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In this work we show that for every $d < \infty$ and the Ising model definedon $G(n,d/n)$, there exists a $\beta_d > 0$, such that for all $\beta <\beta_d$ with probability going to 1 as $n \to \infty$, the mixing time of thedynamics on $G(n,d/n)$ is polynomial in $n$. Our results are the firstpolynomial time mixing results proven for a natural model on $G(n,d/n)$ for $d> 1$ where the parameters of the model do not depend on $n$. They also providea rare example where one can prove a polynomial time mixing of Gibbs sampler ina situation where the actual mixing time is slower than $n \polylog(n)$. Ourproof exploits in novel ways the local treelike structure of Erd\H{o}s-R\'enyirandom graphs, comparison and block dynamics arguments and a recent result ofWeitz.  Our results extend to much more general families of graphs which are sparsein some average sense and to much more general interactions. In particular,they apply to any graph for which every vertex $v$ of the graph has aneighborhood $N(v)$ of radius $O(\log n)$ in which the induced sub-graph is atree union at most $O(\log n)$ edges and where for each simple path in $N(v)$the sum of the vertex degrees along the path is $O(\log n)$. Moreover, ourresult apply also in the case of arbitrary external fields and provide thefirst FPRAS for sampling the Ising distribution in this case. We finallypresent a non Markov Chain algorithm for sampling the distribution which iseffective for a wider range of parameters. In particular, for $G(n,d/n)$ itapplies for all external fields and $\beta < \beta_d$, where $d \tanh(\beta_d)= 1$ is the critical point for decay of correlation for the Ising model on$G(n,d/n)$.

Rapid Mixing of Gibbs Sampling on Graphs that are Sparse on Average