We investigate the systematic mechanisms for designing fast mixing Markov chains Monte Carlo algorithms to sample from discrete point processes. Such processes are defined as probability distributions $\mu(S)\propto \exp(f(S))$ over all subsets $S\subseteq 2^V$ of a finite set $V$ through a bounded set function $f:2^V\rightarrow \mathbb{R}$. In particular, a subclass of discrete point processes characterized by submodular functions (which include determinantal point processes, log-submodular distributions, and submodular 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 it is possible to design fast mixing Markov Chain Monte Carlo methods that provide size-free error bounds on marginal approximations. The conditions that we introduce involve a control on the second order (discrete) derivatives of set functions. We then provide sufficient conditions for fast mixing when the set function is submodular, and specialize our results for canonical examples.