Balram, Ajit C. ; Dhar, Deepak
(2012)
*Non-perturbative corrections to mean-field critical behavior: the spherical model on a spider-web graph*
Journal of Physics A: Mathematical and Theoretical, 45
(12).
Article ID 125006.
ISSN 1751-8113

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Official URL: http://iopscience.iop.org/article/10.1088/1751-811...

Related URL: http://dx.doi.org/10.1088/1751-8113/45/12/125006

## Abstract

We consider the spherical model on a spider-web graph. This graph is effectively infinite dimensional, similar to the Bethe lattice, but has loops. We show that these lead to non-trivial corrections to the simple mean-field behavior. We first determine all normal modes of the coupled springs problem on this graph, using its large symmetry group. In the thermodynamic limit, the spectrum is a set of δ-functions, and all the modes are localized. The fractional number of modes with frequency less than ω varies as exp ( − C/ω) for ω tending to zero, where C is a constant. For an unbiased random walk on the vertices of this graph, this implies that the probability of return to the origin at time t varies as exp ( − C't^{1/3}), for large t, where C' is a constant. For the spherical model, we show that while the critical exponents take the values expected from the mean-field theory, the free energy per site at temperature T, near and above the critical temperature T_{c}, also has an essential singularity of the type exp [ − K(T − T_{c})^{−1/2}].

Item Type: | Article |
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Source: | Copyright of this article belongs to Institute of Physics. |

ID Code: | 112289 |

Deposited On: | 31 Jan 2018 04:29 |

Last Modified: | 31 Jan 2018 04:29 |

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