## 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 |
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Journal | Journal Of Physics A-Mathematical And Theoretical |

Publication status | Accepted/In press - 20 Dec 2019 |