”Fluctuating cluster” equations for a polymer to O(2). (Toward exact Flory equation?)

Although Flory's approximation describes very well the radius r of a polymer, it disagrees sharply with an exact e expansion (derived from an analogy to n = 0 spins). Here the polymer is described explicitly qs a critical cluster of n = 0 spins. While ordinary clusters resemble trees bifurcating into b branches, a cluster of n = 0 spins constitutes a single line, weighted by b phantom bifurcations. Analogous description of the polymer leads to a modified Flory-like equation, which describes r and b, jointly, and to two more equations for r and b in separate. These give very accurate exponents u and ~y, correct to O(e~). In addition, the approach can be extended to a many polymer system, and to other critical clusters.