1} < y < z_{i+1} < \cdots < z_{k} } f( z_{1}, \cdots, z_{i-1}, y, z_{i+1}, \cdots, z_{k}) (y-z_{i-1}) \end{align} $$

(91) $$ \begin{align} &\hspace{9cm} dz_{1} \cdots z_{i-1} z_{i+1}\cdots dz_{k} dy \end{align} $$

(92) $$ \begin{align} &\quad \qquad + \frac{1}{k!} \int_{ z_{1}< \cdots < z_{k-1} < y } f( z_{1},\cdots, z_{k-1}, y) (y-z_{k-1}) \, dz_{1} \cdots dz_{k-1} dy \end{align} $$

(93) $$ \begin{align} &\quad\quad \qquad + \frac{1}{k!} \int_{ y<z_{1} <\cdots < z_{k-1} } f( y,z_{1}, \cdots, z_{k-1}) (1-z_{k-1}) \, dz_{1} \cdots dz_{k-1} dy. \end{align} $$

We then rename y as $z_{i}$ for the first integral above and as $z_{k}$ for the second integral above. For the last integral, we rename y as $z_{1}$ and $z_{i}$ as $z_{i+1}$ for $i=1,\dots ,k-1$ . This gives

(94) $$ \begin{align} &= \frac{1}{k!} \int_{ z_{1} < \cdots < z_{k} } f( z_{1}, \cdots, z_{k}) \left[ (1-z_{k})+\left(\sum_{i=1}^{k-1} z_{i}-z_{i-1} \right) + (z_{k}-z_{k-1})\right] \, dz_{1} \cdots dz_{k} \end{align} $$

(95) $$ \begin{align} &= \mathbb{E}\left[ f( X_{(1)},\cdots , X_{(I-1)}, X_{(I)}, X_{(I+1)}, \cdots , X_{(k)})\right]. \end{align} $$

This shows the assertion.

6.1 Definition of the decoupled carrier process

In this section, we introduce a ‘decoupled version’ of the carrier process $W_{x}$ in (8), which will be critical in proving Theorem 2.3 (ii) as well as Theorems 2.6–2.7.

Figure 8 Illustration of the original circular exclusion rule (left) and its decoupled version (right) for $\kappa =7$ and ball density $\mathbf {p}=(.1,\, .1,\, .25,\, .05,\, .15,\, .2,\, .1,\, .05)$ . We take the set of exceptional colors $\mathcal {C}_{e}$ to be the set of unstable colors $\mathcal {C}_{u}^{\mathbf {p}}=\{2,5,6\}$ . For instance, in the decoupled carrier process, inserting new balls of color $5$ into the carrier only excludes existing balls of colors $2,3$ and $4$ .

To illustrate the idea, consider the carrier process $W_{x}$ with $\kappa =2$ as in Figure 1. While the transition kernel for this Markov chain depends on whether it is in the interior or at the boundary of the state space $\mathbb {Z}^{2}_{\ge 0}$ , we may consider a similar Markov chain on the entire integer lattice $\mathbb {Z}^{2}$ that only uses the kernel in the interior, by allowing the counts of color 1 and 2 balls in $W_{x}$ to be negative. In the general construction of decoupled carrier processes, we will allow the freedom to choose positive colors $ \alpha _{1}<\dots <\alpha _{r}$ in $\{1,\dots ,\kappa \}$ whose count can be negative. Recall that inserting a ball of color i to the carrier $W_{n}$ will exclude the largest color $i_{*}$ in $W_{n}$ that is less than i. In the decoupled carrier process, the color wheel $\mathbb {Z}_{\kappa +1}$ is divided into intervals $[0,\alpha _{1}]$ , $[\alpha _{1},\alpha _{2}],\dots , [\alpha _{r}, \kappa ]$ , and inserting a color i in $(\alpha _{j},\alpha _{j+1}]$ can only exclude a color in the interval $[\alpha _{j},\alpha _{j+1}]$ . Hence, the interaction between colors in distinct intervals is ‘decoupled’. See Figure 8 for an illustration.

Definition 6.1 (Decoupled carrier process).

