1.1 Identifying and tracking solitons

Define the runs of $\eta $ as maximal blocks of successive sites where $\eta $ has a constant value, so that they form a partition of $\mathbb {Z}$ . Assume first that $\eta $ has a finite number of balls, so it has a finite number of finite runs and two semi-infinite runs of zeros, one to the left and one to the right.

A k-soliton is a collection of $2k$ boxes identified by the Takahashi–Satsuma algorithm [Reference Takahashi and SatsumaTS90] running as follows.

Notice that a k-soliton consists of k zeros (possibly nonconsecutive) followed by k ones or vice versa, and letters that do not belong to any soliton are all zero and correspond to the records of $\eta $ ; see Figure 1.1. For a general $\eta \in \mathcal {X}$ , all excursions have a finite number of boxes. To identify the solitons in $\eta $ , we take each excursion of $\eta $ , append infinitely many zeros to the left and right of the excursion and then apply the above algorithm to it.

Figure 1.1

Applying the Takahashi–Satsuma algorithm to a sample configuration. Dots represent records. On the left we have the resulting word after successive iterations. Identified solitons are shown in bold once and then with a color corresponding to their size. The algorithm is applied to each excursion separately, so the rightmost $1$ -soliton in the picture is ignored by this instance of the procedure. (color online)

We define the head and tail of a k-soliton $\gamma $ as follows: the head $\mathcal {H}(\gamma )=\{\mathcal {H}_{1}, \mathcal {H}_{2}, \dots, \mathcal {H}_{k}\}\subseteq \mathbb {Z}$ is the set of k boxes with ones in $\gamma $ and the tail $\mathcal {T}(\gamma )=\{\mathcal {T}_{1}, \mathcal {T}_{2}, \dots, \mathcal {T}_{k}\}\subseteq \mathbb {Z}$ is the set of k boxes with zeroes in $\gamma $ . Namely, $\mathcal {H}(\gamma )\cup \mathcal {T}(\gamma )$ is the set of boxes that are removed when executing the previous algorithm on $\eta $ . Let $\Gamma _k\eta $ be the set of k-solitons of a ball configuration $\eta \in \mathcal {X}$ . The following is proved in Section 7.

By the above proposition, we can track each k-soliton $\gamma $ in the evolution of $\eta $ . For each k-soliton $\gamma \in \Gamma _k \eta $ , call $(\gamma ^t)_{t\geqslant 0}$ the trajectory satisfying $\gamma ^0 =\gamma $ , $\gamma ^t\in \Gamma _kT^t\eta $ and

(1.4) $$ \begin{align} \mathcal{H}(\gamma^{t+1}) = \mathcal{T}(\gamma^t). \end{align} $$

1.2 Soliton nesting and motion

As shown in Figure 1.1, solitons can be nested inside larger solitons. As it turns out, they are nested in a hierarchical way. Moreover, solitons only move to the right, and they are only free to move when they are not nested inside larger solitons. In particular, solitons can only overtake smaller solitons. We now make these statements precise.

Let $ \gamma \in \Gamma _k \eta $ for some $ k\in \mathbb {N} $ . Let us denote by $x(\gamma )$ its leftmost site. Take $ z $ as the first site to the right of $ \gamma $ such that $ z $ is either a record or belongs to another $ m $ -soliton for some $ m \geqslant k $ . The interval spanned by $ \gamma $ is defined as $ I(\gamma ) := [x(\gamma ),z-1] \cap \mathbb {Z} $ .

The interval $ I(\gamma ) $ and the above properties are illustrated in Figure 1.2.

Figure 1.2

Here we show $ I(\gamma ) $ in an example with nine records, a 5-soliton, a 4-soliton, two 3-solitons, two 2-solitons and two 1-solitons, with one color for each size. In this example, a 1-soliton is contained in a 2-soliton, both 2-solitons are contained in the 5-soliton and both 3-solitons are contained in the 4-soliton. $ I(\gamma ) $ is underlined with the same color as $ \gamma $ , and black zeros are records. (color online)

Recall the notation $\gamma ^{1}$ from Proposition 1.3. The following lemma says that a soliton can move forward only if it is not already nested inside a larger one.

Our last observation is that smaller solitons never overtake larger ones.

These three lemmas are also proved in Section 7.

1.3 Asymptotic speeds

We use $\theta $ to denote shift operators on $ \mathbb {Z} $ , its power set $ \mathcal {P}(\mathbb {Z}) $ and $ \{0,1\}^{\mathbb {Z}} $ . Namely,

(1.8) $$ \begin{align} \theta x = x-1, \quad \theta A = \{\theta x : x\in A\}, \quad \theta\eta(y):= \eta(\theta^{-1} y) \text{ for } \eta\in \{0,1\}^{\mathbb{Z}}. \end{align} $$

Let $ \mathcal {B} $ denote the Borel $ \sigma $ -field of $ \{0,1\}^{\mathbb {Z}} $ . We say that a probability measure $\mu $ on $\{0,1\}^{\mathbb {Z}}$ is shift-ergodic if $\mu $ is $\theta $ -invariant and for every event $A \in \mathcal {B}$ satisfying $\theta ^{-1}(A)=A$ we have $\mu (A)=0$ or $1$ . Let $\mu $ be a shift-ergodic measure on $\mathcal {X}$ and denote by $\rho _k$ the mean number of k-solitons per excursion, by $w_0=1+\sum _k 2k\rho _k$ the mean distance between records and by $\bar {\rho }_k = \frac {\rho _k}{w_0}$ the mean number of k-solitons per unit space (precise definitions in Subsection 4.3). Recall that $x(\gamma )$ is the leftmost site of a soliton $\gamma $ and that $\gamma ^t$ is the soliton $\gamma $ at time t. We now state the main result concerning soliton speeds.

Theorem 1.1. Let $\mu $ be a T-invariant and shift-ergodic measure on $\mathcal {X}$ . Then there exist deterministic speeds $(v_k)_k$ such that $\mu $ -almost surely (a.s.) for all $k\geqslant 1$ and $\gamma \in \Gamma _k\eta $ ,

(1.9) $$ \begin{align} \lim_{t\to\infty} \frac{x(\gamma^t)}{t} &= v_k. \end{align} $$

The soliton speeds $ v_k $ are finite, positive, increasing in $ k $ and satisfy the system

(1.10) $$ \begin{align} v_k = k + \sum_{m<k} 2m \bar{\rho}_m (v_k-v_m) - \sum_{m>k} 2k \bar{\rho}_m (v_m-v_k). \end{align} $$

System (1.10) comes from the following. When a k-soliton is isolated, it advances by k units, and when it encounters an m soliton, the encounter causes it to advance $2m$ extra units if $m<k$ or stay put for two units of time if $m>k$ . The term $\bar {\rho }_m |v_k-v_m|$ gives the frequency of such encounters as seen from a k-soliton.

When $ \rho _k=0 $ , the soliton speed $ v_k $ does not come from (1.9) but in principle formally from (1.10). There is still an interpretation for $ v_k $ in terms of the dynamics, as discussed in Subsection 6.5.

When $\rho _k>0$ for finitely many $ k $ , the system has a unique solution [Reference Croydon and SasadaCS21, Lemma 5.1]. We believe it is unique in general, but we have been unable to prove it. The following partial result is proved in Section 7.

As a side remark, $ \sum _m m \bar {\rho }_m = \lambda $ ; that is, the mean number of occupied boxes per unit space. Uniqueness has not been proved to hold in general, and we show that $(v_k)_k$ is determined by the vector $ (\bar {\rho }_k)_k $ under a stronger assumption in terms of soliton components (described in Subsection 1.5).

