Feynman Path Integral Formulation

284 8 Numerical Studiestask, since at large distance the correlations are small and the statistical noise becomeslarge. Still, the first results (Hamber and Williams, 1995) suggest that thepotential has more or less the expected classical form in the vicinity of the criticalpoint, both as far as the mass dependence and perhaps even the distance dependenceare concerned. In particular it is attractive.8.6 Scaling in the Vicinity of the Critical PointIn practice the correlation functions at fixed geodesic distance are difficult to computenumerically, and therefore not the best route to study the critical properties.But scaling arguments allow one to determine the scaling behavior of correlationfunctions from critical exponents characterizing the singular behavior of the freeenergy and various local averages in the vicinity of the critical point. In general adivergence of the correlation length ξξ (k) ≡ ∼k→k cA ξ |k c − k| −ν , (8.50)signals the presence of a phase transition, and leads to the appearance of a singularityin the free energy F(k). The scaling assumption for the free energy postulatesthat a divergent correlation length in the vicinity of the critical point at k c leads tonon-analyticities of the typeF ≡− 1 V lnZ = F reg + F singF sing ∼ ξ −d , (8.51)where the second relationship follows simply from dimensional arguments (the freeenergy is an extensive quantity). The regular part F reg is generally not determinedfrom ξ by purely dimensional considerations, but as the name implies is a regularfunction in the vicinity of the critical point. Combining the definition of ν inEq. (8.50) with the scaling assumption of Eq. (8.51) one obtainsF 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 ininvariant averages, such as the average curvature and its fluctuation. For the averagecurvature one obtainsR(k) ∼ A R |k c − k| dν−1 , (8.53)k→k cup to regular contributions (i.e. constant terms in the vicinity of k c ). An additiveconstant can be added, but numerical evidence sor far points to this constant beingconsistent with zero. Similarly one has for the curvature fluctuation