Let $\xi :=(\xi _{x})_{x\in \mathbb {N}}$ be $\kappa $ -color ball configuration and fix a set $\mathcal {C}_{e}\subseteq \{ 1,\dots ,\kappa \}$ of ‘exceptional colors’. Let

(110) $$ \begin{align} \Omega:=\{ (x_{1},\dots,x_{\kappa})\in \mathbb{Z}^{\kappa}\,:\, x_{i}\ge 0 \text{ if } i\notin \mathcal{C}_{e} \}. \end{align} $$

The decoupled carrier process over $\xi $ associated with $\mathcal {C}_{e}$ is a process $(X_{x})_{x\in \mathbb {N}}$ on the state space $\Omega $ defined as follows. If $\mathcal {C}_{e}=\emptyset $ , then we take $X_{x}\equiv W_{x}$ , where $W_{x}$ is the carrier process in (8). Suppose $\mathcal {C}_{e}=\{ \alpha _{1},\dots ,\alpha _{r} \}$ for some $r\ge 1$ with $\alpha _{1} < \dots < \alpha _{r}$ . Denote $\alpha _{r+1}:=\kappa +1$ . Having defined $X_{1},\dots ,X_{x}$ , denote $i:=\xi _{x+1}$ if $\xi _{x+1}\in \{1,\dots ,\kappa \}$ and $i:=\kappa +1$ if $\xi _{x+1}=0$ . Then

(111) $$ \begin{align} X_{x+1} - X_{x}:= \begin{cases} \mathbf{e}_{i}- \mathbf{1}(i_{*}\ne 0) \, \mathbf{e}_{i_{*}} & \text{if } 1\le i \le \alpha_{1} \\ \mathbf{e}_{i} - \mathbf{e}_{i'} & \text{if } \alpha_{1}< i \le \kappa \\ -\mathbf{e}_{i'} & \text{if } i=\kappa+1, \end{cases} \end{align} $$

where $i_{*}:= \sup \{j\,:\, 1\le j< i ,\,\, X_{x}(j)\ge 1 \}$ (with the convention $\sup \emptyset = 0$ ) and

(112) $$ \begin{align} i':=\begin{cases} \alpha_{j} & \text{if } \alpha_{j}< i\le \alpha_{j+1} \text{ and } X_{x}(\alpha_{j})=\dots=X_{x}(i-1)\le 0\\ q & \text{if } \alpha_{j}< q < i\le \alpha_{j+1} \text{ and } X_{x}(q)\ge 1, X_{x}(q+1)=\dots=X_{x}(i-1)=0. \end{cases} \end{align} $$

Unless otherwise mentioned, we take $X_{0}=\mathbf {0}$ and $\xi =\xi ^{\mathbf {p}}$ with density $\mathbf {p}=(p_{0},\dots ,p_{\kappa })$ .

It is helpful to compare the recursion (111) for the decoupled carrier process to that of the carrier process in (8). Notice that in (8), inserting i into $W_{x}$ can decrease by one at coordinate $i_{*}$ only when $W_{x}(i_{*})\ge 1$ . Hence, $W_{x}$ is confined in the nonnegative orthant $\mathbb {Z}^{\kappa }_{\ge 0}$ . In comparison, when a ball of color i is inserted to the decoupled carrier $X_{x}$ , it decreases by one at coordinate, say $\ell \in \{ i', i_{*} \}$ . If $\ell \notin \mathcal {C}_{e}$ , then the above construction ensures that $X_{x}(\ell )\ge 1$ . From this, one can observe that $X_{x}(j)\ge 0$ for all $x\ge 0$ whenever $j\notin \mathcal {C}_{e}$ . In contrast, if $\ell \in \mathcal {C}_{e}$ , then $X_{x+1}(\ell )=X_{x}(\ell )-1$ regardless of whether $X_{x}(\ell )\ge 1$ . Hence, $X_{x}$ can take negative values on the exceptional colors. We call the recursion in (111) as the ‘decoupled circular exclusion’.

