Independence ratio and random eigenvectors in transitive graphs

Harangi, Viktor ^[1] ; Virág, Bálint ^[1]
1. [1] University of Toronto
  
  University of Toronto
  
  Canadá
Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 5, 2015, págs. 2810-2840
Idioma: inglés
DOI: 10.1214/14-AOP952
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Resumen
- A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum λmin of the spectrum of the adjacency matrix. To complement this result we use random eigenvectors to gain lower bounds in the vertex-transitive case. For example, we prove that the independence ratio of a 3-regular transitive graph is at least q=12−34πarccos(1−λmin4).
  
  The same bound holds for infinite transitive graphs: we construct factor of i.i.d. independent sets for which the probability that any given vertex is in the set is at least q−o(1).
  
  We also show that the set of the distributions of factor of i.i.d. processes is not closed w.r.t. the weak topology provided that the spectrum of the graph is uncountable.