Fluctuation Bounds in the Exponential Bricklayers Process

Abstract

This paper is the continuation of our earlier paper (Balázs et al. in Ann. Inst. Henri Poincaré Probab. Stat. 48(1):151–187, 2012), where we proved t ^1/3-order of current fluctuations across the characteristics in a class of one dimensional interacting systems with one conserved quantity. We also claimed two models with concave hydrodynamic flux which satisfied the assumptions which made our proof work. In the present note we show that the totally asymmetric exponential bricklayers process also satisfies these assumptions. Hence this is the first example with convex hydrodynamics of a model with t ^1/3-order current fluctuations across the characteristics. As such, it further supports the idea of universality regarding this scaling.

Appendix: Covariance Identities for Bricklayer Process with Exponential Rates

The purpose of this appendix is to prove the variance formula for stationary BLP under the following exponential bound assumption on rates: for some C,β<∞,

$$f(\omega _0)\le C\mathrm{e}^{\beta |\omega _0|}.$$

Assume the height process is normalized initially by h ₀(0)=0.

Theorem 3

Fix z∈ℤ. In the stationary infinite volume process with time-marginal distribution $\underline {\omega }(t)\sim \underline {\mu}^{\theta }$,

$$\mathbf {Var}\bigl[h_z(t)\bigr]=\sum_{n\in \mathbb {Z}}|n-z| \mathbf {Cov}\bigl[\omega _n(t),\omega _0(0)\bigr]. $$

(34)

Formula (34) was already proved in [8] for a general class of processes that includes ZRP and BLP. However, the proofs in [8] were carried out under the assumption that certain semigroup and generator calculations work. This presents no problem when the single-site state space is compact (such as exclusion processes) for then one has a strongly continuous semigroup on the Banach space of continuous functions on the state space of the process. For BLP with superlinear rates, only some rudimentary features of the usual semigroup picture have been established in [6]. Hence the need to justify (34). We use the finite-volume auxiliary processes as introduced in [6].

To prove Theorem 3 we show that the infinite-volume stationary process is a limit of finite-volume $(\ell ,\mathfrak{r},\theta )$ processes as $-\ell,\mathfrak{r}\to\infty$. A preliminary form of (34) is true for an $(\ell ,\mathfrak{r},\theta )$ process by simple counting. (See (42) below and its expanded form on lines (44)–(47).) The technical work goes into establishing moment bounds that are uniform over $\ell<0<\mathfrak{r}$. These in turn permit us to take the $-\ell,\mathfrak{r}\to\infty$ limit in the proto-formula (44)–(47).

This appendix is based on the construction of the infinite-volume BLP $\underline {h}(t)$ given in [6]. Article [6] utilized two types of finite-volume processes: the $[\ell ,\mathfrak{r}]$ processes whose height variables were denoted by $\underline {h}^{[\ell ,\mathfrak{r}]}(t)$, and the $(\ell ,\mathfrak{r},\theta )$ processes with height variables denoted by $\underline {g}^{(\ell ,\mathfrak{r},\theta )}(t)$.

The $[\ell ,\mathfrak{r}]$ evolution is a straightforward restriction of the full system into the finite interval $[\ell ,\mathfrak{r}]$, with generator

$$L^{[\ell ,\mathfrak{r}]}\varphi (\underline {\omega })=\sum_{i=\ell}^{\mathfrak{r}-1}\bigl[f(\omega _i)+f(-\omega _{i+1}) \bigr]\cdot \bigl[\varphi \bigl(\underline {\omega }^{(i, i+1)}\bigr)-\varphi (\underline {\omega }) \bigr].$$

This generator defines a countable state space Markov process that evolves over the sites $\ell,\dots,\mathfrak{r}$: the jump $\underline {\omega }\to \underline {\omega }^{(i, i+1)}$ happens at rate f(ω _i )+f(−ω _i+1), independently at different sites i, but only for $\ell\leq i\leq \mathfrak{r}-1$. Columns outside the interval $[\ell, \mathfrak{r}]$ are frozen for all time. The virtue of this process is monotone dependence on the interval $[\ell ,\mathfrak{r}]$. In [6] the infinite-volume process $\underline {h}(\cdot)$ was defined as the a.s. increasing limit of the processes $\underline {h}^{[\ell ,\mathfrak{r}]}(t)$ as $-\ell,\mathfrak{r}\to\infty$.

