Condition 1.2.23 is necessary for the universality observed in 1.2.25. It is known that condition 1.2.23 corresponds to clustering behavior in the model
Theorem 3 in [10]. This means that the components X
N i
t of the system, after a long time, spend most of their time near the boundary of [0, 1]. In fact, according
to Conjecture 1.2.1,
lim
N →∞
P[X
N
N
k
t ∈ dy|X
N,k
N
k
t = x] = K
k x
d y, 1.2.26
where K
g,k x
d y =
· · · ν
F
k −1
g,c
k
x
d z
1
ν
F
k −2
g,c
k −1
z
1
d z
2
· · · ν
F g,c
1
z
k −1
d y. 1.2.27
Thus, the probability measure K
g,k x
· describes the conditional distribution of the urns in a k-block, given that the k-block average is x. It has been shown in [1] that
K
g,k θ
· ⇒ 1 − θδ + θδ
1
as k → ∞,
1.2.28 if and only if 1.2.23 holds. Therefore 1.2.23 corresponds to the situation where
after a long time each urn with probability close to θ contains almost only balls of color 1, and with probability close to 1
− θ almost only balls of color 2.
1.2.4 Higher-dimensional generalizations
The results in Proposition 1.2.2 leave one with a number of questions. Notably, one would like to understand better the origin of the observed universality. What
is so special about the Wright-Fisher diffusion function that all other diffusion functions in the class
H
are attracted to it? In October 1995 my supervisor and me took this question as our motivation to study higher-dimensional equivalents of the
transformation F
c
. For simplicity, we first restricted ourselves to isotropic diffusions. Thus, we
considered a transformation of the form F
c
gx : =
K
gyν
g,c x
d y, 1.2.29
where K ⊂
R
d
is some compact and convex domain, and ν
g,c x
is the unique equi- librium distribution of a diffusion whose generator extends
A
g,c x
f y : =
α
cx
α
− y
α ∂
∂ y
α
f y + gy
α ∂
2
∂ y
α 2
f y. 1.2.30
This Ansatz immediately raised a number of questions.
1. From which class
H
can we choose our diffusion function g so that F
c
g is well-defined? In particular:
a For which g is the martingale problem for the operator in 1.2.30 well- posed?
b For which g does the associated diffusion have a unique equilibrium ν
g,c x
? c Is it true that F
c
g ∈
H
for all g ∈
H
? 2. Do the iterates of the transformation F
c
describe the large space-time scale behavior of a system of interacting diffusions, i.e., can we prove a rigorous
version of Conjecture 1.2.1?
3. Does the transformation F
c
have a unique fixed shape g
∗
that attracts all other g
∈
H
after appropriate scaling? In the search for answers to these questions, we were not completely without
clues. First, of course, we knew that the Wright-Fisher diffusion arises in a special way as the diffusion limit of particle models see section 1.1.2, and this might
have something to do with its special role. Furthermore, we knew that condition 1.2.23, necessary for the universality, corresponds to clustering behavior. The
fact that the components X
i
t of the system, after a long time, spend most of their time near the boundary of the domain K makes one suspect that the long-time
behavior of the system does not ‘feel’ the diffusion function g on the interior of K , and from this the universality could possibly arise. In fact, one would suspect that
an appropriate reasoning would possibly not even need to consider the iterations of a complicated transformation like the one in 1.2.20, but could perhaps understand
the universal behavior by a direct reasoning based on the dynamics of the system.
We will see how much of this intuition turned out to be right. . . and how much wrong.
1.2.5 Renormalization of isotropic diffusions
We started our investigations by defining, in analogy with the one-dimensional case:
H
: = {g : K → [0, ∞ | g Lipschitz, gx = 0 ⇔ x ∈ ∂ K },
1.2.31 where ∂ K :
= K \K
◦
is the boundary of K , with K
◦
the interior of K . Let us for the moment assume that questions 1 and 2 above can be solved, and let us first con-
centrate on question 3, which concerns the problem of understanding universality.
It is not clear from our definitions what function g
∗
could be a fixed shape under F
c
, or in fact whether such a function exists at all. In the one-dimensional case, the fact that gx
= x1 − x is a fixed shape follows from the equilibrium conditions for the time evolution of the first and second moments of solutions to
the martingale problem for A
g,c x
. We found out that this proof immediately extends to the case that
K = {x ∈
R
d
: |x| ≤ 1}
g
∗
x = 1 − |x|
2
. 1.2.32
In view of the interpretation of our model as explained in section 1.1, this case is rather unsatisfactory. We would rather like to be able to treat the p-type p-tuple
model see section 1.1.3 or the 4-type 2 + 2-tuple model see section 1.1.7. For
the latter, K = [0, 1]
2
and the diffusion function gx
= x
1
1 − x
1
x
2
1 − x
2
1.2.33 arises in a natural way as the continuum limit of a discrete model. A natural idea
would be to see if this function is a fixed shape under F
c
. But this turns out not to be the case.
We found out that there is no explicit formula for F
c
in dimensions d ≥ 2. This
has the following reason. The equilibrium ν
g,c x
solves the equation ν
g,c x
|A
g,c x
f = 0
∀ f ∈
C
2
K , 1.2.34
where we use the notation µ| f :=
K
f xµd x 1.2.35
for any probability measure µ on K and any function f ∈
C
K . If ν
g,c x
has a sufficiently differentiable density, and also g is sufficiently smooth, then after an
integration by parts we can rewrite 1.2.34 as
A
g,c x
†
ν
g,c x
| f = 0 ∀ f ∈
C
2
K , 1.2.36
so that for the density ν
g,c x
we find the partial differential equation A
g,c x
†
ν
g,c x
= −
α ∂
∂ y
α
cx
α
− y
α
+
α ∂
2
∂ y
α 2
gy ν
g,c x
y = 0.
1.2.37 In vector notation we can write this equation in the form
∇ · T
g,c x
ν
g,c x
= 0, 1.2.38
