Computer Aided Series Expansions for Critical Phenomena A

Under quite general conditions critical phenomena can be described with high order linked cluster expansions. The coefficients of the series admit a graphical expansion that is generated with the aid of computers. Our generalization of linked cluster expansions from an infinite to a finite volume allows to perform a finite size scaling analysis. We also indicate a generalization to Dynamical Linked Cluster Expansions with possible applications to spin glasses and neural networks with coupled spin and interaction dynamics. 1 Linked Cluster Expansions in the Infinite Volume We will focus on multiscale phenomena that occur as critical phenomena in statistical systems at second order phase transitions. We consider statistical systems with regions of first and second order transition regions in phase space. Our computational tools are analytical rather than numerical calculations. We use convergent series expansions in a parameter called hopping parameter κ. When applied to a calculation of the free energy and connected correlations, the expansions amount to Linked Cluster Expansions (LCEs). The expansion coefficients admit a representation as a sum of connected graphs. The price one has to pay for describing critical phenomena with an accurracy of the order of 1% is a high order in the expansion parameter. Since the number of contributing graphs rapidly increases with increasing order in κ, a computer aided algorithmic generation of graphs becomes unavoidable. Even an optimization a Talk presented by H.