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For random systems subject to a constraint, the microcanonical ensemblerequires the constraint to be met by every realisation ("hard constraint"),while the canonical ensemble requires the constraint to be met only on average("soft constraint"). It is known that for random graphs subject to topologicalconstraints breaking of ensemble equivalence may occur when the size of thegraph tends to infinity, signalled by a non-vanishing specific relative entropyof the two ensembles. We investigate to what extent breaking of ensembleequivalence is manifested through the largest eigenvalue of the adjacencymatrix of the graph. We consider two examples of constraints in the denseregime: (1) fix the degrees of the vertices (= the degree sequence); (2) fixthe sum of the degrees of the vertices (= twice the number of edges). Example(1) imposes an extensive number of local constraints and is known to lead tobreaking of ensemble equivalence. Example (2) imposes a single globalconstraint and is known to lead to ensemble equivalence. Our working hypothesisis that breaking of ensemble equivalence corresponds to a non-vanishingdifference of the expected values of the largest eigenvalue under the twoensembles. We verify that, in the limit as the size of the graph tends toinfinity, the difference between the expected values of the largest eigenvaluein the two ensembles does not vanish for (1) and vanishes for (2). A key toolin our analysis is a transfer method that uses relative entropy to determinewhether probabilistic estimates can be carried over from the canonical ensembleto the microcanonical ensemble, and illustrates how breaking of ensembleequivalence may prevent this from being possible.

A spectral signature of breaking of ensemble equivalence for constrained random graphs