The $(\ell ,\mathfrak{r},\theta )$ process has also the correct boundary currents that make the i.i.d. product measures $\underline {\mu}^{\theta }$ invariant for the finite collection of increment variables . The generator is

For a concrete construction of the processes, we imagine that bricklayer at site i has two unit rate Poisson processes $N_{i}^{(L)}$ and $N_{i}^{(R)}$ on the first quadrant $\mathbb {R}_{+}^{2}$ of the plane. These govern his brick-laying action to the left (L) and right (R). A Poisson point (t,y) in $N_{i}^{(L)}$ such that y≤f(−ω _i (t−)) signals a brick to be laid on [i−1,i] at time t, while a Poisson point (t,y) in $N_{i}^{(R)}$ such that y≤f(ω _i (t−)) signals a brick to be laid on [i,i+1] at time t. Shift of these planar Poisson processes by time t will be denoted by S _t N.

The construction of the finite systems in terms of these Poisson processes provides the usual jump chain-holding time construction of a continuous time Markov chain on a countable state space. After each jump, the holding time and the next state are read from the Poisson processes. By the strong Markov property of the Poisson processes this is equivalent to looking at freshly sampled exponential variables with appropriate rates. In both $[\ell ,\mathfrak{r}]$ processes and $(\ell ,\mathfrak{r},\theta )$ processes the increment H(t)−H(0) of the maximal height

$$H(t)=\max_{\ell-1\leq j\leq \mathfrak{r}}h_j(t) $$

(35)

is bounded by a Poisson process. Hence explosions do not happen [6, Sects. 3.1–3.2].

In the next section we show that the $(\ell ,\mathfrak{r},\theta )$ processes converge to the infinite volume stationary process. Then we develop moment bounds through martingales, uniformly in $\ell<0<\mathfrak{r}$. After Sect. A.1 we drop the superscripts $[\ell ,\mathfrak{r}]$ and $(\ell ,\mathfrak{r},\theta )$ to ease notation.

1.1 A.1 Convergence of $(\ell ,\mathfrak{r},\theta )$ Processes

The infinite-volume process $\underline {h}(\cdot)$ is defined as the a.s. increasing limit of the processes $\underline {h}^{[\ell ,\mathfrak{r}]}(\cdot)$ [6, Theorem 2.2]. Lemma 7.1 in [6] shows that if the initial state $\underline {\omega }(0)$ is $\underline {\mu}^{\theta}$-distributed then the increment process $\underline {\omega }(\cdot)$ is stationary.

Calculations in this appendix are done in a stationary finite-volume $(\ell ,\mathfrak{r},\theta )$ process. Hence we need to show that this process also converges as $-\ell,\mathfrak{r}\to\infty$ to the stationary infinite-volume process.

It will be convenient to represent the processes by measurable mappings of the initial configuration $\underline {h}$ and the Poisson clocks $\underline {N}$ on (0,t]:

$$\underline {g}^{(\ell ,\mathfrak{r},\theta )}(t)=\varPsi_t^{(\ell ,\mathfrak{r},\theta )}( \underline {h}, \underline {N}),\qquad \underline {h}^{[\ell ,\mathfrak{r}]}(t)=\varPhi_t^{[\ell ,\mathfrak{r}]}( \underline {h}, \underline {N}) \quad \mbox{and}\quad \underline {h}(t)=\varPhi_t( \underline {h}, \underline {N}).$$

Then the construction of the process $\underline {h}(t)$ in [6] can be expressed as follows: for any initial $\underline {h}\in \widetilde{\varOmega}$, any m,T<∞,

$$\varPhi_{t,i}( \underline {h}, \underline {N})=\varPhi_{t,i}^{[\ell ,\mathfrak{r}]}( \underline {h}, \underline {N})\quad\mbox{for }{-}m\le i\le m\mbox{ and }0\le t\le T $$

(36)

for all large enough $-\ell,\mathfrak{r}$, almost surely.

