Feynman Path Integral Formulation

284 8 Numerical Studies
task, since at large distance the correlations are small and the statistical noise becomes
large. Still, the first results (Hamber and Williams, 1995) suggest that the
potential has more or less the expected classical form in the vicinity of the critical
point, both as far as the mass dependence and perhaps even the distance dependence
are concerned. In particular it is attractive.
8.6 Scaling in the Vicinity of the Critical Point
In practice the correlation functions at fixed geodesic distance are difficult to compute
numerically, and therefore not the best route to study the critical properties.
But scaling arguments allow one to determine the scaling behavior of correlation
functions from critical exponents characterizing the singular behavior of the free
energy and various local averages in the vicinity of the critical point. In general a
divergence of the correlation length ξ
ξ (k) ≡ ∼
k→k c
A ξ |k c − k| −ν , (8.50)
signals the presence of a phase transition, and leads to the appearance of a singularity
in the free energy F(k). The scaling assumption for the free energy postulates
that a divergent correlation length in the vicinity of the critical point at k c leads to
non-analyticities of the type
F ≡− 1 V lnZ = F reg + F sing
F sing ∼ ξ −d , (8.51)
where the second relationship follows simply from dimensional arguments (the free
energy is an extensive quantity). The regular part F reg is generally not determined
from ξ by purely dimensional considerations, but as the name implies is a regular
function in the vicinity of the critical point. Combining the definition of ν in
Eq. (8.50) with the scaling assumption of Eq. (8.51) one obtains
F sing (k)
∼
k→k c
(const.)|k c − k| dν . (8.52)
The presence of a phase transition can then be inferred from non-analytic terms in
invariant averages, such as the average curvature and its fluctuation. For the average
curvature one obtains
R(k) ∼ A R |k c − k| dν−1 , (8.53)
k→k c
up to regular contributions (i.e. constant terms in the vicinity of k c ). An additive
constant can be added, but numerical evidence sor far points to this constant being
consistent with zero. Similarly one has for the curvature fluctuation