

![0_image_0.png](0_image_0.png)

For strong local fields (subplots a,c), we observe that Z∅**improvements over BP results**
become less significant as β increases. It is important to note that Z∅ **always improves on**
the BP result, even when the couplings are very strong (β **= 10) and BP fails to converge**
for a small percentage of instances. Z∅ **performs slightly better than CVM-Loop4 and**
substantially better than TreeEP for small and intermediate β **. All three methods show**
similar results for strong couplings β > **2. As in the case of attractive interactions, the best** results are attained using CVM-loop6.

For the case of weak local fields (subplots b,d), BP fails to converge near the transition to the spin-glass phase. For β **= 10, BP converges only in less than 25% of the instances.**
In the most difficult domain, β > **22, all methods under consideration give similar results**
(all comparable to BP). Moreover, it may happen that Z∅ **degrades the** Z
BP approximation because loops of alternating signs have major influence **in the series. This result was**
also reported in G´omez et al. (2007) when loop terms, instead of Pfaffian terms, where considered.

## 3.2.3 Scaling With Graph Size

We now study how the accuracy of the Z∅ **approximation changes as we increase the size**
of the grid. We generate random grids with mixed couplings for
√N = {4, ..., 18} and focus on a regime of very weak local fields Θ = 0.01 and strong couplings β **= 1, a difficult**