Lemma 10

Let $\underline {h}(t)$ be the infinite-volume process whose increment process $\underline {\omega }(t)$ is stationary with marginal $\underline {\omega }(t)\sim \underline {\mu}^{\theta}$. As $-\ell,\mathfrak{r}\to\infty$, $\underline {g}^{(\ell ,\mathfrak{r},\theta )}(\cdot)\to \underline {h}(\cdot)$ in the following sense: given any m,T<∞,

$$g^{(\ell ,\mathfrak{r},\theta )}_i(t)=h_i(t)\quad \mbox{\textit{for} }{-}m\le i\le m\mbox{ \textit{and} }0\le t\le T$$

for all large enough $-\ell,\mathfrak{r}$.

Proof

The proof of Lemma 7.1 on p. 1243 of [6] shows that there exists a (nonrandom) time t ₀=t ₀(θ)>0 such that, for any m, $g^{(\ell ,\mathfrak{r},\theta )}_{i}(t)=h^{[\ell ,\mathfrak{r}]}_{i}(t)$ for −m≤i≤m and 0≤t≤t ₀ if −ℓ and $\mathfrak{r}$ are large enough. Combined with (36) we have the statement on the time interval [0,t ₀].

To get to an arbitrary time T, the claim is proved by induction up to time kt ₀>T, for a positive integer k. Since process $\underline {h}(\cdot)$ stays in $\widetilde{\varOmega}$, (36) can be applied to restarted processes.

The other ingredient of the induction step is control of discrepancies between processes $g^{(\ell ,\mathfrak{r},\theta )}_{i}(\cdot)$ and $h^{[\ell ,\mathfrak{r}]}_{i}(\cdot)$. For this purpose let be the event that in the process either all columns in the range {⌊3ℓ/4⌋,…,⌊ℓ/2⌋+1} grew, or all columns in the range $\{\lfloor {\mathfrak{r}/2}\rfloor ,\ldots,\lceil {3\mathfrak{r}/4}\rceil \}$ grew. By [6, Corollary 5.5], , and this bound is independent of a, b.

Discrepancies originate at the edges ℓ and $\mathfrak{r}$. On the event the (a,b,θ) and $[\ell ,\mathfrak{r}]$ processes started from a common initial configuration are indistinguishable inside $(\ell/2,\mathfrak{r}/2)$ because discrepancies have not had a chance to penetrate into $(\ell/2,\mathfrak{r}/2)$. By the monotonicity properties of these processes the columns of the $(\ell ,\mathfrak{r},\theta )$ process never go below those of the $[\ell ,\mathfrak{r}]$ process [6, Lemma 3.2]. Hence it is enough to use the event to suppress growth in the $(\ell ,\mathfrak{r},\theta )$ process. We leave the details of this induction argument to the reader. □

1.2 A.2 Martingales

Since the rates are unbounded, we begin by stating a general lemma about countable Markov chains. Let S be a countable state space, Q a generator matrix, q _x =−q _x,x =∑ _y:y≠x q _x,y the total rate to jump away from state x∈S. Let P(t) be the semigroup associated to Q, in other words the minimal positive solution of the backward equation P′(t)=QP, P(0)=I [9]. The next lemma is proved with standard techniques.