Theorem 1.2. If, when conditioned on having a record at $ x=0 $ , $\mu $ has independent soliton components and each component is independent and identically distributed (i.i.d.), then the soliton speeds $(v_k)_k$ in (1.9) are also given by the unique solution to

(1.12) $$ \begin{align} \begin{aligned} w_k&=1+\sum_{m>k} 2(m-k) \rho_m, & \rho_k &= \alpha_k w_k,\\ s_k &= k+\sum_{m<k}2(k-m) s_m \alpha_m, & v_k &= \frac {s_k}{w_k}, \end{aligned} \end{align} $$

and, in particular, they are determined by $(\rho _k)_k$ .

In system (1.12), $w_k$ is the density of k-slots per excursion (see Subsection 1.4 for the definition of k-slot), $\alpha _k$ is the density of k-solitons per k-slot, $s_k$ is the average size of the head of a k-soliton, $k-m$ is the number of m-slots in the head of a k-soliton and the factor $\frac {1}{w_k}$ is the probability that a typical k-soliton is free to move (see Section 6 for details).

The proof of (1.12) uses independence properties of the components for an explicit computation of the mean jump size of a typical k-soliton $\gamma $ in one iteration. By ergodicity, the mean jump size is $v_k$ , the limit of $x(\gamma ^t)/t$ , as shown in Subsection 6.1.

In the setup of Theorem 1.1, we cannot compute the mean jump size explicitly. However, if we assume that the solution to (1.10) is indeed unique, then by taking an initial measure with independent components and the same vector $ (\rho _k)_k $ , we see that the vector $ (v_k)_k $ must be given by (1.12). So if the solution to (1.10) is indeed unique (as conjectured), the independence assumption in Theorem 1.2 can be waived.

In practice, the soliton speeds $(v_k)_{k}$ can be computed by truncating $\rho $ (replace $\rho _k$ by $0$ for large k) and solving these finite recursions for w, $\alpha $ , s and finally v. When the initial ball configuration consists of i.i.d. Bernoulli random variables, one can find $\alpha _k$ explicitly in terms of the density $\lambda $ by computing partition functions [Reference Ferrari and GabrielliFG20a], substitute the equation for $\rho $ into that for w and then compute s and v. Using this and the above theorem, we have found the asymptotic speeds of the solitons for the simulations shown in Figures 1.3 and 1.4 as well as that on the first page.

Figure 1.3

Simulation for an i.i.d. configuration with density $0.15$ . The transparent red lines have deterministic slopes computed by Theorem 1.2, which have been manually shifted so that they would overlay a soliton. This window covers 2000 sites and 140 time steps going downwards and has been stretched vertically by a factor of $5$ . The figure on the first page is the same except for the density. (high resolution, color online)

Figure 1.4

Simulation for $(\rho _k)_k=(.006,.005,.1,.003,0,0,0,\dots )$ . The initial configuration was obtained by first appending one k-soliton with probability $\rho _k$ after each record and then applying T a number of times in order to mix. As in Figure 1.3, it is a 2000 x 140 window stretched by 5, and red lines are deterministic. (high resolution, color online)

1.4 Slot decomposition

Recall that a k-soliton has a head and a tail, each one consisting of k (possibly nonconsecutive) sites. We say that the jth box of the head or tail of an m-soliton is a k-slot for all $k<j$ and that a record is a k-slot for every k. Roughly speaking, the k-slots are the places where k-solitons can be appended; see Subsection 2.1 for precise definitions and examples.

The set of configurations with a record at the origin is defined by

$$ \begin{align*} \skew8\widehat{\mathcal{X}}:=\{\eta\in\mathcal{X}:0\in R\eta\}. \end{align*} $$

Assume that $\eta \in \skew8\widehat{\mathcal {X}}$ . Enumerate the k-slots from left to right such that the $0$ th k-slot is at position $s_k(\eta,0)=0$ , and let $s_k(\eta,i)$ denote the position of the ith k-slot for $i\in \mathbb {Z}$ . We say that a k-soliton $\gamma $ is appended to the ith k-slot if its head and tail are contained between $s_k(\eta,i)$ and $s_k(\eta,i+1)$ . Define the kth component of $\eta $ as the configuration $M_k\eta $ given by M _k $\eta $ (i):= number of k-solitons appended to the ith k-slot. Define

$$ \begin{align*} \mathcal{M}:=\bigl\{\zeta= (\zeta_k)_{k\geqslant1}, \,\zeta_k \in (\mathbb{Z}_+)^{\mathbb{Z}}: \textstyle{\sum}_k \zeta_k(i)<\infty,\,\text{ for all }i\in\mathbb{Z}\bigr\}. \end{align*} $$

The next result, proved in Subsection 2.2, says that we can recover $\eta $ from its components $(M_{k}\eta )_{k}$ .

The fundamental property of this decomposition is that it makes the BBS dynamics linear. We show a simple case, deferring the full statement until Theorem 3.1 in Section 3.

Theorem 1.4. Suppose $ \eta \equiv 0 $ on $ \mathbb {Z}_- $ and $ \eta $ has infinitely many records on $ \mathbb {Z}_+ $ . Then for all $k\geqslant 1$ ,

$$ \begin{align*} M_k T\eta =\theta^{-k} M_k\eta. \end{align*} $$

So, when we see a configuration $ \eta $ through its components, the kth component is displaced by k units, without interacting with the other components. In the general case, there will be larger solitons arriving from the left, which will disrupt the enumeration of k-slots, making the statement substantially more involved. See the details in Section 3.

1.5 Invariant measures

Using the slot decomposition and the reconstruction map $(M_{k}\eta )_{k}\mapsto \eta $ that will be described in detail in Subsection 2.2, we get the following.

The above theorem says that the family of invariant measures for this dynamics is at least as large as the family of sequences of states of k-soliton configurations. In particular, given a sequence $(\alpha _k)_k$ specifying the density of k-solitons in the kth component, we can construct an infinite number of mutually singular shift-invariant and T-invariant laws $\mu $ on $\mathcal {X}$ , all having the same specified component densities.

The extra assumption needed in Theorem 1.2 is that $\mu $ be of the above form; that is, conditioning on $ \skew8\widehat{\mathcal {X}} $ , each kth component is i.i.d. and they are independent over k. In this case, we can also study the speed of tagged records and the speed of solitons measured in terms of tagged records; see Subsection 6.5. As pointed out above, this condition should not be necessary for Theorem 1.2 to hold.

We should also note that the converse of Theorem 1.5 is false. In particular, there exist invariant measures that are not constructed from independent components; see Examples 4.6 and 4.7 as well as Remark 4.8 in Subsection 4.1.

2.1 Slots and components

We now describe how solitons can be nested inside each other via what we call slots. Intuitively, the idea of k-slot is that it marks a place where a k-soliton can be inserted without interfering with the rest of the configuration in terms of the Takahashi–Satsuma algorithm; see Figure 2.2.

