  We investigate the systematic mechanism for designing fast mixing Markov
chain Monte Carlo algorithms to sample from discrete point processes under the
Dobrushin uniqueness condition for Gibbs measures. Discrete point processes are
defined as probability distributions $\mu(S)\propto \exp(\beta f(S))$ over all
subsets $S\in 2^V$ of a finite set $V$ through a bounded set function
$f:2^V\rightarrow \mathbb{R}$ and a parameter $\beta>0$. A subclass of discrete
point processes characterized by submodular functions (which include
log-submodular distributions, submodular point processes, and determinantal
point processes) has recently gained a lot of interest in machine learning and
shown to be effective for modeling diversity and coverage. We show that if the
set function (not necessarily submodular) displays a natural notion of decay of
correlation, then, for $\beta$ small enough, it is possible to design fast
mixing Markov chain Monte Carlo methods that yield error bounds on marginal
approximations that do not depend on the size of the set $V$. The sufficient
conditions that we derive involve a control on the (discrete) Hessian of set
functions, a quantity that has not been previously considered in the
literature. We specialize our results for submodular functions, and we discuss
canonical examples where the Hessian can be easily controlled.
