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Dewar, R.C.

English

Dewar, R.C

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Configurational studies of scaling phenomena

The renormalisation group approach to problems - involving
scale- invar iance is briefly reviewed. Then, using renormalisation
group ideas ? Lnd techniques, various scaling phenomena are studied
directly in terms of the scaling properties of the configurations or
patterns that underlie them.

(i) we study the universal scaling properties of configurations of
block-spins near the critical point of an 0(n)-symmetric ferromagneto
as characterised by the block-spin probability density function
(p. d. f .). Critical scaling properties of this p. d. f. are obtained
for n>2 and d=2+e within a perturbative calculation for small e.
A non-perturbative approximation to the renormalisation group
transformation is used to calculate the p. d. f. at the transition of
the d=3 XY model (n=2). Then we carry out a Monte-Carlo simulation
of this model on the ICL Distributed Array Processor (DAP), which
supports the result of the calculation. our results suggest that
universal scaling features of the underlying spin patterns should be
revealed in some characteristic distribution of spin angles; we
discuss the possible role played by topological excitations
(vortices) at the d=3 XY transition.

(ii) we study the problem of, percolation in terms of scaling
properties of the underlying clusters. Following a recent
controversy regarding the scaling behaviour of the mean number of
clusters, we present numezical evidence for the validity of the
currently-accepted scaling picture. For our finite-size scaling
analysis we use a purpose-written fast parallel -cluster-counting
algorithmp implemented on the ICL DAP. A noise reduction scheme is
adopted to suppress the considerable fluctuation in the numerical
data. A clear discrepancy with the controversial scaling picture is
thus established.

(iii) we study scaling in a model of random epidemic growth. From
numerical simulations, we find that the interface between healthy and
infected sites exhibits both static and dynamic scaling properties.
Results are in close agreement with the predictions of a
recent ly-proposed Langevin equation for interface growth, from which
universality classes have been extracted using the dynamical
renormalisation group. We thus identify the essential
large-distance, long-time physics of the epidemic model

Configurational studies of scaling phenomena.

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