Lemma 11

Let ν be an initial distribution on S and T<∞. Assume

$$\int_0^T E^\nu[q_{X_s}]\,\mathrm {d}s= \int_0^T \sum_x\nu(x)\sum_y p_{x,y}(s)q_y\,\mathrm {d}s<\infty. $$

(37)

Let φ be a bounded function on S. Then under P ^ν, for t∈[0,T], the process $\sigma(t)=\int_{0}^{t} Q\varphi(X_{s})\,\mathrm {d}s$ is well-defined and in L ¹(P ^ν), and the following process is a martingale:

$$M_t= \varphi(X_t)-\varphi(X_0)-\int_0^t Q\varphi(X_s)\,\mathrm {d}s. $$

(38)

Now apply this to the $[\ell ,\mathfrak{r}]$ and $(\ell ,\mathfrak{r},\theta )$ processes. To simplify notation, let

$$f_i\bigl(\underline {\omega }(s)\bigr)=f\bigl(\omega _i(s)\bigr)+f\bigl(-\omega _{i+1}(s)\bigr)$$

be the rate of growth of the column height h _i (s).

Lemma 12

Fix $\ell<0<\mathfrak{r}$. Consider either the $[\ell ,\mathfrak{r}]$ process or the $(\ell ,\mathfrak{r},\theta )$ process and in either case denote the height variables by h _i (t). Let ν be an initial distribution such that for some c>β

$$ E^\nu \bigl( \mathrm{e}^{c|h_0|+c\sum_{i=\ell}^{\mathfrak{r}}|\omega _i|} \bigr)<\infty. $$

(39)

Then for any 1≤p<∞ and any index i this process is a martingale:

$$M_t=h^p_i(t)-h^p_i(0)-\int_0^t f_i\bigl(\underline {\omega }(s)\bigr) \bigl( \bigl(h_i(s)+1\bigr)^p-h^p_i(s)\bigr)\,\mathrm {d}s. $$

(40)

Proof

In the $[\ell ,\mathfrak{r}]$ process the total jump rate out of state $\underline {h}$ is

$$q_{ \underline {h}} =\sum_{i=\ell}^{\mathfrak{r}-1}f_i(\underline {\omega }) \le2\sum_{i=\ell}^{\mathfrak{r}}\mathrm{e}^{\beta |\omega _i|} \le2(\mathfrak{r}-\ell+1)\mathrm{e}^{\beta\sum_{i=\ell}^{\mathfrak{r}}|\omega _i|}.$$

In the $(\ell ,\mathfrak{r},\theta )$ process the total jump rate out of state $\underline {g}$ is

$$q^\theta _{ \underline {g}}=q_{ \underline {g}}+ f(-\omega _\ell)+f(\omega _\mathfrak{r}) +\mathrm{e}^\theta +\mathrm{e}^{-\theta } \le2(\mathfrak{r}-\ell+1)\mathrm{e}^{\beta\sum_{i=\ell}^{\mathfrak{r}}|\omega _i|} +\mathrm{e}^\theta +\mathrm{e}^{-\theta }.$$

In the $[\ell ,\mathfrak{r}]$ process under a fixed initial configuration $\underline {h}$,

$$\mathbf {E}^{ \underline {h}} \Bigl[\sup_{s\in[0,T]}\mathrm{e}^{\beta\sum_{i=\ell}^{\mathfrak{r}}|\omega _i(s)|} \Bigr] \le \mathrm{e}^{\beta\sum_{i=\ell}^{\mathfrak{r}}|\omega _i|} \mathbf {E}\bigl[\mathrm{e}^{\beta Y(T)}\bigr]$$

where Y(⋅) is a Poisson process of rate $\lambda=2f(0)(\mathfrak{r}-\ell)$. This comes from the observation that the process

$$v(t)= \sum_{i=\ell}^{\mathfrak{r}}\bigl|\omega _i(t)\bigr| - \sum_{i=\ell}^{\mathfrak{r}}\bigl|\omega _i(0)\bigr|$$

increases only when a local maximum column grows, and this happens at rate at most $2f(0)(\mathfrak{r}-\ell)$. Then, under the initial distribution ν,

(41)

In the $(\ell ,\mathfrak{r},\theta )$ process the same idea works except the rate for the process that bounds v(t) is $\lambda^{\theta }=2f(0)(\mathfrak{r}-\ell+1) +\mathrm{e}^{\theta }+\mathrm{e}^{-\theta }$.