Figure 2.2

A red $ 3 $ -soliton can be appended to a blue $ 5 $ -soliton in $ 2 \times (5-3) = 4 $ different places, represented by blue dots. It is also possible to append it to records, represented by black dots. Attempting to insert a $ 3 $ -soliton at a site not marked by a dot will result in erroneous soliton identification. For instance, the 3-soliton in the middle bottom plot should actually start three sites earlier, and in the right bottom plot we should have a 2- and 6-soliton instead of 3 and 5. The green crosses indicate that the coloring is inconsistent with the procedure shown in Figure 1.1. (color online)

Let $\gamma \in \Gamma _{k}(\xi )$ be a k-soliton for the walk representation $\xi $ . We label the sites in the head (respectively in the tail) of $\gamma $ from left to right: $\mathcal {H}(\gamma )=\{\mathcal {H}_1(\gamma ), \dots, \mathcal {H}_k(\gamma )\}$ (respectively $\mathcal {T}(\gamma )= \{\mathcal {T}_1(\gamma ),\dots,\mathcal {T}_k(\gamma )\}$ ). The slot configuration $S\xi :\mathbb {Z}\to \{0,1,2,\dots \}\cup \{\infty \}$ is defined by

$$ \begin{align*} S\xi(x) := \begin{cases} m-1,&\text{if } x = \mathcal{T}_m(\gamma) \text{ or } \mathcal{H}_m(\gamma), \text{ for }\gamma \in\Gamma_k\xi \text{ and } k \geqslant m,\\ \infty,&\text{if } x \text{ is a record for } \xi. \end{cases} \end{align*} $$

For each $k\geqslant 1$ we say that x is a k-slot for $\xi $ if $S\xi (x)\geqslant k$ .

Note that a record is a k-slot for all k, and an m-soliton contains a number $2m-2k$ of k-slots; see Figure 2.3. Because every $\xi \in \mathcal {W}$ has infinitely many records, it also has infinitely many k-slots.

Figure 2.3

Slot configuration of a walk $\xi $ . Different colors correspond to different solitons; records are painted in black. For each site, the number of dots below it indicates its level in slots: k dots indicate a k-slot. (color online)

For $j\in \mathbb {Z}$ , the position of the record at level $-j$ will be called record j and denoted as

$$ \begin{align*} r(\xi,j) := \min\{x\in\mathbb{Z}: \xi(x)=-j\}. \end{align*} $$

This is the leftmost site where the walk $\xi $ takes the value $-j$ . If $\xi \in \mathcal {W}$ , we have $r(\xi,j) \in \mathbb {Z}$ well defined for all $j\in \mathbb {Z}$ . The site $r(\xi, 0)$ will play a central role in the sequel. Note that $T\xi (x) \leqslant \xi (x)$ by (2.2), so $r(T\xi, j)\leqslant r(\xi, j)$ for all j.

We label all k-slots in increasing order: $\cdots <s_{k}(\xi, -1)<s_{k}(\xi, 0)<s_{k}(\xi, 1)<\cdots $ , where $s_k(\xi,0):=r(\xi,0)$ . The set of k-slots of $\xi $ is denoted $S_k\xi := \{s_k(\xi,i):{i\in \mathbb {Z}}\}$ . We then say that a k-soliton $\gamma $ is appended to the ith k-slot $s_{k}(\xi, i)$ if $\gamma \cap [s_k(\xi,i),s_k(\xi,i+1)-1]\ne \varnothing $ . Observe that if that is the case, we necessarily have $\gamma \subseteq [s_k(\xi,i)+1,s_k(\xi,i+1)-1]$ , as a consequence of Lemma 1.5 and the fact that k-slots can only be records or sites of solitons with larger size. It follows that each k-soliton in $\xi $ is appended to a unique k-slot. On the other hand, it is possible to have multiple k-solitons appended to a single k-slot. Finally, we let $M_k\xi (i)$ be the number of k-solitons appended to the ith k-slot and call $M_{k}\xi =(M_{k}\xi (i))_{i\in \mathbb {Z}}$ the kth component of $\xi $ . For instance, in the example of Figure 2.3, we have M ₆ $\xi $ (0) = 1, M ₅ $\xi $ (2) = 1, M ₄ $\xi $ (2) = 1, M ₁ $\xi $ (9) = 1, M ₁ $\xi $ (18) = 1 and $M_k\xi (i) = 0$ otherwise. See also Figure 2.4 for an illustration on how the solitons are nested inside each other via slots.

Figure 2.4

An illustration of how the solitons are nested inside bigger solitons via slots, in the same sample configuration as in Figure 2.3. Solitons are represented by squares and slots by circles. For each $k \geqslant 1$ , each slot with index $m\geqslant k$ is a k-slot. We say it is the nth k-slot, where n is determined by counting how many k-slots appear before it in the depth-first order, and the counting starts from the 0th k-slot present at record 0.

2.2 Reconstructing the configuration from the components

Here we prove Theorem 1.3. In this subsection only, we will work with a larger set of configurations than $\mathcal W$ . Let $\mathcal W_{*}$ be the set of walks $\xi $ such that $r(\xi, j)$ is well defined for all $j\in \mathbb {Z}$ . Even though $\mathcal W_{*}$ is not preserved by T, the Takahashi–Satsuma algorithm still applies, because all of the excursions of $\xi \in \mathcal W_{*}$ are finite. So the kth component $M_{k}$ described in Subsection 2.1 is also defined for all $\xi \in \mathcal W_{*}$ . On the other hand, because the slot decomposition $\xi \mapsto M\xi $ is insensitive to horizontal shifts, it is not possible to determine $\xi $ knowing $(M_k\xi )_{k\geqslant 1}$ . So we introduce

$$ \begin{align*} \skew5\widehat{\mathcal{W}}_{*}:= \{\xi\in\mathcal{W}_{*}: r(\xi,0)=0\}. \end{align*} $$

(As a side remark, $ \mathcal {W}_* $ is the set of simple walks $ \xi $ such that $ \lim _{x\to -\infty } \xi (x)=+\infty $ and $ \liminf _{x\to +\infty } \xi (x) = -\infty $ . We can build a configuration $\xi \in \mathcal {W}_*$ with $ T\xi \not \in \mathcal {W}_* $ by appending a soliton of height $2^n$ between records $ -n $ and $ -n+1 $ , so that $ \liminf _{x\to -\infty } T\xi (x) = -\infty $ .)

Note that, unlike the lift $\xi [\eta ]$ from $\mathcal {X}$ to $\mathcal {W}$ , which was not unique, for $\eta $ in $ \skew8\widehat{\mathcal {X}} $ there is a unique lift $\xi [\eta ]$ that is in $\skew5\widehat {\mathcal {W}}_{*}$ . We denote this unique lift by $\xi ^{\circ }[\eta ]$ . So the maps $ \eta \mapsto M_k \eta $ as seen in Theorem 1.3 are given by $M_k\eta = M_k\xi ^{\circ }[\eta ]$ . But to remain consistent with the previous subsections, we continue to work with $\xi $ instead of $\eta $ .

Denote by M the map $\xi \mapsto M\xi :=(M_k\xi )_{k\geqslant 1}$ . We now show that $M: \skew5\widehat {\mathcal {W}}_{*}\to \mathcal M$ is invertible.

The height of the excursion between record j and record $j+1$ , denoted as $m(j)$ , is defined as $0$ if the excursion is empty or as the largest k such that a k-soliton is contained in the excursion. Denote also $i_k(j)$ the label of the k-slot located at record j. Because k-slots can only be created by solitons of larger sizes, we note that

$$ \begin{align*} m(j) &= \min\{k\geqslant 0: M_{k'}\xi(i_{k'}(j))=0 \text{ for all }k'>k\} \in \{0,1,2,\dots\}. \end{align*} $$

Both $m(j)$ and $i_k(j)$ depend on $\xi $ but we omit it in the notation. Denote the slot decomposition $M\xi $ of $\xi $ by $\zeta $ ; that is,

$$ \begin{align*} \zeta = (\zeta_k)_k = M \xi. \end{align*} $$

Note that each $\xi \in \mathcal {W}_{*}$ and $j\in \mathbb {Z}$ , $i_k(j)=r(\xi,j)$ for all $k\geqslant \max \{m(0), \dots, m(j)\}$ . Because $m(j)$ is finite, we have that

(2.3) $$ \begin{align} \text{ for each } i\in\mathbb{Z},\ \zeta_k(i) = 0 \text{ for all large } k. \end{align} $$

