Doing the iterations at once

for one-dimensional K only, most work on linearly interacting diffusions in the past years has focussed on systems with one-dimensional state space K . These systems all have invariant harmonics. But with the study of higher-dimensional state spaces it has become clear that there are many systems, arising naturally as limits of particle models, for which the harmonic functions are not invariant. For example, the p-type q-tuple model with 2 q p does not have invariant har- monics see the figure on page 29. A natural question is: do such systems exhibit universal behavior on large time andor space scales? As regards the long time behavior, it seems that the answer is no. It seems that invariant harmonics are necessary for the conservation of the harmonic means formula 1.2.57, and this in turn seems to be necessary to guarantee that the dis- tribution of X i ∞ is universal in the class of all diffusion matrices w with the same w -harmonic functions and in all Abelian groups with a recurrent symmetrized random walk. Since the results in Chapter 4 indicate a strong link between the large time universality and the large space-time universality, it seems that the same negative conclusion must be drawn about the action of the renormalization transformations F c on such diffusion matrices w with non-invariant harmonics. In fact, it seems likely that for certain choices of the constants c k the rescaled iterates σ k F c k ◦ · · · ◦ F c 1 w do not converge to any limit at all. However, this does not exclude the possibility that for certain choices of the c k we may still find some form of universality. Consider in particular the case that c k = c k , where c ∈ 0, ∞ is some constant. Then we have recurrence of the symmetrized random walk, and hence clustering, iff c ≤ 1. Because of the scaling relation F λ c g = λF c 1 λ g λ ∈ 0, ∞, 1.3.1 we can express σ k +1 F k +1 w in terms of σ k F k w in the following way: σ k +1 F k +1 w = σ k +1 σ k F σ k c k +1 σ k F k w. 1.3.2 If c 1, then σ k c k +1 = c k +1 k l =1 c −l → c 1 − c as k → ∞, 1.3.3 but in the critically recurrent case c = 1 we see that σ k c k +1 → ∞ as k → ∞. This means that for large k it is only the large c limit of the renormalization trans- formation F c that is important for us. An expansion in c −1 gives F c w αβ x = w αβ x + c −1 1 2 γ δ w γ δ x ∂ 2 ∂ x γ ∂ x δ w αβ x + O c −2 . 1.3.4 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