186,818 research outputs found
Loop expansion around the Bethe approximation through the -layer construction
For every physical model defined on a generic graph or factor graph, the
Bethe -layer construction allows building a different model for which the
Bethe approximation is exact in the large limit and it coincides with the
original model for . The perturbative series is then expressed by a
diagrammatic loop expansion in terms of so-called fat-diagrams. Our motivation
is to study some important second-order phase transitions that do exist on the
Bethe lattice but are either qualitatively different or absent in the
corresponding fully connected case. In this case the standard approach based on
a perturbative expansion around the naive mean field theory (essentially a
fully connected model) fails. On physical grounds, we expect that when the
construction is applied to a lattice in finite dimension there is a small
region of the external parameters close to the Bethe critical point where
strong deviations from mean-field behavior will be observed. In this region,
the expansion for the corrections diverges and it can be the starting
point for determining the correct non-mean-field critical exponents using
renormalization group arguments. In the end, we will show that the critical
series for the generic observable can be expressed as a sum of Feynman diagrams
with the same numerical prefactors of field theories. However, the contribution
of a given diagram is not evaluated associating Gaussian propagators to its
lines as in field theories: one has to consider the graph as a portion of the
original lattice, replacing the internal lines with appropriate one-dimensional
chains, and attaching to the internal points the appropriate number of
infinite-size Bethe trees to restore the correct local connectivity of the
original model
Functional control of network dynamics using designed Laplacian spectra
Complex real-world phenomena across a wide range of scales, from aviation and
internet traffic to signal propagation in electronic and gene regulatory
circuits, can be efficiently described through dynamic network models. In many
such systems, the spectrum of the underlying graph Laplacian plays a key role
in controlling the matter or information flow. Spectral graph theory has
traditionally prioritized unweighted networks. Here, we introduce a
complementary framework, providing a mathematically rigorous weighted graph
construction that exactly realizes any desired spectrum. We illustrate the
broad applicability of this approach by showing how designer spectra can be
used to control the dynamics of various archetypal physical systems.
Specifically, we demonstrate that a strategically placed gap induces chimera
states in Kuramoto-type oscillator networks, completely suppresses pattern
formation in a generic Swift-Hohenberg model, and leads to persistent
localization in a discrete Gross-Pitaevskii quantum network. Our approach can
be generalized to design continuous band gaps through periodic extensions of
finite networks.Comment: 9 pages, 5 figure
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