Compact orbit spaces in Hilbert spaces and limits of edge-colouring models. (February 2016) - Regts, Guus; Schrijver, Alexander
Abstract: Let G be a group of orthogonal transformations of a real Hilbert space H . Let R and W be bounded G -stable subsets of H . Let ‖ . ‖ R be the seminorm on H defined by ‖ x ‖ R : = sup r ∈ R | 〈 r, x 〉 | for x ∈ H . We show that if W is weakly compact and the orbit space R k / G is compact for each k ∈ N, then the orbit space W / G is compact when W is equipped with the norm topology induced by ‖ . ‖ R . As a consequence we derive the existence of limits of edge-colouring models which answers a question posed by Lovász. It forms the edge-colouring counterpart of the graph limits of Lovász and Szegedy, which can be seen as limits of vertex-colouring models. In the terminology of de la Harpe and Jones, vertex- and edge-colouring models are called 'spin models' and 'vertex models' respectively.