Namely, $\zeta \in \mathcal M$ . Conversely, suppose that $\zeta = (\zeta _k)_{k\geqslant 1} \in \mathcal M$ , so, in, particular $\zeta $ satisfies (2.3). We first give an algorithm to reconstruct the excursion of $\xi $ between records 0 and 1. To that end, we introduce the following notation: denote the number of k-slots in the excursion $\varepsilon $ between successive records $y_1<y_2$ by

(2.4) $$ \begin{align} n_k(\varepsilon) := 1 + |S_k \xi \cap \{y_1+1,\dots, y_2-1\}|, \end{align} $$

where the term $1$ refers to the record $y_1$ preceding $\varepsilon $ and the second term counts the number of k-slots belonging to m-solitons of $\varepsilon $ with $m>k$ . Here is the algorithm:

Note that m is well defined by (2.3). In case $m=0$ , the algorithm produces an empty configuration. Let us also note that when inserting solitons in the above algorithm, there is only one way to do it consistently with the soliton decomposition: if a soliton is inserted to the right of the site x with $\eta (x)=0$ , then it has its head on the left and tail on the right (i.e., $11\cdots 100\cdots 0$ ); otherwise, it is inserted with its tail on the left and head on the right (i.e., $00\cdots 011\cdots 1$ ). The procedure is illustrated in Figure 2.5. Call $\varepsilon ^0$ the excursion between record $0$ and record $1$ obtained from $ \zeta $ as just described. Construct $\varepsilon ^1$ , the excursion between records 1 and 2, using the same algorithm but with the data $\zeta ^1 = (\zeta ^1_k)_{k\geqslant 1}$ , where each component is given by

$$ \begin{align*} \zeta^1_k = \big( \zeta_k(n_k+i) \big)_{i \geqslant 0}, \end{align*} $$

which consists of the entries of $\zeta $ with nonnegative indices i not used in the reconstruction of $\varepsilon ^0$ . Note that $\zeta ^1$ also satisfies (2.3). Iterate this procedure to construct an infinite sequence of excursions $(\varepsilon ^j)_{j=0,1,2,\dots }$ . See Figure 2.7.

Figure 2.5

Reconstruction algorithm for a single excursion. This example is obtained using the field $\zeta $ shown in Figure 2.6.

Figure 2.6

Reconstruction of $\xi $ from $\zeta $ . In the lower part we show records $-2$ to $2$ in boldface and the excursions between them. Above we show the parts of the field $\zeta $ used in the reconstruction of $\varepsilon ^{-2},\varepsilon ^{-1},\varepsilon ^{0},\varepsilon ^{1}$ . Reconstruction of $\varepsilon ^0$ was shown in Figure 2.5 and $\varepsilon ^{1},\varepsilon ^{-1},\varepsilon ^{-2}$ is shown in Figure 2.7.

Figure 2.7

Reconstruction algorithm for other excursions. The procedure is the same as in Figure 2.5 but all of the intermediate steps are omitted.

To reconstruct the configuration to the left of record $0$ – that is, to obtain the excursions $\varepsilon ^j$ with negative j – we use an analogous algorithm that uses the entries of $\zeta $ with i-indices starting at $-1$ and moving left instead of starting at $0$ and moving right. First take $\zeta ^{-1} = (\zeta ^{-1}_k)_{k \geqslant 1}$ where each component is given by $\zeta ^{-1}_k = \big ( \zeta _k(i) \big )_{i < 0}$ and use $\zeta ^{-1}$ to construct $\varepsilon ^{-1}$ . Then define $\zeta ^{-2} = (\zeta ^{-2}_k)_{k \geqslant 1}$ where each component is given by $\zeta ^{-2}_k = \big ( \zeta _k(i-n_k) \big )_{i < 0}$ and use it to construct $\varepsilon ^{-2}$ . Iterate this procedure to construct an infinite sequence of excursions $(\varepsilon ^j)_{j=-1,-2,\dots }$ .

Put record $0$ at the origin and concatenate the excursions with one record between each pair of consecutive excursions. This yields a walk denoted as $\xi ^*$ , shown in Figure 2.6. Note that in $\xi ^{*}$ , all excursions are finite and therefore $r(\xi, j)$ is finite for all $j\in \mathbb {Z}$ . Namely, $\xi ^{*}\in \skew5\widehat {\mathcal {W}}_{*}$ .

Call $M^{-1}:\zeta \mapsto \xi ^*$ the resulting transformation. We claim that $M^{-1}$ is the inverse map of M; that is, $M^{-1} M\xi = \xi $ for $\xi \in \skew5\widehat {\mathcal {W}}_{*}$ and $MM^{-1}\zeta =\zeta $ for $\zeta \in \mathcal M$ . The second identity follows from the fact that $M_{k}\xi ^{*}=\zeta _{k}$ for each k. Now let $\varepsilon $ be an excursion of $ \xi $ . For $k\geqslant 0$ , denote by $\varepsilon _{[k]}$ the ball configuration obtained by removing the boxes belonging to an $\ell $ -soliton with $\ell \leqslant k$ . Then $\varepsilon _{[0]}=\varepsilon $ and $\varepsilon _{[k]}$ is the empty excursion for k sufficiently large. Now observe from the previous definitions that $M_m\varepsilon _{[k]}=M_m \varepsilon $ for all $m>k$ , because m-slots are only created by solitons of sizes larger than m. So the reconstruction algorithm correctly finds $\varepsilon _{[k]}$ from $M_{k}\varepsilon $ and $\varepsilon _{[k+1]}$ and hence it correctly finds $\varepsilon $ .

This shows that $M: \skew5\widehat {\mathcal {W}}_{*}\to \mathcal M$ is invertible, and so is its restriction to a subset $M: \skew5\widehat {\mathcal {W}}\to M(\skew5\widehat {\mathcal {W}})\subseteq \mathcal M$ , where $\skew5\widehat {\mathcal {W}}:=\{\xi [\eta ]\in \skew5\widehat {\mathcal {W}}_{*}: \eta \in \skew8\widehat{\mathcal {X}}\}$ . This proves Theorem 1.3.

Let us also point out that the image set $M(\skew5\widehat {\mathcal {W}})$ is not simple to characterise. However, as we will see below, if we sample $\zeta $ with an appropriate measure, the resulting (random) element $M^{-1}\zeta $ does belong a.s. to $\skew5\widehat {\mathcal {W}}$ .

4.1 Construction of the measures

Our recipe to produce invariant measures uses the construction described in Subsection 2.2, and it gives a distribution $\widehat {\mu }$ of configurations ‘seen from a typical record’. The proof of Theorem 1.5 is based on properties of $\widehat {\mu }$ and how they relate to the dynamics. This relationship is given by the Palm theory, which we briefly recall now.

Let $\eta \in \{0,1\}^{\mathbb {Z}}$ . From (1.1) and (1.8), we have $\theta R \eta = R \theta \eta $ . Suppose that $\eta \in \mathcal {X}$ , and denote

$$ \begin{align*} \mathtt r(\eta) := \inf\{x\geqslant 1: x \in R\eta\}. \end{align*} $$

If $ \eta $ is random with law $ \widehat {\mu } $ such that $\widehat {\mu }(\mathtt r):=\int \mathtt r(\eta )\,\widehat {\mu }(\mathrm {d}\eta ) <\infty $ , its inverse-Palm measure $\mu =\operatorname {\mathrm {Palm}}_{R}^{\mathbb {Z}}(\widehat {\mu })$ is defined as follows. For every test function $\varphi $ ,

(4.1) $$ \begin{align} \int \varphi(\eta) \,\mu(\mathrm{d}\eta) := \int \frac{\textstyle\sum_{i=0}^{\mathtt r(\eta)-1} \varphi(\theta^i\eta)} {\mathtt r(\eta)} \cdot \frac{\mathtt r(\eta)}{\widehat{\mu}(\mathtt r)} \widehat{\mu}(\mathrm{d}\eta). \end{align} $$

Informally, to sample a configuration distributed as $\mu $ one can first sample a configuration using the distribution $\widehat {\mu }$ biased by the length of the first excursion (when $ 0\in R\eta $ ) and then choose a site uniformly from this excursion (along with the record preceding it) to place the origin.