In the proposition below, we establish a basic coupling result between the carrier and the decouple carrier processes. For its proof, we will introduce the following notation. Define the following function $f_{W}:\mathbb {Z}^{\kappa }_{\ge 0}\times \{0,\dots ,\kappa \}\rightarrow \{0,\dots ,\kappa \}$ as

(113) $$ \begin{align} f_{W}(\mathbf{w}, y)&:= \begin{cases} 0 & \mathrm{if} \ [W_{0}=\mathbf{w} \ \mathrm{and} \ \xi_{1}=y \ \,\, \Longrightarrow \,\, W_{1}-W_{0}=\mathbf{e}_{y}] \\j & \mathrm{if} \ [W_{0}=\mathbf{w} \ \mathrm{and} \ \xi_{1}=y \,\,\,\, \Longrightarrow \,\, W_{1}-W_{0}=\mathbf{e}_{y}-\mathbf{e}_{j} \ \mathrm{or} \ -\mathbf{e}_{j}]. \end{cases} \end{align} $$

Roughly speaking, if $f_{W}(\mathbf {w}, y)=j$ , then j is the color of the ball that is excluded when a ball of color y is inserted into the carrier of state $\mathbf {w}$ . The circular exclusion rule says $f_{W}(\mathbf {w}, y)=\sup \{i \,:\, 1\le i < y,\, \mathbf {w}(i)\ge 1 \}$ with the convention $\kappa +1\equiv 0$ and $\sup \emptyset = 0$ . Similarly, define a function $f_{X}:\Omega \times \{0,\dots ,\kappa \}\rightarrow \{0,\dots ,\kappa \}$ as

(114) $$ \begin{align} f_{X}(\mathbf{w}, y)&:= \begin{cases} 0 & \mathrm{if} \ [X_{0}=\mathbf{w} \ \mathrm{and} \ \xi_{1}=y \ \,\, \Longrightarrow \,\, \ X_{1}-X_{0}=\mathbf{e}_{y}] \\ j & \mathrm{if} \ [X_{0}=\mathbf{w} \ \mathrm{and} \ \xi_{1}=y \ \,\, \Longrightarrow \,\, \ X_{1}-X_{0}=\mathbf{e}_{y}-\mathbf{e}_{j} \ \mathrm{or} \ -\mathbf{e}_{j}]. \end{cases} \end{align} $$

Intuitively, if $f_{X}(\mathbf {w}, y)=j$ , then j is the color of the ball that is excluded when a ball of color y is inserted into the decoupled carrier of state $\mathbf {w}$ .

For each $x\in \mathbb {N}$ , define $\hat {X}_{x}\in \mathbb {Z}^{\kappa }_{\ge 0}$ by

(115) $$ \begin{align} \hat{X}_{x}(i):=X_{x}(i) - \min_{0\le s \le x} X_{s}(i) \quad \text{for all } i=1,\dots,\kappa. \end{align} $$

Note that $\hat {X}_{x}(i)\ge \max (0, X_{x}(i))$ for all i by definition and $X_{0}=\mathbf {0}$ . Also, $\hat {X}_{x}(i)\equiv X_{x}(i)$ for all $i\notin \mathcal {C}_{e}$ since $X_{x}(i)\ge 0$ for all $x\in \mathbb {N}$ and all $i\notin \mathcal {C}_{e}$ .

6.2 Proof of the Skorokhod decomposition of the carrier process

Now we give an explicit construction of the Skorokhod decomposition of $(W_{x})_{x\ge 0}$ . First, let R be the $\kappa \times \kappa $ tridiagonal matrix with 0 on the subdiagonal, 1 on the main diagonal and -1 on the superdiagonal entries:

(118) $$ \begin{align} R :=\text{tridiag}_{\kappa}(0,1,-1) = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 & 0 \\ \vdots && \ddots \\ 0 & \cdots &0& 1 & -1 \\ 0 & & \cdots & 0 & 1 \end{bmatrix} = I - Q, \end{align} $$

where I is the $\kappa \times \kappa $ identity matrix and $Q=I-R$ . Notice that the spectral radius of Q is zero for all $\kappa \ge 2$ being an upper triangular matrix with zero diagonal entries. The above reflection matrix also has the property of being ‘completely- $\mathcal {S}$ ’; see Definition 10.2 and the proof of Theorem 2.5 for justification.

