Thus, one is tempted to look for ‘asymptotic fixed shapes’, which would have to solve the equation
γ δ
w
∗ γ δ
x
∂
2
∂ x
γ
∂ x
δ
w
∗ αβ
x = λw
∗ αβ
x x
∈ K 1.3.5
for some λ ∈ 0, ∞.
Although all this is rather speculative, it seems that if any form of universality holds for systems with non-invariant harmonics, then we are most likely to find it
in the class of critically recurrent systems. Such universality would be interesting, because it would be the first example of universality that does not follow from
invariant harmonics.
1.3.2 Renormalization on other lattices
The critically recurrent symmetrized random walk also becomes important if one tries to prove large space-time results for other lattices than the high N limit of
the hierarchical group
N
. One such result has been derived by Klenke [23]. For interacting diffusions with one-dimensional state space, indexed by the hierarchical
lattice
N
with N fixed, he was able to give a description of the law of large block averages in terms of a Wright-Fisher diffusion process. The important object to
look at in this case is the so-called interaction chain. This is the chain of all block averages up to a certain size, observed at a given time βt:
X
N,k
β t, . . . , X
N,0
β t,
1.3.6 introduced in formula 1.2.15. If one lets N tend to infinity, then in the right time
scale β this chain converges to a ‘backward’ Markov chain, as explained in Con- jecture 1.2.1. For finite N , the interaction chain does not have the Markov property,
but in the critically recurrent case one can let k tend to infinity with the result that 1.3.6 in the right scaling converges to a diffusion process. These facts are so far
known for one-dimensional K only. Their proofs depend on moment calculations involving a dual model, which are not available for the isotropic models treated in
this dissertation. It seems worthwhile to investigate if for isotropic models they can be obtained by alternative methods.
1.3.3 Discrete models
In section 1.1 we have seen how certain discrete particle models, closely related to the voter model, have a continuum limit: the diffusion models discussed in this
dissertation. If we could rigorously prove the convergence of a given particle model
to the associated diffusion model, then our results would make direct contact with the theory of these particle models. This is certainly something worth trying. Apart
from this, we can more generally take the results in this dissertation as a motivation to try to prove analogous results for the particle models.
For example, we should expect that the discrete p-type p-tuple model, just like its diffusion counterpart, clusters if and only if the symmetrized random walk
is recurrent. Moreover, for infinite Abelian groups but not for finite , we expect the large-time frequencies of the remaining p
− 1 colors to be given by the distribution of Brownian motion, starting from θ with θ as in 1.1.55, when it
first hits the boundary ∂ K . However, it seems that a proof of this claim is more difficult for the particle model than for the diffusion model.
We see that although the diffusion models are a lot harder to define than the discrete particle models, because of the use of rather heavy diffusion theory, they
also make certain things easier.
1.3.4 Outlook and conclusion
We have seen how certain systems of linearly interacting diffusions exhibit univer- sal behavior on large time scales and on large space-time scales. We have come to
understand this universality as a phenomenon that is caused by a special property of the systems, which we have called ‘invariant harmonics’.
Technical difficulties often forced us to prove weaker theorems than we origi- nally planned, but a nice aspect of the models considered is that they show, so to
say, a wide variety in tractability. For the 2-type 2-tuple model, there are coupling techniques available and there is a duality. For general p-type 2-tuple models, the
coupling techniques seem to fail, but there is still a duality, while for the p-type
p-tuple models there is no duality, but there are calculations involving covariances and harmonic functions that are in some way a substitute. Finally, there are models
like the 4-type 3-tuple model, for which we are still hardly able to prove anything.
Chapter 2
Renormalization of Hierarchically Interacting
Isotropic Diffusions
Abstract
We study a renormalization transformation arising in an infinite system of interact- ing diffusions. The components of the system are labeled by the N -dimensional
hierarchical lattice N ≥ 2 and take values in a compact convex set D ⊂
R
d
d ≥ 1. Each component starts at some θ ∈ D and is subject to two motions:
1 an isotropic diffusion according to a local diffusion rate g : D → [0, ∞
chosen from an appropriate class; 2 a linear drift towards an average of the sur- rounding components weighted according to their hierarchical distance. In the
local mean-field limit N → ∞, block averages of diffusions within a hierarchi-
cal distance k, on an appropriate time scale, are expected to perform a diffusion with local diffusion rate F
k
g, where F
k
g = F
c
k
◦ · · · ◦ F
c
1
g is the k-th iterate of renormalization transformations F
c
c 0 applied to g. Here the c
k
mea- sure the strength of the interaction at hierarchical distance k. We identify F
c
and study its orbit F
k
g
k ≥0
. We show that there exists a ‘fixed shape’ g
∗
such that lim
k →∞
σ
k
F
k
g = g
∗
for all g, where the σ
k
are normalizing constants. In terms of the infinite system, this property means that there is complete universal behavior
on large space-time scales. Our results extend earlier work for d
= 1 and D = [0, 1] resp. [0, ∞. The renormalization transformation F
c
is defined in terms of the ergodic measure of a d-dimensional diffusion. In d
= 1 this diffusion allows a Yamada-Watanabe-type coupling, its ergodic measure is reversible and the renormalization transformation
37
