The lattice outburst process

In an outburst process, an infected site can spread infection to a whole ball around it at once. Every newly infected site can then cause outbursts of its own. The number of clocks grows with the infected region, but many of them ring in its interior and add nothing. We would like to understand how this overlap limits growth, especially when arbitrarily large outbursts are possible.

We came to the lattice version while studying cluster–cluster aggregation. A cluster can gain many particles in a single collision. Replacing each incoming cluster by a ball that contains it gives us a simpler process whose growth bounds the original one. There is already a substantial theory of outburst growth in the continuum; here we use some of those ideas to bound moments of the infected set’s size.

The lattice model

Let $G_t$ be the infected set in $\mathbb Z^d$, starting from $G_0=\{0\}$. At every infected site $x$, and for every integer radius $R\geq1$, an independent clock rings at rate $R^{-\alpha}$. When it rings, we add the entire $\ell^1$ ball:

$$G_t=G_{t^-}\cup B_1(x,R), \qquad B_1(x,R)=\{y\in\mathbb Z^d:\|y-x\|_1\leq R\}.$$

In two dimensions these balls are diamonds. Once a site is infected, it stays infected. An outburst whose whole diamond is already infected has no effect. Larger values of $\alpha$ make large outbursts less frequent.

For $\alpha>1$, the total clock rate at each site is finite. Writing $\lambda_\alpha=\sum_{R\geq1}R^{-\alpha}$, we can describe the same process as follows: events occur at rate $\lambda_\alpha|G_t|$, their centres are uniform in $G_t$, and their radii have distribution $\mathbb P(R=r)=r^{-\alpha}/\lambda_\alpha$. The normalization matters: $R^{-\alpha}$ itself is a rate, not a probability distribution.

The continuum model

Maria Deijfen introduced a continuum version, with a shape theorem published in 2003. The infected region lies in $\mathbb R^d$, outbursts occur at a rate equal to its volume, and their centres are chosen uniformly from it. Each outburst adds a Euclidean ball with a radius drawn from a prescribed distribution.

Starting from a bounded region of positive volume, Deijfen showed that bounded radius distributions give linear growth: after rescaling space by time, the infected region approaches a deterministic ball. Even though the number of possible outburst centres keeps increasing, the region has a finite asymptotic speed.

Gouéré and Marchand extended this result in 2008 to certain unbounded radius distributions. Their sufficient condition holds, in particular, if the radius has a finite moment of order $d+1+s$ for some $s>0$. Their proof relates unusually rapid growth to paths that collect unusually large random weights, an idea we will also use.

A shape theorem concerns what happens eventually in almost every realization. We want bounds on $\mathbb E[|G_t|^k]$ that hold at every time. For this, we also have to control the rare realizations in which growth is much faster than usual. Those matter increasingly as $k$ gets larger.

Why we follow rooted clusters

In cluster–cluster aggregation, clusters move and merge. Give each initial particle a root label. When a moving cluster merges into a passive cluster, keep the passive root and delete the moving cluster’s root.

Follow one root for as long as it survives. Its cluster grows by accepting incoming clusters without moving at the moment of attachment. Between attachments it may translate, but that translation disappears if we use coordinates centred at the root. We stop following the root if its cluster becomes the source of a merger.

Suppose an incoming cluster of $R$ vertices attaches at a site $x$. Because the incoming cluster is connected, all of it lies within $B_1(x,R)$. The rate of size-$R$ attachments at $x$ is at most $R^{-\alpha}$: there are at most $2d$ contact bonds, each contributing at most $R^{-\alpha}/(2d)$.

We can therefore couple each attachment to an outburst of radius $R$ at $x$. Filling in the whole ball, and allowing extra outbursts, gives a set containing the rooted cluster throughout its lifetime. A mass-transport argument transfers this comparison from roots followed forward in time to roots observed at a fixed time. Bounds for the outburst process then give bounds for cluster-size moments.

A bound on moments

Our result. For $d\geq1$ and $k>0$, if

$$\alpha>d(k+1)+2,$$

then the lattice outburst process is nonexplosive and

$$\boxed{\mathbb E[|G_t|^k]\leq C_{\alpha,d,k}(1+t)^{dk},\qquad t\geq0.}$$

Nonexplosion means that infinitely many outbursts cannot occur in a finite time interval. The exponent $dk$ is what we would expect from a region with radius of order $t$: its volume is of order $t^d$. We do not identify a limiting shape or growth constant here.

For example, in two dimensions we obtain $\mathbb E[|G_t|]\leq C(1+t)^2$ when $\alpha>6$, and a second-moment bound of $C(1+t)^4$ when $\alpha>8$. These are sufficient conditions from the proof.

The proof separates slow outbursts, which cannot travel far in limited time, from rare fast ones. Greedy lattice-animal estimates control how much a possible infection path can gain from the fast outbursts, including the rare paths needed for the moment bounds. A formal writeup is coming soon.

References

1. M. Deijfen. Asymptotic shape in a continuum growth model. Advances in Applied Probability 35(2), 303–318 (2003).
2. J.-B. Gouéré and R. Marchand. Continuous first-passage percolation and continuous greedy paths model: linear growth. Annals of Applied Probability 18(6), 2300–2319 (2008).
3. J. T. Cox, A. Gandolfi, P. S. Griffin and H. Kesten. Greedy lattice animals. I. Upper bounds. Annals of Applied Probability 3(4), 1151–1169 (1993).
4. A. Gandolfi and H. Kesten. Greedy lattice animals. II. Linear growth. Annals of Applied Probability 4(1), 76–107 (1994).

The lattice moment estimate stated as our result above refers to the forthcoming writeup; the cited papers provide the continuum model and the greedy lattice animal framework.