Next, we define the pushing process $(Y_{x})_{x\ge 0}$ on $\mathbb {Z}^{\kappa }_{\ge 0}$ recursively as follows: Set $Y_{0}=\mathbf {0}$ . Having defined $Y_{x}$ , denoting $y_{W}:=f_{W}(W_{x}, \xi _{x+1})$ (see (113)) and $y_{X}:=f_{X}(X_{x}, \xi _{x+1})$ (see (114)), define

(119) $$ \begin{align} Y_{x+1} - Y_{x} := \begin{cases} \mathbf{0} & \text{if } y_{W}=y_{X} \\ \mathbf{e}_{y_{W}+1}+\dots+\mathbf{e}_{y_{X}} & \text{if } y_{W}<y_{X}. \end{cases} \end{align} $$

Note that (119) covers all cases since $y_{W}\le y_{X}$ due to Proposition 6.2. From the definition, it is clear that every coordinate of $Y_{x}$ is nondecreasing. Also, clearly, $Y_{x}$ is determined by the first x ball colors $\xi _{1},\dots ,\xi _{x}$ .

Proof. Let $y:=\xi _{x+1}$ if $\xi _{x+1}\ne 0$ and $y:=\kappa +1$ if $\xi _{x+1}=0$ . Also let $y_{W}:=f_{W}(W_{x}, \xi _{x+1})$ and $y_{X}:=f_{X}(X_{x}, \xi _{x+1})$ (see (113) and (114)). We first show (ii). According to (117) in Proposition 6.2, we have $y_{W}\le y_{X}< y$ . Also, by the definition of $y_{W}$ , we have $W_{x}(y_{W}+1)=\dots =W_{x}(y-1)=0$ . Hence, if $Y_{x+1}(i)-Y_{x}(i)>0$ , then $i\in \{ y_{W}+1,\dots , y-1 \}$ , and hence, $W_{x}(i)=0$ . This shows (ii).

Next, we show (i) by induction on $x\ge 0$ . It holds trivially when $x=0$ , so suppose for the induction step that it holds for some $x\ge 0$ . We wish to show that

(120) $$ \begin{align} W_{x+1} = X_{x+1} + R Y_{x+1}. \end{align} $$

From (113)–(114), note that

(121) $$ \begin{align} (W_{x+1}-W_{x}) - (X_{x+1}-X_{x}) = \begin{cases} \mathbf{0} & \text{if } y_{W}=y_{X} \\ \mathbf{e}_{y_{X}} - \mathbf{e}_{y_{W}} & \text{if } 1\le y_{W}<y_{X} \\ \mathbf{e}_{y_{X}} & \text{if } 0= y_{W}<y_{X}. \end{cases} \end{align} $$

If $y_{W}=y_{X}$ , then $R(Y_{x+1}-Y_{x}) = \mathbf {0}$ , so (120) holds by the induction hypothesis. Next, suppose $1\le y_{W}<y_{X}$ . Note that

(122) $$ \begin{align} R(Y_{x+1}-Y_{x}) & = R(\mathbf{e}_{y_{W}+1}+\dots+\mathbf{e}_{y_{X}}) \end{align} $$

(123) $$ \begin{align} &= (\mathbf{e}_{y_{W}+1}-\mathbf{e}_{y_{W}}) + (\mathbf{e}_{y_{W}+2}-\mathbf{e}_{y_{W}+1}) + \dots + (\mathbf{e}_{y_{X}}-\mathbf{e}_{y_{X}-1}) \end{align} $$

(124) $$ \begin{align} &= \mathbf{e}_{y_{X}} - \mathbf{e}_{y_{W}}. \end{align} $$

Lastly, suppose $0= y_{W}<y_{X}$ . Then

(125) $$ \begin{align} R(Y_{x+1}-Y_{x}) & = R(\mathbf{e}_{1}+\dots+\mathbf{e}_{y_{X}}) \end{align} $$

(126) $$ \begin{align} &=\mathbf{e}_{1} + (\mathbf{e}_{2}-\mathbf{e}_{1}) + (\mathbf{e}_{3}-\mathbf{e}_{2}) + \dots + (\mathbf{e}_{y_{X}}-\mathbf{e}_{y_{X}-1}) = \mathbf{e}_{y_{X}}. \end{align} $$

Hence, in all cases, the induction step holds by the induction hypothesis, (121) and (119).

