# Mutually Catalytic Super Branching Random Walks: Large Finite Systems and Renormalization Analysis

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*J. T. Cox; D. A. Dawson; A. Greven*

We study features of the longtime behavior and the
spatial continuum limit for the diffusion limit of the following
particle model. Consider populations consisting of two types of
particles located on sites labeled by a countable group. The
populations of each of the types evolve as follows: Each particle
performs a random walk and dies or splits in two with probability
\(\frac{1}{2}\) and the branching rates of a particle of each
type at a site \(x\) at time \(t\) is proportional to
the size of the population at \(x\) at time \(t\) of the other
type. The diffusion limit of “small mass, large number of initial
particles” is a pair of two coupled countable collections of
interacting diffusions, the mutually catalytic super branching random
walk.

Consider now increasing sequences of finite subsets of
sites and define the corresponding finite versions of the process. We
study the evolution of these large finite spatial systems in
size-dependent time scales and compare them with the behavior of the
infinite systems, which amounts to establishing the so-called finite
system scheme. A dichotomy is known between transient and recurrent
symmetrized migrations for the infinite system, namely, between
convergence to equilibria allowing for coexistence in the first case
and concentration on monotype configurations in the second
case. Correspondingly we show (i) in the recurrent case both large
finite and infinite systems behave similar in all time scales, (ii) in
the transient case we see for small time scales a behavior resembling
the one of the infinite system, whereas for large time scales the
system behaves as in the finite case with fixed size and finally in
intermediate scales interesting behavior is exhibited, the system
diffuses through the equilibria of the infinite system which are
indexed by the pair of intensities and this diffusion process can be
described as mutually catalytic diffusion on \((\mathbb{R}^+)^2\).

At the same time, the above finite system asymptotics
can be applied to mean-field systems of \(N\) exchangeable mutually
catalytic diffusions. This is the building block for a
renormalization analysis of the spatially infinite hierarchical model
and leads to an association of this system with the so-called
interaction chain, which reflects the behavior of the process on large
space-time scales. Similarly we introduce the concept of a continuum
limit in the hierarchical mean field limit and show that this limit
always exists and that the small-scale properties are described by
another Markov chain called small scale characteristics. Both chains
are analyzed in detail and exhibit the following interesting
effects.

The small scale properties of the continuum limit
exhibit the dichotomy, overlap or segregation of densities of the two
populations, as a function of the underlying random walk kernel. A
corresponding concept to study hot spots is presented. Next we look in
the transient regime for global equilibria and their equilibrium
fluctuations and in the recurrent regime on the formation of monotype
regions. For particular migration kernels in the recurrent regime we
exhibit diffusive clustering, which means that the sizes (suitably
defined) of monotype regions have a random order of magnitude as time
proceeds and its distribution is explicitly identifiable. On the other
hand in the regime of very large clusters we identify the
deterministic order of magnitude of monotype regions and determine the
law of the random size. These two regimes occur for different
migration kernels than for the cases of ordinary branching or
Fisher-Wright diffusion. Finally we find a third regime of very rapid
deterministic spatial cluster growth which is not present in other
models just mentioned.

A further consequence of the analysis is that mutually
catalytic branching has a fixed point property under renormalization
and gives a natural example different from the trivial case of
multitype models consisting of two independent versions of the fixed
points for the one type case.

#### Table of Contents

# Table of Contents

## Mutually Catalytic Super Branching Random Walks: Large Finite Systems and Renormalization Analysis

- Contents v6 free
- 0 Introduction 110 free
- 1 Results: Longtime behavior of large finite systems 817
- 2 Results: Renormalization analysis and corresponding basic limiting dynamics 1322
- 3 Results: Application of renormalization to large scale behavior 2635
- 4 Preparation: Key technical tools 3645
- 5 Finite system scheme (Proof of Theorems 1,2) 4655
- 6 Multiple space–time scale analysis (Proof of Theorem 3, 5) 7382
- 7 Analysis of the interaction chain (Proof Theorem 4, 6 – 8) 8594