Let F(x)=(b∧x)∨(−b) be a truncation function. Now that assumption (37) has been verified, for any integer 0<b<∞ and $\ell\le i\le \mathfrak{r}-1$ (38) gives the martingale

$$M^{(b)}_t=F\bigl(h_i(t)\bigr)^p-F\bigl(h_i(0)\bigr)^p -\int _0^t f_i\bigl(\underline {\omega }(s)\bigr)\bigl( F\bigl(h_i(s)+1\bigr)^p-F\bigl(h_i(s)\bigr)^p \bigr) \,\mathrm {d}s.$$

So for an event and s<t

As b↗∞ dominated convergence takes each term to the desired limit. This is justified by the following. Restrict to an interval s,t∈[0,T]. The rate in the last expectation is bounded as in

$$f_i\bigl(\underline {\omega }(u)\bigr)\le\sup_{s\in[0,T]}\mathrm{e}^{\beta(|\omega _{i}(s)|+|\omega _{i+1}(s)|) }$$

and the random variable on the right has a finite L ^p-norm for some p>1 by a bound of the type in (41) and the assumption that c>β in (39). For the height we have

$$\bigl|h_i(t)\bigr|=h_i^+(t)+h_i^-(t)\le H(t)+h_i^-(0) \le \bigl(H(t)-H(0)\bigr) +H(0)+\bigl|h_i(0)\bigr|$$

where H(t) is the maximal height of (35). The increment H(t)−H(0) is stochastically bounded by a Poisson process while H(0) and |h _i (0)| have all moments by assumption (39). We conclude that (40) is a martingale. □

1.3 A.3 Bounds for the $(\ell ,\mathfrak{r},\theta )$ Process

Henceforth consider stationary $(\ell ,\mathfrak{r},\theta )$ processes with μ ^θ marginals and initial height normalized by h ₀(0)=0. The increment process is denoted by $\underline {\omega }(t)$ and the height process by $\underline {h}(t)$. In this process the column heights h _i (t) for i≤ℓ−2 and $i\ge \mathfrak{r}+1$ are frozen at their initial values. We start with moment bounds that hold uniformly in $\ell<0<\mathfrak{r}$.

Lemma 13

Fix θ. For 1≤p<∞ there is a constant C=C(p,θ) such that in all stationary $(\ell ,\mathfrak{r},\theta )$ processes with marginal distribution μ ^θ, for all t≥0 and i∈ℤ,

$$\mathbf {E}\bigl[\bigl(h_i(t)-h_i(0)\bigr)^p\bigr] \le \mathrm{e}^{Ct}.$$

The bound is valid also for the boundary columns i=ℓ−1 and $i= \mathfrak{r}$, and also for the infinite-volume stationary process with marginal distribution μ ^θ.

Proof

Abbreviate $\bar{h}_{i}(t)=h_{i}(t)-h_{i}(0)$. It suffices to consider the $(\ell ,\mathfrak{r},\theta )$ processes because the bound extends to the $-\ell,\mathfrak{r}\to\infty$ limit by Fatou’s lemma and Lemma 10. The increments ω _i have all exponential moments under μ ^θ so assumption (39) is satisfied. In particular, $\bar{h}_{i}(t)$ has all moments.

Fix $\ell<0<\mathfrak{r}$. Columns h _i for $i\notin[\ell-1,\mathfrak{r}]$ are frozen at their initial values and need no argument. Consider the case $\ell\le i\le \mathfrak{r}-1$. It suffices to consider a positive integer p. In the next calculation, use martingale (40), the fact that $\bar{h}_{i}(s)$ is a nonnegative integer to bound a lower power with a higher one, and apply Hölder’s inequality:

Rewrite this as

$$\mathbf {E}\bar{h}_i(t)^p+1 \le1+ C_{p,\theta } \int _0^t \bigl(\mathbf {E}\bar{h}_i^p(s)+1\bigr) \,\mathrm {d}s$$

and now Gronwall’s inequality gives the conclusion.

