Abstract
We present an analytical approach for describing spectrally
constrained maximum entropy ensembles of nitely connected regular loopy
graphs, valid in the regime of weak loop-loop interactions. We derive an expression
for the leading two orders of the expected eigenvalue spectrum, through the use of
innitely many replica indices taking imaginary values. We apply the method to
models in which the spectral constraint reduces to a soft constraint on the number
of triangles, which exhibit `shattering' transitions to phases with extensively many
disconnected cliques, to models with controlled numbers of triangles and squares,
and to models where the spectral constraint reduces to a count of the number of
adjacency matrix eigenvalues in a given interval. Our predictions are supported by
MCMC simulations based on edge swaps with nontrivial acceptance probabilities.
constrained maximum entropy ensembles of nitely connected regular loopy
graphs, valid in the regime of weak loop-loop interactions. We derive an expression
for the leading two orders of the expected eigenvalue spectrum, through the use of
innitely many replica indices taking imaginary values. We apply the method to
models in which the spectral constraint reduces to a soft constraint on the number
of triangles, which exhibit `shattering' transitions to phases with extensively many
disconnected cliques, to models with controlled numbers of triangles and squares,
and to models where the spectral constraint reduces to a count of the number of
adjacency matrix eigenvalues in a given interval. Our predictions are supported by
MCMC simulations based on edge swaps with nontrivial acceptance probabilities.
| Original language | English |
|---|---|
| Journal | Journal Of Physics A-Mathematical And Theoretical |
| Publication status | Accepted/In press - 20 Dec 2019 |