Exact exponents in two dimensions

The asymptotic behaviour of a renormalization factor is obtained in the same way by integrating (12.3.91) in the form 

for instance, taking (12.3.103) into account, we find (for M = 0) 

Other quantities can be examined and similar logarithmic corrections can be found for them. 

In practice, it seems difficult to verify the existence of such corrections. Nevertheless, by computer simulation Havlin and Ben-Avraham (1981),49 and also Aragao de Carvalho and Caracciolo (1983),50 have been able to observe logarithmic effects for that purpose, the first-named authors used Monte Carlo methods to construct self-avoiding walks on a four-dimensional lattice. 

Around 1984, it became clear that critical phenomena in two dimensions can be described exactly and that two different approaches are possible. 

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It is well-known that in a polymer melt as in a semi-dilute solution, the size exponent has the trivial value v = 1/2. However, for a melt, it is also possible to define exponents aM as was done for dilute solutions (see Section 3.3) as before, these exponents are directly related to the partition functions associated with a network [see eqns (12.3.64) and (12.3.65)]. The values of aM in the present case are not obvious. However, the exact values in two-dimensions... 

Tricritical swelling exponent in two dimensions estimations and exact values... 

H. Saleur and B. Duplantier, Exact determination of the percolation hull exponent in two dimensions,... 

B. Duplantier, H. Saleur. Exact tricritical exponents for polymers at the 0-point in two dimensions. Phys Rev Lett 59 539-542, 1987. 

Note that whenever a new exponent is defined, there is also a scaling relation that calculates this exponent from t and a. There are only two independent exponents that describe the distribution of molar masses near the gelation transition, with the other exponents determined from scaling relations. Table 6.4 summarizes the exponents in different dimensions that have been determined numerically, along with the exact results for 1, d=2, and d>6. It turns out that t/=6 is the upper critical dimension for percolation, and the mean-field theory applies for all dimensions d>6. 

Moreover, in 1970, Polyakov discovered that the correlation functions of a critical magnetic system have invariance properties for transformations belonging to the special conformal group. Then, in 1984, Belavin, Polyakov, and Zamolodchikov showed that these conformal transformations are really important in two dimensions. Since that time, research in this domain led to very interesting results. Thus exact values of the main exponents associated with two-dimensional polymer solutions, were found in 1986 by Saleur and Duplantier. 

As early as 1970, Polyakov51 showed that at the critical point, a critical system is invariant for transformations belonging to the conformal group. Thus, as this group is rather rich in two dimensions, these invariance properties could later52 be used to make precise studies of critical phenomena and, in particular, to find exact values of critical exponents. 

The conformal invariance properties of critical systems are certainly important for any space dimension, but even more so in two dimensions because the two-dimensional conformal group is very rich. This essential fact was clearly recognized and exploited by Belavin, Polyakov, and Zamolodchikov in 1984.52 Their theory is not simple, but very interesting it is presently developing (1990) and consequendy certain points remain obscure. Here, we shall give only an idea of this method, in spite of the fact that it has already led to fundamental results. In particular, as we shall see, it provided the means for the calculation the exact values of the exponents aM defined in Section 3.3 of the present chapter [see (12.3.68)]. 

Finally, in 1987, Duplantier and Saleur found the exact values21 of the tricritical exponents v, and y, in two dimensions. These values are... 

In two dimensions, however, the situation will prove to be very different from what it is in three or more dimensions whereas van der Waals theory predicts o) = 1, which follows at once from the postulated identification a> = v, with v the critical exponent describing the divergence of the bulk correlation length and the exact... 

Between method c and d is the den Nijs-Pearson-Nienhuis et al. conjecture about the Potts model in two dimensions (which includes percolation and the lattice gas as special cases) It assumes a simple form for the variation of the exponents and fits its parameters for exactly solved models. Since... 

For the exact solution of A -electron atoms at the large dimension limit, the symmetry breaking is shown to be a first-order phase transition. For the special case of two-electron atoms, the first-order transition shows a triple point where three phases with different symmetry exist. Treatment of the Hartree-Fock solution reveals a different kind of symmetry breaking where a second-order phase transition exists for N — 2. The Hartree-Fock two-electron atoms in weak external electric field exhibit a critical point with mean-field critical exponents ( = j, a = Odis, 5 = 3, and y — 1). ... 

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