For $ \eta \in \mathcal {X} $ , we define the record-shift operator $\widehat {\theta }$ as

$$ \begin{align*} \widehat{\theta} \eta := \theta^{\mathtt r(\eta)}\eta. \end{align*} $$

We also define the dynamics seen from a record $\widehat {T}:\skew8\widehat{\mathcal {X}}\to \skew8\widehat{\mathcal {X}}$ by

(4.2) $$ \begin{align} \widehat{T} \eta := \theta^{r(T \xi^{\circ}[\eta],0)}T\eta, \end{align} $$

where $\xi ^{\circ }[\eta ]$ was defined in Subsection 2.2 as the unique lift of $\eta $ with record $0$ at $0$ . In words, $ \widehat {T} $ means apply $ T $ and recenter the configuration at the new position of record 0. The following lemma follows from standard properties of Palm measures and is proved in Section 5.

Lemma 4.3. Let $\widehat {\mu }$ be a probability measure of $\{0, 1\}^{\mathbb {Z}}$ . Suppose that $\widehat {\mu }(0\in R\eta )=1$ , $\widehat {\mu }(\mathtt r)< \infty $ , $\widehat {\mu }$ is $\widehat {\theta }$ -invariant and $\widehat {\mu }(R\eta =\mathbb {Z})=0$ . Then $\widehat {\mu }$ is supported on $\skew8\widehat{\mathcal {X}}$ , and $\operatorname {\mathrm {Palm}}_{R}^{\mathbb {Z}}(\widehat {\mu })$ is $\theta $ -invariant, supported on $\mathcal {X}$ and satisfies

$$ \begin{align*} \operatorname{\mathrm{Palm}}_{R}^{\mathbb{Z}}(\widehat{\mu}) \circ T^{-1} = \operatorname{\mathrm{Palm}}_{R}^{\mathbb{Z}}(\widehat{\mu} \circ \widehat{T}^{-1}). \end{align*} $$

If, moreover, $\widehat {\mu }$ is $\widehat {\theta }$ -ergodic, then $\operatorname {\mathrm {Palm}}_{R}^{\mathbb {Z}}(\widehat {\mu })$ is $\theta $ -ergodic.

We now introduce the assumptions and notation used throughout the rest of this section.

Let $\zeta =(\zeta _k)_{k\geqslant 1}$ be a sequence of independent random elements of $(\mathbb {Z}_{+})^{\mathbb {Z}}$ with shift-invariant distributions. Let $ P $ and $ E $ denote the underlying probability measure and expectation and suppose that the law of $ \zeta $ satisfies $\sum _k k E[\zeta _k(0)] <\infty $ and $P(\sum _{k,i}\zeta _{k}(i)>0)=1$ . In particular, this implies that the random field $\zeta $ a.s. satisfies (2.3) by Borel–Cantelli. Take $\xi =M^{-1} \zeta $ as the walk reconstructed from $\zeta $ according to the algorithm described in Subsection 2.2 and depicted in Figure 2.6. Let $\widehat {\mu }$ denote the resulting law of $\eta =\eta [\xi ]$ .

Before giving the proof, let us see how it implies Theorem 1.5.

Proof of Theorem 1.5. Let $\widehat {\mu }$ be the law of $\eta = M^{-1}\zeta $ , and define

$$ \begin{align*} \mu:=\operatorname{\mathrm{Palm}}_{R}^{\mathbb{Z}}(\widehat{\mu}). \end{align*} $$

By Proposition 4.4 and Lemma 4.3, $\mu $ is $\theta $ -invariant and supported on $\mathcal {X}$ . Also,

$$ \begin{align*}\mu\circ T^{-1} = \operatorname{\mathrm{Palm}}_{R}^{\mathbb{Z}}(\widehat{\mu}\circ\widehat{T}^{-1})=\operatorname{\mathrm{Palm}}_{R}^{\mathbb{Z}}(\widehat{\mu})=\mu,\end{align*} $$

so $\mu $ is also T-invariant. Moreover, under the i.i.d. assumption, the second part of Proposition 4.4 combined with the second part of Lemma 4.3 implies that $\mu $ is $\theta $ -ergodic.

Example 4.6 Ergodicity without independent components

It is possible for law $\mu $ of $\eta $ to be $\theta $ -ergodic and T-invariant while its components $M_k \eta $ are not independent under $\widehat {\mu }$ . Let $\zeta '$ be the configuration . Let $\zeta _1=\zeta _4$ be a configuration chosen uniformly at random in the set $\{\zeta ', \theta \zeta ', \theta ^2\zeta '\}$ ; let $\zeta _k\equiv 0$ for all $k\notin \{1,4\}$ and $\zeta =(\zeta _k)_{k\geqslant 1}$ . The reader can check that this example satisfies the stated properties. The resulting configuration is periodic for $ \theta $ and invariant for $ T $ :

$$ \begin{align*} \cdots 00{\color{blue} 10}{\color{red} 1111}{{\color{blue} 01}}{\color{red} 0000}{{\color{blue} 10}}000{{\color{blue} 10}}{\color{red} 1111}{{\color{blue} 01}}{\color{red} 0000}{{\color{blue} 10}}0 \cdots, \end{align*} $$

where colors (online) represent records, $ 4 $ -solitons and $ 1 $ -solitons.

Example 4.7 Independent components without ergodicity

It is also possible for $\zeta $ to be independent over k, $\theta $ -ergodic for each k but produce (by the above procedure) a configuration $\eta $ whose law is not $\theta $ -ergodic. To see that, take $\zeta _5(x)\equiv 1$ , $\zeta _1$ as in the previous example, $\zeta _k\equiv 0$ for all $k\notin \{1,5\}$ and $\zeta =(\zeta _k)_{k\geqslant 1}$ . The resulting configurations give three classes periodic for $ \theta $ and cyclic over $ T $ :

$$ \begin{align*} \cdots 0{{\color{blue} 10}}{\color{red} 1111}{{\color{blue} 01}}{\color{red} 1000}{{\color{blue} 10}}{\color{red} 00} 0{{\color{blue} 10}}{\color{red} 1111}{{\color{blue} 01}}{\color{red} 1000}{{\color{blue} 10}}{\color{red} 00} \cdots \\ \cdots 0{\color{red} 111}{{\color{blue} 01}}{\color{red} 1100}{{\color{blue} 10}}{\color{red} 000}{{\color{blue} 10}} 0{\color{red} 111}{{\color{blue} 01}}{\color{red} 1100}{{\color{blue} 10}}{\color{red} 000}{{\color{blue} 10}} \cdots \\ \cdots 0{\color{red} 11}{{\color{blue} 01}}{\color{red} 111}{{\color{blue} 01}}{\color{red} 0000}{{\color{blue} 10}}{\color{red} 0} 0{\color{red} 11}{{\color{blue} 01}}{\color{red} 111}{{\color{blue} 01}}{\color{red} 0000}{{\color{blue} 10}}{\color{red} 0} \cdots \end{align*} $$

where colors (online) represent records, $ 5 $ -solitons and $ 1 $ -solitons.

