Integral equations for cluster expansion sums of polymer models with (soft) repulsion
Seminar Room 1, Newton Institute
In this note, we derive integral equations for quantities a(P) defined as sums of cluster (Mayer) expansion terms taken over clusters "touching" a given polymer P. Using these quantities, many of the interesting properties of the model, like correlation decay can be evaluated. These relations have a slightly more complicated structure than usual exponential, "K.P." inequalities for quantities a(P) (developed successively by many authors: Cammarota, Seiler, Kotecky, Preiss, Dobrushin, Sokal, Scott, Fernandez, Ueltschi, and others) but they are equations. Their derivation employs quantities a(P,t) of auxiliary models with repulsions around P "softened" by an artificial parameter t, 0 < t < 1. Then at least some of the inequalities mentioned above can be interpreted as sufficient conditions (on the "smallness" of the complex polymer weights w(P)) for the convergence of the fixpoint method of solving such an integral equation.
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