Spectral atoms of unimodular random trees

Abstract : 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).
Type de document :
Pré-publication, Document de travail
2016
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https://hal.archives-ouvertes.fr/hal-01374519
Contributeur : Justin Salez <>
Soumis le : vendredi 30 septembre 2016 - 15:19:13
Dernière modification le : lundi 29 mai 2017 - 14:21:38
Document(s) archivé(s) le : samedi 31 décembre 2016 - 15:47:06

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  • HAL Id : hal-01374519, version 1
  • ARXIV : 1609.09374

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UPMC | PMA | INSMI | USPC

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Justin Salez. Spectral atoms of unimodular random trees. 2016. <hal-01374519>

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