%0 Book Section %1 statphys23_0115 %A Sacksteder, V.E. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K approximation bethe complex field geometries mean over pseudo-hermitian statphys23 sum systems theory topic-9 %T Sums over Geometries and Improvements on the Mean Field Approximation %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=115 %X We analyze the saddle points of a Lagrangian that Efetov proposed as a tool for calculating systematic corrections to the Bethe approximation, a mean-field approximation which is important in statistical mechanics, glasses, and combinatorial optimization. We find that the trivial saddle point generates a sum over geometries reminiscent of dynamically triangulated quantum gravity, which suggests new possibilities to improve mean field approximations to the $D$-dimensional theories by using sums over geometries. In the case of the Efetov theory, the dominant geometries are locally tree-like, and the sum over geometries diverges in a way that is similar to quantum gravity's divergence when all topologies are included. We also analyze the other saddle points of the Efetov Lagrangian and find that the Hessian at these points is non-normal and pseudo-hermitian, which is a novelty for bosonic theories.

@incollection{statphys23_0115, abstract = {We analyze the saddle points of a Lagrangian that Efetov proposed as a tool for calculating systematic corrections to the Bethe approximation, a mean-field approximation which is important in statistical mechanics, glasses, and combinatorial optimization. We find that the trivial saddle point generates a sum over geometries reminiscent of dynamically triangulated quantum gravity, which suggests new possibilities to improve mean field approximations to the $D$-dimensional theories by using sums over geometries. In the case of the Efetov theory, the dominant geometries are locally tree-like, and the sum over geometries diverges in a way that is similar to quantum gravity's divergence when all topologies are included. We also analyze the other saddle points of the Efetov Lagrangian and find that the Hessian at these points is non-normal and pseudo-hermitian, which is a novelty for bosonic theories.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Sacksteder, V.E.}, biburl = {https://www.bibsonomy.org/bibtex/2dda055c83dde02c97310745bea510c68/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {759de374153a4449503677e13c867417}, intrahash = {dda055c83dde02c97310745bea510c68}, keywords = {approximation bethe complex field geometries mean over pseudo-hermitian statphys23 sum systems theory topic-9}, month = {9-13 July}, timestamp = {2007-06-20T10:16:12.000+0200}, title = {Sums over Geometries and Improvements on the Mean Field Approximation}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=115}, year = 2007 }