We provide a full and rigorous proof of a theorem attributed to Żuk, stating that random groups in the Gromov density model for true0d>13 have property (T) with high probability. The original paper had numerous gaps; in particular, crucial steps involving passing between different models of random groups were not described. We fix the gaps using combinatorial arguments and a recent result concerning perfect matchings in random hypergraphs. We also provide an alternative proof, avoiding combinatorial difficulties and relying solely on the spectral properties of random graphs in the G(n, p) model.

We present precise multilevel exponential concentration inequalities for polynomials in Ising models satisfying the Dobrushin condition. The estimates have the same form as two-sided tail estimates for polynomials in Gaussian variables due to Latała. In particular, for quadratic forms we obtain a Hanson-Wright type inequality.We also prove concentration results for convex functions and estimates for nonnegative definite quadratic forms, analogous as for quadratic forms in i.i.d. Rademacher variables, for more general random vectors satisfying the approximate tensorization property for entropy.2010 Mathematics Subject Classification. 60E15, 82B99.

We give a universal recipe for constructing nonlinear entanglement witnesses able to detect non-classical correlations in arbitrary systems of distinguishable and/or identical particles for an arbitrary number of constituents. The constructed witnesses are expressed in terms of expectation values of observables. As such they are, at least in principle, measurable in experimentsComment: 14 page