Papers Published
Abstract:
Approximating the partition function of the ferromagnetic Ising model with
general external fields is known to be #BIS-hard in the worst case, even for
bounded-degree graphs, and it is widely believed that no polynomial-time
approximation scheme exists. This motivates an average-case question: are there
classes of instances for which polynomial-time approximation schemes exist? We
investigate this question for the random field Ising model on graphs with
maximum degree $\Delta$. We establish the existence of fully polynomial-time
approximation schemes and samplers with high probability over the random fields
if the external fields are IID Gaussians with variance larger than a constant
depending only on the inverse temperature and $\Delta$. The main challenge
comes from the positive density of vertices at which the external field is
small. These regions, which may have connected components of size $\Theta(\log
n)$, are a barrier to algorithms based on establishing a zero-free region, and
cause worst-case analyses of Glauber dynamics to fail. The analysis of our
algorithm is based on percolation on a self-avoiding walk tree.