The boundary columns i=ℓ−1 and $i=\mathfrak{r}$ are handled similarly, with the only difference that the rates include also the constant terms e^θ or e^−θ. □

Fix a path z(t) in ℤ with z(0)=0 and since t is fixed abbreviate z=z(t). Define

$$I_+(t)=\sum_{n= z(t)+1}^{\mathfrak{r}+1}\omega _n(t) \quad\mbox{and}\quad I_-(t)=\sum _{n=\ell-1}^{z(t)}\omega _n(t).$$

Due to the frozen columns at the boundaries and the normalization h ₀(0)=0 we have the identity

$$h_z(t)=I_+(t)-I_+(0)=-I_-(t)+I_-(0)$$

from which

$$\mathbf {Var}\bigl[h_z(t)\bigr]= \mathbf {Cov}\bigl[ -I_-(t)+I_-(0),I_+(t)-I_+(0) \bigr]. $$

(42)

For each time t the increment variables are i.i.d. μ ^θ-distributed. The process is independent of the initial values ω _ℓ−1(0) and $\omega _{\mathfrak{r}+1}(0)$ of the boundary increments. This is because the bricklayer at site ℓ−1 lays bricks to his right at rate e^θ regardless of the value of the increment at site ℓ−1, and similarly the left action of the bricklayer at site $\mathfrak{r}+1$ has constant rate e^−θ. But the time increment

$$\omega _{\ell-1}(t)-\omega _{\ell-1}(0)= -h_{\ell-1}(t)+h_{\ell-1}(0) $$

(43)

and the same for $\omega _{\mathfrak{r}+1}(t)$ are needed for h _z (t). Expanding and removing vanishing covariances from (42) leaves

(44)

(45)

(46)

(47)

We argue that the sums on lines (45)–(47) vanish as $-\ell,\mathfrak{r}\to\infty$ by showing that the covariances decay exponentially in the spatial distance.

We illustrate with a term $\mathbf {Cov}(\omega _{i_{0}}(t),\omega _{\mathfrak{r}+1}(t))$ for a fixed i ₀≤z from the first sum on line (46). Consider $-\ell,\mathfrak{r}$ large so that $\ell<i_{0}<z<\mathfrak{r}$. Let $w=\lfloor {(i_{0}+\mathfrak{r})/2}\rfloor$ be a lattice point at or next to the midpoint between i ₀ and $\mathfrak{r}$. Let ζ(t) and $\underline {\xi}(t)$ be two further processes whose initial configurations agree with those of $\underline {\omega }(t)$, both for the increments and for the heights:

$$ \underline {\zeta }(0)=\underline {\xi}(0)=\underline {\omega }(0)\quad\mbox{and}\quad \underline {h}^\zeta (0)= \underline {h}^\xi(0)= \underline {h}(0). $$

(48)

Processes ζ(t) and $\underline {\xi}(t)$ follow the $(\ell ,\mathfrak{r},\theta )$ evolution, except that the columns $\{ \underline {h}^{\zeta }_{i}(t)\colon\allowbreak i\ge\nobreak w-1\}$ with bases in [w−1,∞) are not permitted to grow, and the columns based in (−∞,w] are similarly frozen. (Equivalently, replace the Poisson clocks of the corresponding bricklayers with empty point measures.) To determine their dynamics, in addition to disjoint collections of Poisson clocks, ζ(t) requires initial increments {ω _i (0):i≤w−1} while $\underline {\xi}(t)$ requires initial increments {ω _i (0):i≥w+1}. Thus {ζ _i (t):i≤w−1} and {ξ _i (t):i≥w+1} are independent processes. The height processes and are not independent. For example if z>0 the initial heights are very much influenced by the initial increments {ζ _i (0):0<i≤z}.

