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Article Dans Une Revue Journal of the European Mathematical Society Année : 2020

Spectral atoms of unimodular random trees

Résumé

We use the Mass Transport Principle to analyze the local recursion governing the resolvent (A−z) −1 of the adjacency operator of unimodular random trees. In the limit where the complex parameter z approaches a given location λ on the real axis, we show that this recursion induces a decomposition of the tree into finite blocks whose geometry directly determines the spectral mass at λ. We then exploit this correspondence to obtain precise information on the pure-point support of the spectrum, in terms of expansion properties of the tree. In particular, we deduce that the pure-point support of the spectrum of any unimodular random tree with minimum degree δ ≥ 3 and maximum degree ∆ is restricted to finitely many points, namely the eigenvalues of trees of size less than ∆−2 δ−2. More generally, we show that the restriction δ ≥ 3 can be weakened to δ ≥ 2, as long as the anchored isoperimetric constant of the tree remains bounded away from 0. This applies in particular to any unimodular Galton-Watson tree without leaves, allowing us to settle a conjecture of Bordenave, Sen and Virág (2013).
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Dates et versions

hal-01374519 , version 1 (30-09-2016)
hal-01374519 , version 2 (02-12-2020)

Identifiants

Citer

Justin Salez. Spectral atoms of unimodular random trees. Journal of the European Mathematical Society, 2020, 22 (2), pp.345-363. ⟨10.4171/JEMS/923⟩. ⟨hal-01374519v2⟩
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