7.1 Decomposition of the decoupled carrier process

Let $\mathbf {p}=(p_{0},\dots ,p_{\kappa })$ be the ball density at each site. We choose the set of exceptional colors $\mathcal {C}_{e}$ so that it satisfies the following ‘stability condition’:

(127) $$ \begin{align} \text{For all } 1\le j \le r, \quad \max \{p_{i}\,:\, \alpha_{j}<i<\alpha_{j+1} \} < p_{\alpha_{j+1}}, \end{align} $$

where we set $\alpha _{0}=0=\alpha _{r+1}$ . Since balls of a nonexceptional color i in $(\alpha _{j}, \alpha _{j+1})$ can be excluded by balls of color $\alpha _{j+1}$ in the decoupled carrier, the above condition ensures that $(X_{x}(i))_{x\ge 0}$ do not blow up. A canonical choice of such $\mathcal {C}_{e}$ is the set of unstable colors $\mathcal {C}^{\mathbf {p}}_{u}$ that we defined above the statement of Theorem 2.5.

Define the following processes:

(128) $$ \begin{align} \begin{cases} X_{x}:=\text{The decoupled carrier process over } \xi=\xi^{p} \text{ associated with } \mathcal{C}_{e} \text{ satisfying (127)} \\ X_{x}^{s} := \left(\mathbf{1}(i \notin \mathcal{C}_{e}) \, X_{x}(i) \,;\, i=1,\dots,\kappa \right) \hspace{2cm} (\triangleright \text{ The `stable part' of } X_{x}) \\ X_{x}^{u} := \left(\mathbf{1}(i \in \mathcal{C}_{e}) \, X_{x}(i) \,;\, i=1,\dots,\kappa \right) \hspace{1.97cm} (\triangleright \text{ The `unstable part' of } X_{x}). \end{cases} \end{align} $$

Namely, $X^{s}_{x}$ (resp., $X^{u}_{x}$ ) agrees with $X_{x}$ on the nonexceptional (resp., exceptional) colors but its coordinates on exceptional (resp., nonexceptional) colors are zero. Clearly, we have the following decomposition:

(129) $$ \begin{align} X_{x} = X_{x}^{s} + X_{x}^{u} \qquad \text{for all } x\ge 0. \end{align} $$

In Lemma 7.1, we will show that $(X^{s}_{x})_{x\ge 0}$ defines an irreducible Markov chain whose empirical distribution converges to its unique stationary distribution $\pi ^{s}$ defined as

(130) $$ \begin{align} \pi^{s} \big(n_{1}, \dots, n_{\kappa}\big) = \prod_{j\in \mathcal{C}^{\mathbf{p}}_{u}} \mathbf{1}(n_{j}=0) \prod_{j=0}^{r}\left[ \prod_{\alpha_{j}<i<\alpha_{j+1}} \left(1-\frac{p_{i}}{p_{\alpha_{j+1}}} \right) \left( \frac{p_{i}}{p_{\alpha_{j+1}}} \right)^{n_{i}} \right], \end{align} $$

where we set $\alpha _{0}=0=\alpha _{r+1}$ . Hence, the expression in the bracket above is a nondegenerate geometric distribution. Thus, the above is the product of $\kappa -r$ geometric distributions, so it is indeed a probability distribution on $\Omega ^{s}$ . Comparing (130) with (9), we see that the exceptional color $\alpha _{j+1}$ plays the role of color 0 for the nonexceptional colors in the interval $(\alpha _{j},\dots ,\alpha _{j+1})$ .

Lemma 7.1. Let $(X^{s}_{x})_{x\ge 0}$ be the process defined in (128). Then it is an aperiodic Markov chain on the state space $\mathbb {Z}_{\ge 0}^{\kappa }$ and has a unique communicating class with unique stationary distribution $\pi ^{s}$ defined in (130). Furthermore, if we denote the distribution of $X^{s}_{x}$ by $\pi ^{s}_{x}$ , then

(131) $$ \begin{align} \lim_{x\rightarrow \infty} d_{TV}(\pi^{s}_{x}, \pi^{s} ) = 0. \end{align} $$