4.2 Invariance of the reconstructed configuration

We now prove the main part of Proposition 4.4, namely, $\widehat {\theta }$ -invariance and $\widehat {T}$ -invariance of $\widehat {\mu }$ , as well as $\theta $ -ergodicity in case of i.i.d. components. The proof of $\widehat {\mu }(\mathtt r)<\infty $ is given in Subsection 4.3. The condition $\widehat {\mu }(R\eta =\mathbb {Z})=0$ is easily checked from the construction of $M^{-1}\zeta $ and the assumption $\sum _{k,i}\zeta _{k}(i)>0$ a.s. Denote by $\mathcal {F}_k$ the sigma-field generated by $(\zeta _m)_{m>k}$ .

Because one can write $\eta = M^{-1}M\eta $ and $\widehat {T} \eta = M^{-1}M\widehat {T}\eta $ by Theorem 1.3, it suffices to show that the slot decompositions $M\eta $ and $M\widehat {T}\eta $ have the same law. More precisely, it suffices to show

(4.9) $$ \begin{align} E\left(\prod_{k=1}^n \varphi_k(M_k\widehat{T}\eta)\right) = \prod_{k=1}^n E \varphi_k(\zeta_k) \end{align} $$

for test functions $\varphi _1,\dots,\varphi _n$ and $n \in \mathbb {N}$ .

We proceed by an induction on n. First, recall the notation from (4.2) and note that

$$ \begin{align*} &E\big(\varphi_k(M_k\widehat{T}\eta)\,\big|\, \mathcal{F}_k \big)\\ &\qquad= E\big(\varphi_k(\theta^{-o^1_k(\xi^{\circ}[\eta])-k}M_k\eta)\,\big|\, \mathcal{F}_k \big) \qquad (\text{by Theorem~3.1}) \\ &\qquad= E\big(\varphi_k(\theta^{-o^1_k(\xi^{\circ}[\eta])-k}\zeta_k)\,\big|\, \mathcal{F}_k \big) \qquad (\text{because }M_k\eta=\zeta_k) \\ &\qquad= E\varphi_k(\zeta_k), \end{align*} $$

because $\zeta _k$ is shift-invariant and independent of $(\zeta _m)_{m>k}$ , whereas $o^1_k(\xi ^{\circ }[\eta ])$ is determined by these elements. The inductive step to show (4.9) is then

$$ \begin{align*} E\left(\prod_{i=k}^n \varphi_i(M_i\widehat{T}\eta)\right) &= E\left(E\left(\prod_{i=k}^n \varphi_i(M_i\widehat{T}\eta)\bigg| \mathcal{F}_k \right)\right),\\ \phantom{E\Bigl(\prod_{i=k}^n \varphi_i(M_i\widehat{T}\eta)\Bigr)} &= E\left( \prod_{i=k+1}^n \varphi_i(M_i\widehat{T}\eta) E\left(\varphi_k(M_k\widehat{T}\eta)\Big| \mathcal{F}_k \right)\right),\\ \phantom{E\left(\prod_{i=k}^n \varphi_i(M_i\widehat{T}\eta)\right)} &= E \varphi_k(\zeta_k) \ E\left(\prod_{i=k+1}^{n} \varphi_i(M_i\widehat{T}\eta)\right); \end{align*} $$

in the second identity we have used that $M_i\widehat {T}\eta $ is determined by $(\zeta _m)_{m\geqslant i}$ by the last statement of Theorem 3.1. This shows that $\widehat {\mu }$ is $\widehat {T}$ -invariant.

We now prove $ \widehat {\theta } $ -invariance. Consider the transformation $M^{-1} : \zeta \mapsto \xi ^*$ defined in Subsection 2.2 and denote $\eta ^{*}=\eta [\xi ^{*}]$ the corresponding ball configuration. Call $\varepsilon ^0_*$ the excursion of $\eta ^*$ between records 0 and 1. The construction of Subsection 2.2 gives

$$ \begin{align*} \widehat{\theta} \eta^* = \theta^{r(\xi^{\circ}[\eta^*],1)}\eta^* = M^{-1} \big(\theta^{n_k(\varepsilon^0_*)}\zeta_k: k\geqslant 1\big). \end{align*} $$

So it suffices to show that $( \theta ^{n_k(\varepsilon ^0_*)}\zeta _k )_{k\geqslant 1}$ has the same law as $(\zeta _k)_{k \geqslant 1}$ . But $n_k(\varepsilon ^0_*)$ is determined by $(\zeta _m)_{m>k}$ and thus is independent of $\zeta _k$ . Hence, the law of $\zeta _k$ is invariant by the random shift of $n_k(\varepsilon ^0_*)$ and it is independent of $(\zeta _m)_{m>k}$ . This shows that $ \eta ^* $ and $ \widehat {\theta } \eta ^* $ have the same law.

Finally, under the extra assumption that $(\zeta _k(i))_{i\in \mathbb {Z}}$ is i.i.d. for each k, the reconstruction map mentioned above will produce an i.i.d. sequence of excursions separated by records. This in turn implies that the resulting configuration $\eta $ is $\widehat {\theta }$ -ergodic.

4.3 Expected excursion length

We continue with the proof of Proposition 4.4 to prove that $\widehat {\mu }(\mathtt r)<\infty $ . Let

(4.10) $$ \begin{align} \alpha_k := E[\zeta_k(0)]. \end{align} $$

The proof consists of two steps: first, we show that the following system

(4.11) $$ \begin{align} w_k = 1 + {\sum_{m>k}} 2(m-k) w_m\alpha_m,\qquad k=0,1,2,\dots \end{align} $$

has a unique finite nonnegative solution $w=(w_k)_{k\geqslant 0}$ ; that is, $ w \in [0,\infty )^{\mathbb {N}_0} $ ; second, we show that the average number of k-slots per excursion in $M^{-1} \zeta $ is $w_k$ , whence the average number of k-solitons per excursion satisfies

(4.12) $$ \begin{align} \rho_k=\alpha_k w_k. \end{align} $$

In particular, this will imply that the average size of the excursions (along with the record preceding them) satisfies

(4.13) $$ \begin{align} \widehat{\mu}(\mathtt r) = w_0 = 1 + \sum_{m\geqslant1} 2m\, \rho_m <\infty. \end{align} $$

So we start by studying (4.11). Let c _k := 2 $\sum$ _m $>$ _k (m $-$ k) $\alpha$ _m . Because $\sum _{k} k E[\zeta _{k}(0)]<\infty $ , we can take $\tilde {k}$ such that $\sum$ _m $>$ _k~4m $\alpha$ _m $<$ 1, so $c_k< \frac {1}{2}$ for $k\geqslant \tilde k$ . Let $K:= \{k\in \mathbb {N}:k\geqslant \tilde k\}\cup \{\aleph \}$ and consider a Markov chain $(X_n)_{n\geqslant 0}$ on K with absorbing state $\aleph $ and transition probabilities

; $q_{k,\aleph } = 1-c_k$ ; $q_{\aleph,\aleph }=1$ and $q_{k,m}=0$ otherwise. Define the absorption time by $\tau := \inf \{n\geqslant 0: X_n=\aleph \}$ . Denote by $P_k$ the law of $(X_n)_{n\geqslant 0}$ starting from k and by $ E_k $ the expectation. Applying the Markov property at time $1$ , we see that $w_k := E_k\tau $ satisfies

That is, $(w_{k})_{k\geqslant \tilde k}$ verifies the system (4.11) with $k= \tilde k, \tilde k+1, \dots $ . Because $c_k\geqslant c_{k+1}$ , we have $P_k(\tau>n) \leqslant c_k^{n}$ and thus w _k = E _k $\tau\leq 11-$ c _k $<2<\infty$ for $k \geq \tilde{k}$ . Because $w_{\tilde {k}}<\infty $ , we can use (4.11) with $k=\tilde {k}-1$ to define $w_{\tilde {k}-1}<\infty $ , and iterating this argument we get $w_k < \infty $ for all k. This proves the existence of a finite solution to (4.11).