On the sites where ζ(t) and $\underline {\xi}(t)$ lay bricks they obey the same realizations of Poisson clocks as the $(\ell ,\mathfrak{r},\theta )$ process $\underline {\omega }(t)$. This coupling preserves the inequalities

$$ \underline {h}^\zeta (t)\le \underline {h}(t)\quad \mbox{and}\quad \underline {h}^\xi(t)\le \underline {h}(t). $$

(49)

Let

$$A=\bigl\{ \omega _i(t)=\zeta _i(t)\mbox{ for }i\le i_0\bigr\}\quad\mbox{and}\quad B=\bigl\{\omega _{\mathfrak{r}+1}(t)=\xi_{\mathfrak{r}+1}(t)\bigr\}. $$

(50)

Let us say that the space-time window [a,b]×(s ₀,s ₁] is a block if h _i (s ₁)=h _i (s ₀) for some ⌈a⌉≤i≤⌊b⌋−1, in other words, if some column failed to grow inside the space interval [a,b] during time interval (s ₀,s ₁]. The space-time window is not a block if every column inside grew during (s ₀,s ₁]. The sense of the terminology is that a block does not permit a discrepancy to pass. For convenience we do not require a,b to be integers. We restate Corollary 5.5 from [6]. This lemma is valid in the stationary process because there is control over the spatial averages of the increments ω _i and thereby control over rates.

Lemma 14

[6, Cor. 5.5]

There exist constants k ₀<∞ and t ₀>0 and a function 0<G(s)<∞ for s∈(0,t ₀) such that G(s)↗∞ as s↘0, and this bound holds: if s ₁−s ₀≤t ₀ and b−a≥k ₀ then

$$\mathbf {P}\bigl\{[a,b]\times(s_0,s_1]\mbox{ \textit{is not a block}}\bigr\} \le \mathrm{e}^{-(b-a)\cdot G(s_1-s_0)}.$$

For a given θ this bound is valid for all $(\ell ,\mathfrak{r},\theta )$ processes.

Recalling that t is fixed in the present discussion, fix a positive integer m and real τ>0 so that

$$t=m\tau \quad \mbox{and}\quad \tau\in(0, t_0].$$

On the event A ^c there must exist a sequence of times $0<t_{w-2}<t_{w-3}<t_{w-4}<\cdots<t_{i_{0}}\le t$ such that column h _i grew at time t _i , i ₀≤i≤w−2. This results from the observation that in the range i≤w−2 the first discrepancy $h_{i}-h^{\zeta }_{i}>0$ in column heights at i can appear only next to an existing discrepancy. Thus the leftmost discrepancy starts from the value Q(0)=∞ due to (48), arrives at the boundary w−1 of the frozen region of $\underline {h}^{\zeta }$ at time t _w−1 when h _w−1 first grows, and then moves to the left one step at a time and always with a jump of the column h _Q(t)−1(t) that is not matched by a jump in $h_{Q(t)-1}^{\zeta }(t)$.

Consequently event A ^c implies that at least one of the windows

$$\biggl[w-2-j\frac{w-2-i_0}{m}, w-2-(j-1)\frac{w-2-i_0}{m} \biggr] \times \bigl[(j-1)\tau, j\tau\bigr], \quad1\le j\le m,$$

is not a block. For if all these windows were blocks, the sequence of growths over intervals [w−i,w−i+1] at times t _w−i , 2≤i≤w−i ₀, could not happen. This gives the bound

$$\mathbf {P}\bigl(A^c\bigr)\le m\mathrm{e}^{-C(w-2-i_0)/m}\le m\mathrm{e}^{-C(\mathfrak{r}-i_0)/m}$$

where for the second inequality we assumed that $\mathfrak{r}$ is large enough relative to z.

The same argument for the propagation of discrepancies can be repeated for event B in (50) to improve the bound to

$$\mathbf {P}(A^c\cup B^c)\le2m\mathrm{e}^{-C(\mathfrak{r}-i_0)/m} \le C_1\mathrm{e}^{-C_2(\mathfrak{r}-i_0)} $$

(51)

where we subsumed m in the constants.

Next we turn these bounds into bounds on covariances.

