Invariant random subgroups of strictly diagonal limits of finite symmetric groups

We classify the ergodic invariant random subgroups of strictly diagonal limits of finite symmetric groups

Approximations of standard equivalence relations and Bernoulli percolation at p\_u

The goal of this note is to announce certain results in orbit equivalence theory, especially concerning the approximation of p.m.p. standard equivalence relations by increasing sequence of sub-relations, with applications to the behavior of the Bernoulli percolation on graphs at the threshold pu. R\'esum\'e en Fran\c{c}ais. Approximations de relations d'\'equivalence standards et percolation de Bernoulli \`a p\_u

Classical and Quantum Implications of the Causality Structure of Two-Dimensional Spacetimes with Degenerate Metrics

The causality structure of two-dimensional manifolds with degenerate metrics is analysed in terms of global solutions of the massless wave equation. Certain novel features emerge. Despite the absence of a traditional Lorentzian Cauchy surface on manifolds with a Euclidean domain it is possible to uniquely determine a global solution (if it exists), satisfying well defined matching conditions at the degeneracy curve, from Cauchy data on certain spacelike curves in the Lorentzian region. In general, however, no global solution satisfying such matching conditions will be consistent with this data. Attention is drawn to a number of obstructions that arise prohibiting the construction of a bounded operator connecting asymptotic single particle states. The implications of these results for the existence of a unitary quantum field theory are discussed.Comment: 27 pages LaTex (6 Figures), Journal of Mathematical Physics (Accepted