For uniqueness, suppose that $(\widetilde w_{k})_{k\geqslant 0}$ is a finite nonnegative solution to (4.11). In particular, $\sum _{m>k}\widetilde w_{m}q_{k,m}=\sum _{m>k}2(m-k)\widetilde w_{m}\alpha _{m}<\infty $ , $k\geqslant 0$ . Moreover, the previous choice of $\tilde k$ ensures that $\sum _{m>k}q_{k,m}<\infty $ for all $k\geqslant \tilde k$ . It follows that for all $k\geqslant \tilde k$ ,

$$ \begin{align*} |w_{k}-\widetilde w_{k}| \leqslant \sum_{m>k} q_{k,m} |w_{m}-\widetilde w_{m} \big| \leqslant c_{k}\sup_{m>k} |w_{m}-\widetilde w_{m}| \leqslant \tfrac{1}{2} \sup_{m>k} |w_{m}-\widetilde w_{m}|. \end{align*} $$

Taking the supremum over $k \geqslant \tilde {k}$ , we get

$$ \begin{align*} \sup_{k\geqslant \tilde k}|w_{k}-\widetilde w_{k}| \leqslant \frac12 \sup_{k \geqslant \tilde{k}}|w_{k}-\widetilde w_{k}|. \end{align*} $$

By (4.11), $\widetilde w_{k}$ and $ w_k $ are decreasing in $ k $ , so the above supremum is finite; hence, it is zero. That is, $\widetilde w_{k}=E_{k}\tau $ , $k\geqslant \tilde k$ . Given $(\widetilde w_{k})_{k\geqslant \tilde k}$ , (4.11) determines in a unique way the values of $(\widetilde w_{k})_{k\leqslant \tilde k}$ ; this completes the proof of uniqueness.

We now consider truncated approximations for the reconstruction algorithm of Subsection 2.2. Let

$$ \begin{align*} \zeta^{[n]}_k(i) := \begin{cases} \zeta_k(i), & k \leqslant n, \\ 0, & k>n. \end{cases} \end{align*} $$

Let $\varepsilon ^{[n]}$ denote the first excursion (i.e., the one between records 0 and 1) of $M^{-1} \zeta ^{[n]}$ . Let $W^n_k\zeta = n_k(\varepsilon ^{[n]})$ be the number of k-slots in $\varepsilon ^{[n]}$ ; see (2.4). Then $W^n_k\zeta \nearrow W_k\zeta $ a.s., where $W_k \zeta := W_k^{\infty } \zeta $ . Letting $w_k^n:=E[W^n_k\zeta ]$ and $w_k:=E[W_k\zeta ]$ , by monotone convergence we have $w^n_k\nearrow w_k$ as $n\to \infty $ . On the other hand, because each m-soliton contains $2(m-k) \ k$ -slots,

$$ \begin{align*} W_k^n \zeta = 1 + \sum_{m>k} 2(m-k) \times (\text{number of } m\text{-solitons in } \varepsilon^{[n]}) \end{align*} $$

and thus

$$ \begin{align*} w_k^n=1+\sum_{m>k} 2(m-k) E(\text{number of } m\text{-solitons in } \varepsilon^{[n]}). \end{align*} $$

Let denote the expected number of m-solitons per m-slot in $\zeta ^{[n]}$ . Because $W^n_k\zeta $ is a function of $(\zeta _m:m>k)$ that is independent of $\zeta _k$ , the expected number of m-solitons in $\varepsilon ^{[n]}$ is $w_m^n \times \alpha _m^n$ . Therefore, $(w_k^n)_{k \geqslant 0}$ and $(\alpha _k^n)_{k \geqslant 1}$ satisfy relation (4.12) and the system (4.11). Finally, because $w^n_k < 2$ for all $k \geqslant \tilde {k}$ and $n\in \mathbb {N}$ , $w_k$ is finite for every k and therefore (4.13) is satisfied, concluding the proof that $\widehat {\mu }(\mathtt r)<\infty $ .

6.1 Existence of speeds via Palm measure and ergodicity

Here we show that $\lim _t \frac {1}{t}x(\gamma ^t)$ exists $ \mu $ -a.s. and we give an explicit formula for it. Recall the definition of $x(\gamma )$ given before the statement of Theorem 1.1. Let $\Gamma _k^{\circ }\eta :=\{x(\gamma ): \gamma \in \Gamma _k\eta \}$ denote the set of leftmost sites of k-solitons of $\eta $ . If $\rho _{k}=0$ , then $\Gamma _{k}$ is empty and (1.9) holds for all $ \gamma \in \Gamma _k(\eta ) $ by vacuity, for any value of $ v_k $ .

From now on, we assume $\rho _{k}>0$ , which implies $\mu (0\in \Gamma _k^{\circ }\eta )>0$ . Because $\Gamma _{k}^{\circ }\theta \eta =\theta \Gamma _{k}^{\circ }\eta $ , the Palm construction described in Section 5 applies.

For $\gamma \in \Gamma _k\eta $ and $z=x(\gamma )$ , we define $\Delta _\eta ^k(z) := x(\gamma ^1)-x(\gamma )$ , the size of the jump of k-soliton $\gamma $ after one iteration of T. For $z\not \in \Gamma _k^{\circ }\eta $ we set $\Delta _\eta ^k(z)=0$ . With this notation, the displacement of a tagged k-soliton after $t+1$ iterations of T can be decomposed as

$$ \begin{align*} x(\gamma^{t+1})-x(\gamma) = \Delta^k_\eta(x(\gamma)) + \Delta^k_{T\eta}(x(\gamma^1)) + \cdots + \Delta^k_{T^t\eta}(x(\gamma^t)). \end{align*} $$

We want to divide both sides by t and use the ergodic theorem. This will require a couple of subtle observations. The first step is to consider the system as seen from a typical k-soliton.

Let $\skew8\widehat{\mathcal {X}}^k$ be the set of configurations in $\mathcal {X}$ such that $\Gamma _k^{\circ }\eta $ is doubly infinite and contains $ 0 $ . Let $\widehat {\mu }_k:=\operatorname {\mathrm {Palm}}_{\mathbb {Z}}^{\Gamma _k^{\circ }}(\mu )$ be the Palm measure of $\mu $ with respect to $\Gamma _k^{\circ }\eta \subseteq \mathbb {Z}$ ; that is, $\widehat {\mu }_k=\mu (\,\cdot \,|\,0\in \Gamma _k^{\circ }\eta )$ . For $\eta \in \skew8\widehat{\mathcal {X}}^k$ , let $\mathtt r^k(\eta ) := \inf \{x\geqslant 1: x\in \Gamma _k^{\circ }\eta \}$ and define $\widehat {\theta }_k : \skew8\widehat{\mathcal {X}}^k \to \skew8\widehat{\mathcal {X}}^k$ as the ‘shift to the next k-soliton’ given by $\widehat {\theta }_k \eta := \theta ^{\mathtt r^k(\eta )}\eta $ . Also, for a k-soliton $\gamma $ such that $x(\gamma )=0$ , let $\widehat {T}_k\eta := \theta ^{x(\gamma ^1)} T\eta $ denote the dynamics as seen from a tagged k-soliton.

