  We investigate concentration properties of functions of random vectors with
values in the discrete cube, satisfying the stochastic covering property (SCP)
or the strong Rayleigh property (SRP).
  Our result for SCP measures include subgaussian inequalities of
bounded-difference type extending classical results by Pemantle and Peres and
their counterparts for matrix-valued setting strengthening recent estimates by
Aoun, Banna and Youssef. Under a stronger assumption of the SRP we obtain
Bernstein-type inequalities for matrix-valued functions, generalizing recent
bounds for linear combinations of positive definite matrices due to Kyng and
Song.
  We also treat in detail the special case of independent Bernoulli random
variables conditioned on their sum for which we obtain strengthened estimates,
deriving in particular modified log-Sobolev inequalities, Talagrand's convex
distance inequality and, as corollaries, concentration results for convex
functions and polynomials, as well as improved estimates for matrix-valued
functions. These results generalize inequalities for the uniform measure on
slices of the discrete cube, studied extensively by many authors. Our approach
is based on recent results by Hermon and Salez and a general framework
involving modified log-Sobolev inequalities on the discrete cube, which is of
independent interest.

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