(52)

By independence of $\zeta _{i_{0}}(t)$ and $\xi_{\mathfrak{r}+1}(t)$ the first covariance after the equality sign vanishes. We claim that for $\ell\le i\le \mathfrak{r}$,

$$\mathbf {E}\bigl|\omega _{i}(t)\bigr|^p +\mathbf {E}\bigl|\zeta _{i}(t)\bigr|^p +\mathbf {E}\bigl|\omega _{\mathfrak{r}+1}(t)\bigr|^p +\mathbf {E}\bigl|\xi_{\mathfrak{r}+1}(t)\bigr|^p\le \mathrm{e}^{Ct}. $$

(53)

Granting this for the moment, we apply Hölder’s inequality to the other expectations in (52), then (51) and the moment bounds (53) to arrive at

$$|\mathbf {Cov}\bigl(\omega _{i_0}(t),\omega _{\mathfrak{r}+1}(t)\bigr) |\le C_1\mathrm{e}^{-C_2(\mathfrak{r}-i_0)}. $$

(54)

Now to verify (53) term by term. For ω _i (t) this is simply a finite moment under the μ ^θ distribution. The other increment variables we express so that Lemma 13 applies to each case. For the boundary increment $\omega _{\mathfrak{r}+1}(t)$ write

(55)

utilizing the frozen column $h_{\mathfrak{r}+1}$ as in (43). Utilizing (48) and (49)

$${\zeta _{i}(t)}={h^\zeta _{i-1}(t)-h^\zeta _{i}(t)}\le{h_{i-1}(t)}-{h_{i}(0)} ={h_{i-1}(t)}-{h_{i-1}(0)}+\omega _i(0)$$

with a similar upper bound for −ζ _i (t), which together give

$$\bigl|\zeta _{i}(t)\bigr| \le \bigl({h_{i-1}(t)}-{h_{i-1}(0)}\bigr) \vee \bigl({h_{i}(t)}-{h_{i}(0)} \bigr) +\bigl|\omega _i(0)\bigr|.$$

For $\xi_{\mathfrak{r}+1}(t)$ use stochastic monotonicity of columns to begin with

$${\xi_{\mathfrak{r}+1}(t)}={h^\zeta _{\mathfrak{r}}(t)-h^\zeta _{\mathfrak{r}+1}(t)}\le{h_{\mathfrak{r}}(t)}-{h_{\mathfrak{r}+1}(0)}$$

and continue as in (55). The completes the verification of (53).

We have proved the exponential decay of the covariance in (54). The same arguments work for all the covariances on lines (44)–(47), and we state the lemma in this generality. The bounds extend also to the infinite-volume stationary process because Lemma 13 gives moment bounds that ensure uniform integrability.

Lemma 15

There exist constants C _i =C _i (t,θ)∈(0,∞) such that, for all stationary $(\ell ,\mathfrak{r},\theta )$ processes and all i,j∈ℤ and s∈[0,t],

$$\mathbf {Cov}\bigl(\omega _{i}(s),\omega _{j}(t)\bigr) \le C_1 \mathrm{e}^{-C_2|i-j|}. $$

(56)

The bound is also valid for the infinite-volume stationary process.

Now we can complete the proof of the covariance formula.

Proof of Theorem 3

Since the bound (56) hold uniformly as $-\ell,\mathfrak{r}\to\infty$, the sums on lines (45)–(47) vanish while the sums on line (44) converge to

$$\sum_{i\le0,\,j>z} \mathbf {Cov}\bigl(\omega _i(0),\omega _j(t)\bigr) + \sum_{i\le z,\,j>0} \mathbf {Cov}\bigl(\omega _i(t),\omega _j(0)\bigr).$$

Translation invariance of the stationary infinite volume process turns the above sum into the right-hand side of (34). The left-hand side of (44) converges to the left-hand side of (34) as $-\ell,\mathfrak{r}\to\infty$ by Lemma 10 and by the uniform integrability given by Lemma 13. □