From Lemma 5.8, $ \widehat {\mu }_k $ is $\widehat {T}_k$ -invariant. Now for $\eta \in \skew8\widehat{\mathcal {X}}^k$ and $\gamma $ with $x(\gamma )=0$ , the above decomposition becomes

$$ \begin{align*} x(\gamma^{t+1}) = \Delta^k_{\eta}(0) + \Delta^k_{\widehat{T}_k\eta}(0) + \Delta^k_{\widehat{T}_k^2\eta}(0) + \cdots + \Delta^k_{\widehat{T}_k^t\eta}(0). \end{align*} $$

By the ergodic theorem for $(\widehat {T}_k^t\eta )_{t \in \mathbb {Z}}$ , $\lim \limits _{t\to \infty } \frac {1}{t}x(\gamma ^t)$ exists $\widehat {\mu }_k$ -a.s., and on average it equals

(6.1) $$ \begin{align} v_k := \int \Delta_\eta^k(0) \, \widehat{\mu}_k(\mathrm{d}\eta) \end{align} $$

(to avoid integrability issues, note that by Lemma 1.6 we have $ \Delta _\eta ^k(0) \geqslant 0 $ ).

It remains to show that this limit is in fact nonrandom.

Consider the field

$$ \begin{align*} \tilde{v}_k(\eta,z) := \begin{cases} \lim\limits_{t\to\infty} \dfrac{x(\gamma^t)}{t}, & \text{if } z = x(\gamma) \text{ for some } \gamma\in\Gamma_k(\eta), \\ 0,&\text{otherwise.} \end{cases} \end{align*} $$

By Proposition 1.3 and Lemma 1.5, if $\gamma, \tilde \gamma $ are two k-solitons with $x(\gamma )\leqslant x(\tilde \gamma )$ , then $x(\gamma ^{t})\leqslant x(\tilde \gamma ^{t})$ for all t. Hence, we have

(6.2) $$ \begin{align} \tilde{v}_k(\eta,x) \leqslant \tilde{v}_k(\eta,y) \text{ for all } x\leqslant y \text{ in } \Gamma_k^{\circ}\eta. \end{align} $$

On the other hand, by Lemma 5.6 the measure $ \widehat {\mu }_k $ is $\widehat {\theta }_k$ -ergodic. So, under $ \widehat {\mu }_k $ the sequence $ \big ( \tilde {v}_k(\widehat {\theta }^j_k \eta,0) \big )_{j \in \mathbb {Z}} \in [0,+\infty ]^{\mathbb {Z}} $ is a.s. nondecreasing by (6.2) and its law is $ \theta $ -ergodic, which implies that it is $ \widehat {\mu }_k $ -a.s. equal to some constant $ v_k \geqslant 0 $ . So $ \tilde {v}_k (\eta,x) = v_k $ for all $ x \in \Gamma ^{\circ }_k(\eta ) $ , for $ \widehat {\mu }_k $ -a.e. $ \eta $ . By Lemma 5.5, it also holds for $ \mu $ -a.e. $ \eta $ .

We conclude with a short proof that $ v_k<\infty $ . From (5.7) with $\Gamma _k^{\circ }$ in the role of $ Z $ , we get

(6.3)

Denote by $h(\gamma )$ the leftmost site of the tail of $\gamma $ and by $t(\gamma )$ the leftmost site of its head. Then Proposition 1.3 implies that $\Delta _{\eta }^{k}(x)\leqslant |h(\gamma )-t(\gamma )|$ . Combined with the fact that intervals spanned by different k-solitons do not overlap (Lemma 1.5), this yields $\sum _{y=1}^{\mathtt r(\eta )} \Delta _{\eta }^k(y) \leqslant \mathtt r(\eta )$ , from which we get $v_k \leqslant \frac {\widehat {\mu }(\mathtt r)}{\rho _k} = \frac {w_0}{\rho _k} < \infty $ by (4.13).

6.2 Equation for speeds from soliton interactions

We now prove that the soliton speeds satisfy (1.10). It is enough to consider $ k $ such that $ \rho _k>0 $ and otherwise take (1.10) as the definition of $ v_k $ . We start by proving the following identity on the displacement of solitons.

Proposition 6.4. We have

(6.5) $$ \begin{align} x(\gamma^{t})-x(\gamma) = k\, t - 2k \sum_{m>k} \tfrac12 M_t^{\gamma}(m) + \sum_{m<k} 2m \, N_t^{\gamma}(m), \end{align} $$

where

$$ \begin{align*} M_{t}^{\gamma}(m)&=\#\{(\tilde\gamma, s): \tilde\gamma\in \Gamma_{m}\eta, 0\leqslant s < t, I(\gamma^{s})\subseteq I(\tilde\gamma^{s}) \text{ and } x(\tilde\gamma^{s+1}) \ne x (\tilde\gamma^{s}) \}, \\ N_{t}^{\gamma}(m)&=\#\{(\tilde\gamma, s): \tilde\gamma\in \Gamma_{m}\eta, 0\leqslant s<t, I(\tilde\gamma^{s})\subseteq I(\gamma^{s}) \text{ and } I(\tilde\gamma^{s+1})\cap I(\gamma^{s+1})=\varnothing\}. \end{align*} $$

Proof. Before proving it, let us first explain the intuition underpinning (6.5). From Lemma 1.6, we know that as long as there exists some $\tilde \gamma $ satisfying $I(\gamma )\subseteq I(\tilde \gamma )$ , $\gamma $ is prevented from moving. We will see that it takes two time steps for $\tilde \gamma $ to overtake $\gamma $ : the first step occurs when $\gamma $ is nested in the right half of $\tilde {\gamma }$ , after which it will be nested in the left half of $\tilde {\gamma }$ ; the second step occurs when $\gamma $ is nested in the left half of $\tilde {\gamma }$ , after which it will no longer be nested inside $\tilde {\gamma }$ . Because $\tilde \gamma $ can be inside an even larger soliton, during which it stays frozen, these two times are not necessarily consecutive. On the other hand, when $\gamma $ overtakes a smaller soliton of size $\ell $ say, it causes $\gamma $ to move $2\ell $ units more than it normally would have. Put formally, we have the following identity:

(6.6)

The proof of this identity is elementary and is postponed to Section 7. Proceeding with the proof of (6.5), we note that if $\tilde \gamma $ exists in (6.6), then it has to be unique: solitons $\tilde \gamma $ satisfying $I(\tilde \gamma )\supseteq I(\gamma )$ are ordered by their sizes and only the largest one will move forward. Iterating the above t times then yields (6.5).

Now take $\tilde \gamma $ to be the leftmost m-soliton in $[1, \infty )$ and $\gamma $ the rightmost k-soliton in $(-\infty, 0]$ . We have shown previously that $v_{m}=\lim _{t\to \infty } \frac {1}{t}x(\tilde \gamma ^{t})$ and $v_{k}=\lim _{t\to \infty } \frac {1}{t}x(\gamma ^{t})$ , $\mu $ -a.s. Combined with Lemma 1.7, this yields

$$ \begin{align*} v_{k}\leqslant v_{m} \quad \text{if} \quad k<m \text{ and } \rho_{k}, \rho_{m}>0. \end{align*} $$

From now on, we will work under $\widehat {\mu }_{k}$ and always take $\gamma $ to be the soliton with $x(\gamma )=0$ . To lighten the notation, we will drop the superscript ${\gamma }$ from $ M_t $ and $ N_t $ . Recall that $\bar {\rho }_m = \frac {\rho _m}{w_0}$ is the mean number of m-solitons per unit space.

Let us assume for the moment that $\bar {\rho }_m=0$ for all large m.

By (1.9) and (6.5), to get (1.10) it suffices to show the following. For $m>k$ and $\ell <k$ :