Statistical mechanics of polymer networks of any topology

The statistical mechanics is considered of any polymer network with a prescribed topology, in dimension d, which was introduced previously. The basic direct renormalization theory of the associated continuum model is established. It has a very simple multiplicative structure in terms of the partition functions of the star polymers constituting the vertices of the network. A calculation is made to O(ε2), where d = 4 -ε, of the basic critical dimensions σL associated with any L=leg vertex (L ≥ 1). From this infinite series of critical exponents, any topology-dependent critical exponent can be derived. This is applied to the configuration exponent γG of any network G to O(ε2), including L-leg star polymers. The infinite sets of contact critical exponents θ between multiple points of polymers or between the cores of several star polymers are also deduced. As a particular case, the three exponents θ0, θ1, θ2 calculated by des Cloizeaux by field-theoretic methods are recovered. The limiting exact logarithmic laws are derived at the upper critical dimension d = 4. The results are generalized to the series of topological exponents of polymer networks near a surface and of tricritical polymers at the Θ-point. Intersection properties of networks of random walks can be studied similarly. The above factorization theory of the partition function of any polymer network over its constituting L-vertices also applies to two dimensions, where it can be related to conformal invariance. The basic critical exponents σL and thus any topological polymer exponents are then exactly known. Principal results published elsewhere are recalled