The statistical mechanics of spanning forests

This work collects the results of my research activity of the last three years. The main part is the study of the spanning forest model in a mean eld approximation by the use of a recently introduced fermionic representation. These results have been published in the following papers in collaboration with S. Caracciolo and A. Sportiello. A. Bedini, S. Caracciolo, and A. Sportiello, Hyperforests on the complete hypergraph by Grassmann integral representation, Journal of Physics A: Mathematical and Theoretical 41, 205003 (2008). A. Bedini, S. Caracciolo, and A. Sportiello, Phase transition in the spanning-hyperforest model on complete hypergraphs, Nuclear Physics B 822, 493 (2009). A second an more recent subject of research concerns the development of a new general algorithm for the exact computation of statistical mechanics partition function on arbitrary graphs. This work has been done in collaboration with J. L. Jacobsen and its publication is still in progress. A. Bedini, J. L. Jacobsen, Fast solution of NP-hard problems on large random graphs, in progress.